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+{"text":"\\section{Introduction}\n\n\nThe concept of deformation quantization has been introduced by Bayen,\nFlato, Fronsdal, Lichnerowicz and Sternheimer in their seminal paper\n\\cite{bayen.et.al:1978a} based on the theory of associative\ndeformations of algebras \\cite{gerstenhaber:1964a}.  A formal star\nproduct on a Poisson manifold $M$ is defined as a formal associative\ndeformation of the algebra of smooth functions $\\Cinfty (M)$\non $M$ (the name comes from the notation $\\star$ for the deformed product) and its existence has been proved as a corollary of the\nso-called \\emph{formality theorem} in \\cite{kontsevich:2003a} (for\nmore details in deformation quantization we refer to the textbooks\n\\cite{esposito:2015a, waldmann:2007a}). On the other hand, Drinfeld\nintroduced the notion of quantum groups in the setting of formal\ndeformations, see e.g. the textbooks \\cite{chari.pressley:1994a,\n  etingof.schiffmann:1998a} for a detailed discussion.  Drinfeld also\nintroduced the idea of using symmetries to get formal\ndeformations. More explicitely, given an action by derivations of a\nLie algebra $\\lie{g}$ on an associative algebra $(\\algebra{A},\nm_{\\algebra A})$, the definition of the so-called \\emph{Drinfeld\n  twist} \\cite{drinfeld:1983a, drinfeld:1988a} $J \\in\n(\\algebra{U}({\\lie{g}}) \\tensor \\algebra{U}(\\lie g))[\\![\\hbar]\\!]$ allows us\nto obtain an associative formal deformation of $\\algebra{A}$ by means\nof a \\emph{universal deformation formula}\n\\begin{equation}\n    \\label{eq:TheUDF}\n    a \\star_{J} b\n    =\n    m_{\\algebra A} (J\\acts (a \\tensor b))\n\\end{equation}\nfor $a, b \\in \\algebra{A}[\\![\\hbar]\\!]$. Here $\\acts$ is the action of\n$\\lie{g}$ extended to the universal enveloping algebra\n$\\algebra{U}(\\lie g)$ and then to $\\algebra{U}(\\lie g) \\tensor\n\\algebra{U}(\\lie g)$ acting on $\\algebra{A} \\tensor \\algebra{A}$.  The\ndeformed algebra $(\\algebra A[\\![\\hbar]\\!], \\star_J)$ is then a\nmodule-algebra for the quantum group:\n\\begin{equation}\n   \\algebra U_J(\\lie{g}) \n   := (\\algebra U(\\lie{g})[\\![\\hbar]\\!], \\Delta_J \n   := J \\Delta J^{-1}).\\end{equation}\nIn other words, Drinfeld obtains a quantized action. \nWe mention here that the relevance of deformations induced via symmetries has been deeply investigated in\n\\cite{giaquinto.zhang:1998a} and in a non-formal setting in\n\\cite{bieliavsky.gayral:2015a}. \n\nThe aim of this paper consists in obtaining a more general notion of\ndeformation through symmetry, by using formality theory. We focus on\nthe quantization of Lie algebra actions in the particular case of\ntriangular Lie algebras. Such actions can be regarded as the\ninfinitesimal version of Poisson Lie group actions (see e.g.\n\\cite{Kosmann-Schwarzbach2004,Semenov1985}) and they are very\nimportant in the context of integrable systems. Triangular Lie\nalgebras and their quantizations have been studied by many authors,\nsee e.g. \\cite{Calaque2006,enriquez.etingof:2005,xu:2002a}.\nThe idea of applying formality to actions has also been used \nin \\cite{Arnal2007}, where\nthe authors use the Kontsevich formality on a Poisson manifold to construct\nfor each Poisson vector field a derivation of the star product. We recover this result.\n\nThe formality theorem states the existence of an\n$L_\\infty$-quasi-isomorphism from polyvectorfields to polydifferential\noperators on a manifold $M$. In \\cite{Dolgushev2005, Dolgushev2005a}\nDolgushev proves the theorem for general $M$ using the proof for\n$M=\\mathbb{R}^n$. In order to construct such $L_\\infty$-quasi-isomorphisms,\nDolgushev uses Fedosov's methods \\cite{fedosov:1994a} concerning\nformal geometry, Kontsevich's quasi-isomorphism\n\\cite{kontsevich:2003a} and the twisting procedure inspired by Quillen\n\\cite{Quillen69}. Following the construction provided by Dolgushev,\nCalaque proved a formality theorem for Lie algebroids\n\\cite{Calaque2005}.\nWe consider an infinitesimal action of $\\lie g$ on $M$, i.e. a Lie\nalgebra homomorphism $\\varphi\\colon\n\\mathfrak{g}\\rightarrow\\Secinfty(TM)$. This can immediately be\nextended to a DGLA morphism\n\\begin{equation}\n  \\Tpoly{\\mathfrak{g}}{} \\longrightarrow \\Tpoly{}{}(M),\n\\end{equation}\nwhere $\\Tpoly{\\mathfrak{g}}{} = \\wedge^\\bullet \\lie g$ and\n$\\Tpoly{}{}(M) = \\Secinfty(\\wedge^\\bullet TM)$  with the brackets extended via a Leibniz rule. From the formality\ntheorem we know that we have the following $L_\\infty$-quasi-isomorphisms\n\\begin{equation}\n  \\Tpoly{\\mathfrak{g}}{} \\longrightarrow \\Dpoly{\\mathfrak{g}}{}\n  \\quad \\mbox{and} \\quad\n  \\Tpoly{}{}(M) \\longrightarrow \\Dpoly{}{}(M).\n\\end{equation}\nUsing the quasi-invertibility of $L_\\infty$-quasi-isomorphisms we obtain\nthe existence of an $L_\\infty$-morphism \n\\begin{equation}\n\\label{eq:totwist}\n  \\Dpoly{\\mathfrak{g}}{} \\longrightarrow \\Dpoly{}{}(M).\n\\end{equation}\nIf the Lie algebra $\\lie g$ is endowed with an $r$-matrix, i.e.  an\nelement $r \\in \\lie g \\wedge \\lie g$ satisfying the Maurer--Cartan\nequation $\\Schouten{r, r} = 0$, the action always induces a Poisson\nstructure on $M$ and it is automatically a Poisson action.\n\\begin{lemma}\n  \\leavevmode\n  \\begin{lemmalist}\n  \\item Given the formal Maurer--Cartan element $\\hbar r \\in\n    \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]$, we obtain via formality a\n    Maurer--Cartan element $\\rho_\\hbar$. This yields a quantum group\n    $\\algebra U_{\\rho_\\hbar} (\\lie g)$ with deformed coproduct $\\Delta_{1\\otimes 1+\\rho_\\hbar}$.\n  \\item Given the  Maurer--Cartan element $\\hbar \\pi =\n    \\varphi\\wedge \\varphi(r) \\in \\Tpoly{}{}(M)[\\![\\hbar]\\!]$, we obtain via\n    formality a Maurer--Cartan element $B_\\hbar$. This induces a\n    formal deformation $(\\Cinfty(M)[\\![\\hbar]\\!], \\star_{B_\\hbar})$ of the Poisson\n    algebra $(\\Cinfty(M), \\pi)$.\n  \\end{lemmalist}\n\\end{lemma}\nThe DGLA obtained from twisting the DGLA of $\\lie g$-polydifferential\noperators on the point $\\Dpoly{\\lie g}{}$ as given in \\cite{Calaque2005} turns\nout to be a special case of a DGLA canonically associated to any Hopf algebra\n$H$, which we call $H_{poly}$. Given any Maurer-Cartan element $F\\in H_{poly}$ \nthere is the associated Drinfeld twist $J=1\\otimes 1 + F$ (in the formal\nsense). It turns out that the twisted DGLA $H_{poly}^F$ is canonically\nisomorphic to $(H_J)_{poly}$ where $H_J$ denotes the Hopf algebra twisted by\n$J$. Thus, using the twisting procedure on the $L_\\infty$-morphism\n\\eqref{eq:totwist} we prove the following theorem.\n\n\\begin{theorem}\n  Let $\\lie g$ be a Lie algebra endowed with a classical $r$-matrix\n  and a Lie algebra action $\\varphi\\colon\n  \\mathfrak{g}\\rightarrow\\Secinfty(TM)$ inducing a Poisson structure\n  on $M$ by $\\pi := \\varphi\\wedge\\varphi(r)$.  Then, there exists an\n  $L_\\infty$-morphism $(\\algebra U_F(\\lie g)[\\![\\hbar]\\!])_{poly} \\to\n  C(\\algebra A_\\hbar;\\algebra A_\\hbar)$ between the DGLA associated to the quantum group $\\algebra U_{\\rho_\\hbar}(\\lie g)[\\![\\hbar]\\!]$ and the Hochschild complex of the\n  deformation quantization $\\algebra A_\\hbar$ of $\\Cinfty(M)$.\n\\end{theorem}\nThis theorem motivates a definition, which generalizes \nDrinfeld quantized action. \n \\begin{definition}[Deformation Symmetry]\n   A deformation symmetry of a Hopf algebra $H$ in \n   a unital associative algebra $\\algebra A$ is a map\n   \\begin{equation} \n     \\Phi\n     \\colon \n     H_{poly}\\longrightarrow C(\\algebra A)\n   \\end{equation}\n   of $L_\\infty$-algebras. \n\\end{definition}\nComparing the quantized structures obtained with our approach, it is \neasy to see that we recover Drinfeld's universal deformation formulas. \n\n\\vspace{0.3cm}\n\nThe paper is organized as follows. In Section~\\ref{sec:Preliminaries}\nwe recall the language of $L_\\infty$-algebras and the theorem, due to\nKontsevich, stating the existence of an $L_\\infty$-quasi-isomorphism\nbetween polyvector fields and polydifferential operators on the formal\ncompletion at $0\\in \\mathbb{R}^d$.  In Section~\\ref{sec:formality} we briefly\ndiscuss the proof of formality for Lie algebroids, following \n\\cite{Calaque2006, Dolgushev2005, Dolgushev2005a}. In particular, we \nrecall the twisting procedure in the curved context.\nSection~\\ref{sec:FormalityActions}\ncontains the main\nresults of the paper, i.e. the construction of an $L_\\infty$-morphism\nout of a Poisson action and the discussion on twisted structures and\ndeformation symmetry. Finally we compare our approach with Drinfeld's\ndeformation formulas.\n\n\\section*{Acknoledgments}\nThe authors are grateful to Ryszard Nest and Stefan Waldmann \nfor the inspiring discussions. \n\n\n\n  \n\n\\section{Preliminaries}\n\\label{sec:Preliminaries}\n\nGiven a graded vector space $V^\\bullet$ over $\\mathbb{K}$ \nwe denote the $k$-\\emph{shifted} vector space by $V[k]^\\bullet$, it\nis given by\n\\begin{equation}\n  V^\\bullet[k]^l=V^{l+k}\n\\end{equation}\n\n\n\n\n\\subsection{$L_\\infty$-setting}\n\\label{sec:Linfty}\n\nWe shall recall the definitions of $L_\\infty$-algebra and\n$L_\\infty$-morphisms for the convenience of the reader (and to fix\ncertain conventions). For the rest of this section we consider a\nfield $\\mathbb{K}$ of characteristic $0$. Although many constructions \nwill also allow for replacement of $\\mathbb{K}$ by a PID containing the rationals.\n\\begin{definition}[$L_\\infty$-algebra]\n  A degree $+1$ coderivation $Q$ on the co-unital conilpotent\n  cocommutative coalgebra $S^c(\\mathfrak{L})$ cofreely cogenerated by\n  the graded vector space $\\mathfrak{L}[1]^\\bullet$ over $\\mathbb{K}$ is\n  called an $L_\\infty$-structure on the graded vector space $\\mathfrak{L}$ if\n  $Q^2=0$.\n\\end{definition}\nIn more explicit terms we have\n\\begin{equation}\n  S^c(\\mathfrak{L})=\\bigoplus_{k=0}^\\infty \\Anti^{k}\n  \\left(\\mathfrak{L}[1]\\right)\n\\end{equation}\nequipped with the coproduct $\\Delta$ given by \n\\begin{align}\n  \\Delta(1)\n  &=\n  1\\otimes 1\n  \\qquad \\mbox{and}\n  \\\\\n  \\Delta(\\gamma_1\\wedge\\ldots\\wedge \\gamma_k)\n  &=\n  1\\otimes\\gamma_1\\wedge\\ldots\\wedge\\gamma_k\n  +\n  \\gamma_1\\wedge\\ldots\\wedge\\gamma_k\\otimes 1 \n  +\n  \\overline{\\Delta}(\\gamma_1\\wedge\\ldots\\wedge\\gamma_k)\n\\end{align}\nfor $k\\geq1$ and any $\\gamma_i\\in \\mathfrak{L}[1]$. Here we have\n\\begin{equation}\n  \\overline{\\Delta}(\\gamma_1\\wedge\\ldots\\wedge \\gamma_k)=\n  \\sum_{i=1}^{k-1}\\sum_{\\sigma\\in\\mbox{\\tiny Sh($i$,$k-i$)}}\\epsilon(\\sigma)\n  \\gamma_{\\sigma(1)}\\wedge\\ldots\\wedge \\gamma_{\\sigma(i)}\\bigotimes\n  \\gamma_{\\sigma(i+1)}\\wedge\\ldots\\wedge \\gamma_{\\sigma(k)},\n\\end{equation}\nwhere $\\mathrm{Sh}(i,k-i)$ denotes the $(i,k-i)$ shuffles in the symmetric\ngroup $S_k$ in $k$ letters and the \\emph{Koszul sign}\n$\\epsilon(\\sigma)=\\epsilon(\\sigma, \\gamma_1,\\ldots, \\gamma_k)$ is\ndetermined by the rule\n\\begin{equation}\n  \\gamma_1\\wedge\\ldots\\wedge\\gamma_k=\\epsilon(\\sigma)\n  \\gamma_{\\sigma(1)}\\wedge\\ldots\\wedge\\gamma_{\\sigma(k)}.\n\\end{equation}\nRecall that $S^c(\\mathfrak{L})$ is given by the co-invariants of the\ntensor algebra for the action of the symmetric groups generated by\n\\begin{equation}\n  (i \\ i+1)(\\gamma_1\\otimes\\ldots\\otimes \\gamma_k)\n  =\n  (-1)^{|\\gamma_i||\\gamma_{i+1}|}\\gamma_{1}\\otimes\\ldots\\otimes \n  \\gamma_{i-1}\\otimes \\gamma_{i+1}\\otimes \\gamma_i\\otimes \\gamma_{i+2}\\otimes\\ldots \\gamma_{k},\n\\end{equation}\nwhere we use the vertical bars to denote the shifted degree, i.e. the\ndegree of $\\gamma_i$ in $\\mathfrak{L}[1]$. The co-unit is given by the\nprojection $\\pr_\\mathbb{K}$ onto the ground field $\\mathbb{K}$.\n\\begin{remark}\n  \\label{rem:bialgebra}\n  A direct computation shows that, denoting the flip $a\\otimes\n  b\\mapsto (-1)^{|a||b|}b\\otimes a$ by $\\tau$, we have\n  %\n  \\begin{equation}\n    \\Delta\\circ\n    (\\argument\\wedge\\argument)\n    =\n    (\\argument\\wedge\\argument)\n    \\otimes\n    (\\argument\\wedge\\argument)  \n    \\circ\n    (\\id\\otimes\\tau\\otimes\\id)\\circ\\Delta\\otimes\\Delta.\n  \\end{equation}\n \n  So we obtain the unital and co-unital bialgebra\n  $(S^c(\\mathfrak{L}),\\argument\\wedge\\argument, 1\\in \\mathbb{K},\n  \\Delta,\\pr_\\mathbb{K})$, i.e. $1\\wedge X=X$ for all $X\\in\n  S^c(\\mathfrak{L})$.  We sometimes abuse notation by omitting\n  $\\wedge$ in favor of simple concatenation or superscripts, e.g.  $ab\n  := a\\wedge b$ and $x^3 := x\\wedge x\\wedge x$.\n\\end{remark}\n\\begin{lemma}[Characterization of coderivations]\n  Every degree $+1$ coderivation $Q$ on $S^c(\\mathfrak{L})$ is uniquely\n  determined by the components\n  %\n  \\begin{equation}\n    Q_n\\colon \\Anti^n(\\mathfrak{L}[1])\\longrightarrow \\mathfrak{L}[2]\n  \\end{equation}\n  %\n  by the formula \n  %\n  \\begin{equation}\n    Q(\\gamma_1\\wedge\\ldots\\wedge\\gamma_n)\n    =\n    \\sum_{k=0}^n\\sum_{\\sigma\\in\\mbox{\\tiny Sh($k$,$n-k$)}}\n    \\epsilon(\\sigma)Q_k(\\gamma_{\\sigma(1)}\\wedge\\ldots\\wedge\n    \\gamma_{\\sigma(k)})\\wedge\\gamma_{\\sigma(k+1)}\\wedge\n    \\ldots\\wedge\\gamma_{\\sigma(n)},\n  \\end{equation} \n  %\n  where we use the conventions that $\\mathrm{Sh}(n,0) = \\mathrm{Sh}(0,n) = \\{\\id\\}$\n  and that the empty product equals the unit. \n\\end{lemma}\n\\begin{proof}\n  It follows by simply writing out both sides of the defining equation\n  \\begin{equation}\n    \\Delta\\circ Q\n    =\n    (Q\\otimes \\id+\\id\\otimes Q)\\circ\\Delta.\n  \\end{equation}\n\\end{proof}\nNote that $Q_0(1)$ is of degree $1$ in $\\mathfrak{L}[1]$ (thus of\ndegree $2$ in $\\mathfrak{L}$).  The condition $Q^2=0$ can now be\nexpressed in terms of a quadratic equation in the components $Q_n$.\n\\begin{example}[Curved Lie algebras]\n  \\label{ex:CurvedLie}\n  Our main example of an $L_\\infty$-algebra is given by (curved) Lie\n  algebras, i.e. the tuple $(\\mathfrak{L},R,\\D,[\\argument,\\argument])$ where\n  we set $Q_0(1) = R$, $Q_1 = \\D$, $Q_2 = [\\argument,\\argument]$ and $Q_i=0$\n  for all $i\\geq 3$. The condition $Q^2=0$ amounts to:\n  \\begin{itemize} \n  \\item $\\D R=0$, \n  \\item $\\D^2(\\argument)=[R,\\argument]$,\n  \\item $\\D$ is a derivation of $[\\argument,\\argument]$,\n  \\item The graded Jacobi identity for $[\\argument,\\argument]$.\n  \\end{itemize}\n\\end{example}\n\\begin{remark}\n  We should note that the our definition of $L_\\infty$-algebra is\n  usually called \\emph{curved} $L_\\infty$-algebra (see e.g.~\n  \\cite{Markl100}).  Although this definition is also not set in\n  stone, see for instance \\cite{GradyGwilliam} for yet another notion\n  of curved $L_\\infty$-algebra.  For the purpose of this paper it is,\n  however, more convenient to call the curved version simply\n  $L_\\infty$-algebra.  The only $L_\\infty$-algebras playing a role in this paper are, however, the flat $L_\\infty$-algebras, i.e.\n  those having $Q_0=0$.  The usual definition for an\n  $L_\\infty$-algebra thus coincides with our definition of flat\n  $L_\\infty$-algebra.\n\\end{remark}\n\\begin{remark}\n  \\label{rem:filtration}\n  In the following we have to deal with various infinite sums. In\n  order for this to make sense, we always consider only\n  $L_\\infty$-algebras $\\mathfrak{L}$ that are equipped with a\n  decreasing filtration\n  \\begin{equation}\n    \\mathfrak{L}\n    =\n    \\mathcal{F}^0\\mathfrak{L}\n    \\supset\\mathcal{F}^1\\mathfrak{L}\n    \\supset\\ldots\\supset\\mathcal{F}^k\\mathfrak{L}\\supset\\ldots,\n  \\end{equation}\n  respecting the $L_\\infty$-structure and which is moreover\n  \\emph{complete}, i.e.\n  \\begin{equation}\n    \\bigcap_k\\mathcal{F}^k\\mathfrak{L}\n    =\n    \\{0\\}.\n  \\end{equation}\n  This yields a corresponding complete metric topology and we consider\n  convergence of infinite sums in terms of this topology.\n\\end{remark}\n\\begin{definition}[$L_\\infty$-morphisms]\n  \\label{def:Linftymorph}\n  Let $\\mathfrak{L}$ and $\\widetilde{\\mathfrak{L}}$ be\n  $L_\\infty$-algebras.  A degree $0$ filtration preserving co-unital\n  co-algebra morphism\n  \\begin{equation}\n    F\\colon \n    S^c(\\mathfrak{L})\n    \\longrightarrow \n    S^c(\\widetilde{\\mathfrak{L}})\n  \\end{equation}\n  such that $FQ = \\widetilde{Q}F$ is called\n  an $L_\\infty$-morphism.\n\\end{definition}\n\\begin{lemma}[Characterization of co-algebra morphisms]\n  A co-algebra morphism $F$ from\\\\ $S^c(\\mathfrak{L})$ to\n  $S^c(\\widetilde{\\mathfrak{L}})$ is uniquely determined by its\n  components, also called Taylor coefficients,\n  \\begin{equation}\n    F_n\n    \\colon \n    \\Anti^n(\\mathfrak{L}[1])\n    \\longrightarrow \n    \\widetilde{\\mathfrak{L}}[1],\n  \\end{equation}\n  where $n\\geq 1$. Namely, we set $F(1)=1$ and use the formula \n  \\begin{equation}\n    \\label{eq:coalgebramorphism}\n    \\begin{gathered}\n      F(\\gamma_1\\wedge\\ldots\\wedge\\gamma_n)= \n      \\\\     \n      \\sum_{p\\geq1}\\sum_{\\substack{k_1,\\ldots, k_p\\geq1\\\\k_1+\\ldots+k_p=n}}\n      \\sum_{\\sigma\\in \\mbox{\\tiny Sh($k_1$,..., $k_p$)}}\\frac{\\epsilon(\\sigma)}{p!}\n      F_{k_1}(\\gamma_{\\sigma(1)}\\wedge\\ldots\\gamma_{\\sigma(k_1)})\\wedge\\ldots\\wedge \n      F_{k_p}(\\gamma_{\\sigma(n-k_p+1)}\\wedge\\ldots\\wedge\\gamma_{\\sigma(n)}),\n    \\end{gathered}\n  \\end{equation}\n  where $Sh(k_1,...,k_p)$ denotes the set of $(k_1,\\ldots,\n  k_p)$-shuffles in $S_n$ and $Sh(n) = \\{\\id\\}$.\n\\end{lemma}\n\\begin{proof} \n  It simply follows by writing out the defining equation\n  \\begin{equation}\n    \\Delta\\circ F\n    =\n    F\\otimes F\\circ \\Delta.\n  \\end{equation}\n\\end{proof}\n\\begin{example} \n  Let $(\\mathfrak{L},R,\\D,[\\argument,\\argument])$ and\n  $(\\mathfrak{L}',R',\\D',[\\argument,\\argument]')$ be two curved Lie\n  algebras and consider the morphism $C\\colon\n  \\mathfrak{L}\\longrightarrow \\mathfrak{L}'$ of curved Lie algebras,\n  i.e. $C(R)=R'$, $C\\D = \\D'C$ and $C$ is a morphism of the underlying\n  Lie algebras.  Then the map $F$ given by applying the formula\n  \\eqref{eq:coalgebramorphism} to the components $F_1=C$ and $F_i=0$\n  for $i\\geq 2$ is an $L_\\infty$-morphism. In general, if $F\\colon\n  \\mathfrak{L}\\longrightarrow \\mathfrak{L}'$ is an\n  $L_\\infty$-morphism, then $F_1(R)=R'$, but we only have\n  $\\D'F_1(\\gamma)=F_1\\D(\\gamma)+F_2(R\\wedge\\gamma)$.\n\\end{example}\nNote that, given an $L_\\infty$-morphism of \\emph{flat}\n$L_\\infty$-algebras $\\mathfrak{L}$ and $\\widetilde{\\mathfrak{L}}$, we\nobtain the map of complexes\n\\begin{equation}\n  F_1\n  \\colon \n  (\\mathfrak{L},Q_1)\n  \\longrightarrow \n  (\\widetilde{\\mathfrak{L}},\\widetilde{Q}_1).\n\\end{equation}\n\\begin{definition}[$L_\\infty$-quasi-isomorphism]\n  \\label{def:Linftyquis}\n  An $L_\\infty$-morphism $F$ is called $L_\\infty$-quasi-isomorphism if\n  $F_1$ is a quasi-isomorphism of complexes.\n\\end{definition}\nThe $L_\\infty$-quasi-isomorphisms we deal with in this paper happen to\nbe the ones witnessing \\emph{formality}, let us therefore introduce\nthe notion of formal $L_\\infty$-algebras here.\n\\begin{definition}[Formal $L_\\infty$-algebra]\n  \\label{defformal}\n  An $L_\\infty$-algebra $\\mathfrak{L}$ is called formal if it is flat\n  and admits an $L_\\infty$-quasi-isomorphism\n  \\begin{equation}\n    F\n    \\colon \n    \\mathrm{H}(\\mathfrak{L})\\longrightarrow \\mathfrak{L}\n  \\end{equation}\n  for the $L_\\infty$-structure canonically induced on the cohomology\n  $\\mathrm{H}(\\mathfrak{L})$ of $\\mathfrak{L}$.\n\\end{definition}\nFinally, a crucial concept for this paper is the one of Maurer--Cartan\nelements, that we define below.\n\\begin{definition}[Maurer-Cartan element]\n  \\label{MCdef} \n  Given an $L_\\infty$-algebra $(\\mathfrak{L}, Q)$, an element $\\pi\\in\n  \\mathcal{F}^1\\mathfrak{L}[1]^0$ is called a Maurer-Cartan or MC\n  element if it satisfies the following equation\n  \\begin{equation}\n    \\label{eq:MC}\n    \\sum_{n=0}^\\infty \\frac{Q_n(\\pi^n)}{n!}=0.\n  \\end{equation}\n\\end{definition}\n\n\n\n\n\n\n\n\\subsection{Local Formality}\n\\label{sec:LocalFormality}\n\nLet us denote the formal completion at $0\\in \\mathbb{R}^d$ by $\\Rf$. The\nsmooth functions $\\Cinfty(\\Rf)$ on $\\Rf$ are given by the algebra\n\\begin{equation}\n  \\Cinfty(\\Rf)\n  :=\n  \\varprojlim_{k\\rightarrow \\infty} \\Cinfty(\\mathbb{R}^d)\/\\mathcal{I}_0^k,\n\\end{equation} \nwhere $\\mathcal{I}_0$ denotes the ideal of functions vanishing at\n$0\\in\\mathbb{R}^d$.  Note that $\\Cinfty(\\Rf)$ comes equipped with the complete\ndecreasing filtration\n\\begin{equation}\n  \\Cinfty(\\Rf)\\supset\\mathcal{I}_0\\supset\\mathcal{I}_0^2\\supset\\ldots\n\\end{equation}\nand its corresponding (metric) topology. The Lie algebra of continuous\nderivations of $\\Cinfty(\\Rf)$ is denoted by\n$\\Tpoly{}{0}(\\Rf)$. By setting $\\Tpoly{}{-1} := \\Cinfty(\\Rf)$ we obtain\nthe Lie--Rinehart pair $(\\Tpoly{}{-1},\\Tpoly{}{0})$ and the graded\nvector space\n\\begin{equation}\n  \\Tpoly{}{} (\\Rf)\n  :=\n  \\bigoplus_{k\\geq -1}\\Tpoly{}{k} (\\Rf),\n\\end{equation}\nwhere $\\Tpoly{}{k} (\\Rf) := \\Anti^{k+1}\\Tpoly{}{0} (\\Rf)$ for $k\\geq\n0$.  Here the tensor product is understood to be over\n$\\Tpoly{}{-1}(\\Rf)$ and completed.  Notice that there is no confusion\nabout grading here although it may seem unnatural at first glance. It\nis actually obtained by shifting the natural grading.  The natural\nstructure is that of \\emph{Gerstenhaber algebra}, but we are only\nconsidering the underlying graded Lie algebra.  The Lie bracket\n$\\Schouten{\\argument,\\argument}$ on $\\Tpoly{}{0}(\\Rf)$ extends to a\ngraded Lie algebra structure on $\\Tpoly{}{}(\\Rf)$ by the rules\n\\begin{equation}\n  \\begin{aligned}\n    \\Schouten{f,g} \n    &= \n    0 ,\n    \\\\\n    \\Schouten{X_0,f}  \n    &=  \n    X_0(f), \n    \\\\\n    \\Schouten{X_0\\wedge\\ldots\\wedge X_k,Y} \n    &=      \n    \\sum_{j=0}^k(-1)^{kl+j}\\Schouten{X_j,Y}\\wedge X_0\\wedge\\ldots\\wedge\n    X_{j-1}\\wedge X_{j+1}\\wedge\\ldots\\wedge X_k\n  \\end{aligned}\n\\end{equation} \nfor all $f,g \\in \\Tpoly{}{-1}(\\Rf)$, $X_0,\\ldots, X_k \\in\n\\Tpoly{}{0}(\\Rf)$ and $Y \\in \\Tpoly{}{l}(\\Rf)$.\n\nThe universal enveloping algebra of the Lie-Rinehart pair\n$(\\Tpoly{}{-1}(\\Rf),\\Tpoly{}{0}(\\Rf))$ is denoted by\n$\\Dpoly{}{0}(\\Rf)$. Recall that $\\Dpoly{}{0}(\\Rf)$ is naturally\nequipped with the structures of a bialgebra (see\ne.g. \\cite{Moerdijk2010}). More precisely, $\\Dpoly{}{0}(\\Rf)$ allows\nan $\\mathbb{R}$-algebra structure $\\cdot$ and an $\\mathbb{R}$-coalgebra structure\n$\\Delta$. We extend the algebra structure in the obvious\n(componentwise) way to\n\\begin{equation}\n  \\Dpoly{}{}(\\Rf)\n  :=\n  \\bigoplus_{k\\geq -1}\\Dpoly{}{k}(\\Rf),\n\\end{equation} \nwhere $\\Dpoly{}{-1} (\\Rf):= \\Tpoly{}{-1}(\\Rf)$ and $\\Dpoly{}{k} (\\Rf)\n:= \\left(\\Dpoly{}{0}(\\Rf)\\right)^{\\otimes k+1}$. Again the\ntensor product is understood to be over $\\Dpoly{}{-1}(\\Rf)$ and\ncompleted. This allows us to define two $\\mathbb{R}$-bilinear operations\n$\\bullet$ and $[\\argument,\\argument]_G$ given by\n\\begin{equation}\n\\label{eq:Pre-Lie}\n  P_1\\bullet P_2\n  :=\n  \\sum_{i=0}^{k_1}\n  (-1)^{ik_2}(\\id^{\\otimes i}\\otimes\\Delta^{(k_2)}\\otimes \\id^{\\otimes k_1-i})(P_1)\\cdot (1^{\\otimes i}\\otimes P_2\\otimes 1^{\\otimes k_1-i})\n\\end{equation}\nand \n\\begin{equation}\n \\label{eq:Lie}\n  [P_1,P_2]_G\n  :=\n  P_1\\bullet P_2-(-1)^{k_1k_2}P_2\\bullet P_1\n\\end{equation}\nwhere $P_1\\in \\Dpoly{}{k_1}(\\Rf)$, $P_2\\in\\Dpoly{}{k_2}(\\Rf)$ and\n$\\Delta^{(k)}$ denotes the $k$-th iteration of $\\Delta$ given by \n$(\\Delta\\otimes\\id^{\\otimes k-1})(\\Delta\\otimes \\id^{\\otimes\n  k-2})\\ldots(\\Delta\\otimes \\id)\\Delta$. Note that the\nbracket $[\\argument,\\argument]_G$ defines a graded Lie algebra\nstructure on $\\Dpoly{}{}(\\Rf)$.\n\\begin{theorem}[Kontsevich\\cite{kontsevich:2003a}]\n  \\label{thm:kontsevich}\n  There exists an $L_\\infty$-quasi-isomorphism between DGLA's\n  \\begin{equation}\n    \\label{eq:kontsevich}\n    \\mathscr{K}\n    \\colon \n    \\left(\\Tpoly{}{}(\\Rf),0,\\Schouten{\\argument,\\argument}\\right)\n    \\longrightarrow \n    \\left(\\Dpoly{}{}(\\Rf),\\partial,[\\argument,\\argument]_G\\right)\n  \\end{equation} \n  where $\\partial = [\\mu,\\argument]_G$ for $\\mu=1\\otimes 1\\in\n  \\Dpoly{}{1}(\\Rf)$. Moreover\n  \\begin{enumerate} \n  \\item $\\mathscr{K}$ is $\\group{GL}(d,\\mathbb{R})$ equivariant;\n  \\item\\label{thm:kontsevich2} $\\mathscr{K}_n(X_1,\\ldots, X_n)=0$ for all $X_i\\in\n    \\Tpoly{}{0}(\\Rf)$ and $n>1$;\n  \\item $\\mathscr{K}_n(X,Y_2,\\ldots, Y_n)=0$ for all\n    $Y_i\\in\\Tpoly{}{}(\\Rf)$ and $n\\geq 2$ whenever\n    $X\\in\\Tpoly{}{0}(\\Rf)$ is induced by the action of\n    $\\mathfrak{gl}(d,\\mathbb{R})$.\n  \\end{enumerate}\n\\end{theorem}\n\n\n\n\n\\section{Formality for Lie algebroids}\n\\label{sec:formality}\n\nIn this section we recall the formality theorem for Lie\nalgebroids, which is due to Calaque, see \\cite{Calaque2005}. The proof\nof this theorem follows the lines of Dolgushev's construction\n\\cite{Dolgushev2005a, Dolgushev2005} of the\n$L_\\infty$-quasi-isomorphism from polyvectorfields to polydifferential\noperators. The main ingredients are Fedosov's methods\n\\cite{fedosov:1994a} concerning formal geometry, Kontsevich's\nquasi-isomorphism \\cite{kontsevich:2003a} and the twisting procedure\ninspired by Quillen \\cite{Quillen69} (although we use Dolgushev's\nversion \\cite{Dolgushev2005}).  Since we only need\nthe result and not in fact the details of the construction we are \nrather brief here and refer to \\cite{Calaque2005} for details.\n\n\\subsection{Fedosov resolutions}\n\n\nAs a first step, Calaque constructs\nFedosov resolutions of polyvector fields and polydifferential operators\nof Lie algebroids.\n\nLet us recall that a Lie algebroid is a vector bundle $E \\to M$ over a\nmanifold $M$, equipped with a Lie bracket on sections $\\Secinfty (E)$\nand an anchor map $\\rho \\colon E \\to TM$, preserving the Lie bracket,\nsuch that\n\\begin{equation}\n  [v, f w]_E\n  =\n  f[v, w]_E + (\\rho(v)f)w,\n\\end{equation}\nfor any $v, w \\in \\Secinfty (E)$ and $f\\in \\Cinfty (M)$.\nEquivalently, we can consider the algebra of $E$-differential forms\n$\\Secinfty(\\wedge^\\bullet E^*)$ endowed with the differential $\\D_E$\ngiven by $(\\D_Ef)(v)=\\rho(v)(f)$ for $f\\in\\Cinfty(M)$ and $X\\in\n\\Secinfty(E)$, by\n$(\\D_E\\alpha)(v,w)=\\rho(v)(\\alpha(w))-\\rho(w)(\\alpha(v))-\\alpha([v,w]_E)$\nfor $X,Y\\in \\Secinfty(E)$ and $\\alpha\\in\\Secinfty(E^*)$ and extended\nas a derivation for the wedge product.\n\nThe definitions of the DGLA's $\\Tpoly{}{}(\\Rf)$ and $\\Dpoly{}{}(\\Rf)$\ngiven in Section~\\ref{sec:LocalFormality}\ngo through mutatis mutandis to define the DGLA's $ \\Tpoly{E}{}(M)$ and\n$ \\Dpoly{E}{}(M)$ starting from the Lie-Rinehart pair\n$\\left(\\Cinfty(M), \\Secinfty(E)\\right)$. Notice that the resulting\nspaces $ \\Dpoly{E}{k}(M)$ can be identified with the spaces of\n$E$-polydifferential operators of order $k+1$.\nIn order to extend the result of Theorem~\\ref{thm:kontsevich} to any\nLie algebroid, we need to consider the so-called \\emph{Fedosov\n  resolutions}.  The idea (coming from formal\ngeometry) consists in replacing the DGLA's $\\Tpoly{}{}(\\Rf)$ and\n$\\Dpoly{}{}(\\Rf)$ by quasi-isomorphic DGLA's (using DGLA morphisms in\nthis case). For the rest of this section we consider a Lie algebroid\n$E$ of rank $d$.\nWe denote by $ ^E\\FibT{}$ the bundle of formal fiberwise $E$-polyvector\nfields over $M$, this is the bundle \nassociated to the principal bundle of general linear frames in\n$E$ with fiber $\\Tpoly{}{}(\\Rf)$.\nSimilarly, the bundle $ ^E\\FibD{}$ of formal fiberwise\n$E$-polydifferential operators is the bundle over $M$ associated to the\nprincipal bundle of general linear frames in $E$ with fiber\n$\\Dpoly{}{}(\\Rf)$.\nThe Fedosov resolutions are given on the level of vector spaces by the\n$E$-differential forms with values in \n$^E\\FibT{}$ and $^E\\FibD{}$ respectively. We denote these spaces\nby $\\FormsT{}{M}{}$ and $\\FormsD{}{M}{}$ respectively. Note that these\nspaces carry a natural DGLA structure, namely the one induced by the\nstructure on fibers (which is $\\group{GL}(d,\\mathbb{R})$-equivariant).\n\\begin{lemma}\n  \\label{explicit}\n  There exist $\\group{GL}(d,\\mathbb{R})$-equivariant\n  isomorphisms of algebras\n  \\begin{equation}\n    \\Cinfty(\\Rf)\\simeq \\prod_{k=0}^\\infty \\Sym^kT^*_0\\mathbb{R}^d\\simeq \\mathbb{R}\\Schouten{\\hat{x}_1,\\ldots, \\hat{x}_d}\n  \\end{equation}\n  \\begin{equation}\n    \\Tpoly{}{0}(\\Rf)\\simeq \\Cinfty(\\Rf)\\otimes T_0\\mathbb{R}^d.\n  \\end{equation}\n  Here the Lie algebra structure on $\\Cinfty(\\Rf)\\otimes T_0\\mathbb{R}^d$ is\n  induced from the action of $T_0\\mathbb{R}^d$ as derivations at $0$ on\n  $\\Cinfty(\\mathbb{R}^d)$.\n\\end{lemma}\nThe proof of the above lemma can be found in \\cite[Prop. 2.1.10]{Niek} and \\cite[Theorem\n  1.1.3]{MoerdijkReyes}. It implies that\n\\begin{equation}\n  \\FormsT{}{M}{}\n  \\simeq \n  \\Secinfty\\left(\\prod_{k=0}^\\infty \\Anti^\\bullet E\\otimes \\Sym^k E^*\\otimes \\Anti^\\bullet E^*\\right)\n\\end{equation}\nand similarly \n\\begin{equation}\n  \\FormsD{}{M}{}\n  \\simeq \n  \\Secinfty\\left(\\prod_{k=0}^\\infty\\left(\\bigoplus_{l=0}^\\infty \\Sym^l E\\right)^{\\otimes\\bullet}\\otimes \\Sym^k E^*\\otimes\\Anti^\\bullet E^*\\right),\n\\end{equation} \nwhere $\\Anti$ and $\\Sym$\ndenote the anti-symmetric and symmetric algebra, respectively.\n\nThe next step consists in finding a differential on the Fedosov\nresolutions which is compatible with the graded Lie algebra structure\nand which makes them into DGLA's quasi-isomorphic to $\\Tpoly{E}{}(M)$\nand $\\Dpoly{E}{}(M)$, respectively.\nGiven some trivializing coordinate neighborhood $V\\subset M$ of $E$, a\nlocal frame $e_1, \\dots, e_d$ and its dual frame $(x_1,\\ldots, x_d)$,\nwe can define the operators\n\\begin{equation}\n  \\delta\\colon \\FormsT{}{V}{}\\rightarrow \\FormsT{}{V}{}\n\\end{equation} \nby the formula \n\\begin{equation}\n  \\label{delta}\\delta(Y)\n  =\n  \\sum_{i=1}^d x_i\\wedge\\Schouten{{e_i},Y},\n\\end{equation} \nIn other words $\\delta=\\Schouten{A_{-1},\\argument}$ where\n$A_{-1}\\in\\FormsT{1}{V}{0}$ is the one-form\n$A_{-1}=\\sum_{i=1}^dx_i\\otimes e_i$.  One easily checks that $\\delta$\ndoes not depend on the choice of coordinates, since $A_{-1}$ is\nindependent of coordinates, and therefore extends to all of $M$.\nBy replacing $\\Schouten{\\argument,\\argument}$ by\n$[\\argument,\\argument]_G$ in \\eqref{delta} we obtain the operators\n\\begin{equation}\n  \\delta\\colon \\FormsD{}{M}{}\\longrightarrow\\FormsD{}{M}{}.\n\\end{equation}\nNote that, since $\\Schouten{A_{-1},A_{-1}}=[A_{-1},A_{-1}]_G = 0$, we\nhave $\\delta^2 = 0$. Furthermore, since it is given by an inner\nderivation and $\\delta\\mu = 0$, $\\delta$ is compatible with the\nfiberwise Lie structures and thus yields DGLA structures.\n\n\nThe cohomology of the complexes $\\left( \\FormsT{l}{M}{},\\delta\\right)$\nand $\\left( \\FormsD{l}{M}{},\\delta\\right)$ is given by the following\nproposition (proved e.g in \\cite[Prop.~2.1]{Calaque2005}).\n\\begin{proposition}\n  We have that \n  \\begin{equation}\n    \\Cohom{0}( \\FormsT{}{M}{},\\delta)\n    \\simeq\n    \\Gamma^\\infty(\\Anti^\\bullet E)\n    \\hspace{0.3cm} \\mbox{and}\\hspace{0.3cm}\n    \\Cohom{0}( \\FormsD{}{M}{},\\delta)\n    \\simeq\n    \\Secinfty\\left(\\left(\\bigoplus_{l=0}^\\infty S^lE\\right)^{\\otimes\\bullet}\\right)\n  \\end{equation} \n  while \n  \\begin{equation}\n    \\Cohom{>0}( \\FormsT{}{M}{},\\delta)=0\n    \\hspace{0.3cm}\\mbox{and}\\hspace{0.3cm}\n    \\Cohom{>0}( \\FormsD{}{M}{},\\delta)=0\n  \\end{equation}\n\\end{proposition}\nNotice that\n\\begin{equation}\n  \\Cohom{0}( \\FormsT{}{M}{},\\delta)\n  \\simeq \\hspace*{0.05cm}\n   \\Tpoly{E}{}(M)\n  \\hspace{0.3cm}\\mbox{and}\\hspace{0.3cm}\n   \\Cohom{0}( \\FormsD{}{M}{},\\delta)\n  \\simeq \\hspace*{0.05cm}\n   \\Dpoly{E}{}(M)\n\\end{equation}\nas vector spaces. This does not provide us with the quasi-isomorphisms\nwe are looking for, since $\\Cohom{0}( \\FormsT{}{M}{},\\delta)$ carries\nthe trivial Lie algebra structure. To correct it, the idea is to\nconstruct a perturbation of the differential $\\delta$ that does not\naffect the size of the cohomology, but only the Lie\nalgebra structure on cohomology.\nNotice that the operator $\\delta$ is of degree $-1$ in\nterms of the filtration and so we may start perturbing at order $0$,\ni.e. adding a connection in the bundle $E$.  The fact that the resulting\nperturbation should square to zero forces us to choose a torsion-free\nconnection $\\nabla\\!\\!\\!\\!\\nabla$.  This gives us the operators\n\\begin{equation}\n  \\nabla\\!\\!\\!\\!\\nabla\n  \\colon \n  \\FormsT{}{M}{} \\longrightarrow \\FormsT{}{M}{}\n  \\hspace{0.3cm}\\mbox{and}\\hspace{0.3cm} \n  \\nabla\\!\\!\\!\\!\\nabla\n  \\colon \n  \\FormsD{}{M}{}\\longrightarrow\\FormsD{}{M}{}.\n\\end{equation} \nThus we consider the corresponding operators $-\\delta+\\nabla\\!\\!\\!\\!\\nabla$ and\n$-\\delta+ \\nabla\\!\\!\\!\\!\\nabla +\\partial$. This leads to the problem that there\nis no reason to assume that we can find $\\nabla\\!\\!\\!\\!\\nabla$ such that $\\nabla\\!\\!\\!\\!\\nabla^2=0$\n(since not every Lie algebroid is flat). Following the idea of\nFedosov, we correct $-\\delta + \\nabla\\!\\!\\!\\!\\nabla$ by an inner derivation and make\nthe ansatz\n\\begin{equation}\n  D\n  :=\n  -\\delta + \\nabla\\!\\!\\!\\!\\nabla + [A, \\argument]\n\\end{equation}\nwith $A \\in \\FormsT{1}{M}{0}\\hookrightarrow \\FormsD{1}{M}{0}$ and\nwhere $[A,\\argument]$ means $\\Schouten{A,\\argument}$ or\n$[A,\\argument]_G$ depending on the situation.  The trick is to find\n$A$ such that $D^2 = 0$, as proved in \\cite[Prop.~2.2]{Calaque2005}.\n\\begin{lemma}\n\\label{lem:A}\n  There exists a unique $A$ such that\n  \\begin{lemmalist}\n   \\item $\\delta A = R + \\nabla\\!\\!\\!\\!\\nabla A + \\frac{1}{2} [A, A]$\n   \\item $\\delta ^{-1} A = 0$.\n  \\end{lemmalist}\n\\end{lemma}\nHere $R$ denotes the curvature of $\\nabla\\!\\!\\!\\!\\nabla$ expressed in terms of the\nbundle $^E\\FibT{}(M)$ (or $ ^E\\FibD{}(M)$), i.e. it is given by the\nequation\n\\begin{equation}\n  \\nabla\\!\\!\\!\\!\\nabla^2Y\n  =\n  [R,Y]\n\\end{equation}\nand $\\delta^{-1}$ is a particular $\\delta$-homotopy from the\nprojection onto degree $0$, denoted $\\sigma$, to the identity, i.e.\n\\begin{equation}\n  \\delta^{-1}\\delta + \\delta\\delta^{-1} + \\sigma\n  =\n  \\id.\n\\end{equation} \nThe condition $\\delta^{-1}A = 0$ is simply a normalization condition\nensuring uniqueness of the solution.\n\\begin{proposition}\n  \\label{prop:DcohomM}\n  We have  \n  \\begin{equation}\n    \\Cohom{>0}( \\FormsT{}{M}{},D)\n    =\n    0\n    \\qquad \\mbox{and} \\qquad\n   \n    \\Cohom{>0}( \\FormsD{}{M}{},D)\n    =\n    0.\n  \\end{equation} \n  Furthermore we have \n  \\begin{equation}\n  \\begin{gathered}\n    \\Cohom{0}( \\FormsT{}{M}{},D)\\simeq \\Cohom{0}( \\FormsT{}{M}{},\\delta)\n    \\\\\n    \\Cohom{0}( \\FormsD{}{M}{},D)\\simeq \\Cohom{0}( \\FormsD{}{M}{},\\delta).\n  \\end{gathered}\n  \\end{equation}\n\\end{proposition}\n\\begin{proof}\n  \\cite[Thm.~2.3]{Calaque2005}\n\\end{proof}\nLet us denote the isomorphisms from the above Proposition by $\\tau$.\nThen, using a Poincar\\'e-Birkhoff-Witt-type isomorphism, Calaque constructs an\nisomorphism (see \\cite[Sec.~2.3]{Calaque2005})\n\\begin{equation}\n  \\nu \n  \\colon \n  \\Ker\\delta\\cap\\FormsD{0}{M}{}\\longrightarrow \\Dpoly{E}{}(M)\n\\end{equation}\nof filtered vector spaces. \n\nSimilarly, but in an easier way, we obtain an isomorphism \n\\begin{equation}\n  \\nu\n  \\colon\n  \\Ker\\delta\\cap \\FormsT{0}{M}{}\\longrightarrow \\Tpoly{E}{}(M)\n\\end{equation}\nof graded vector spaces.\nFinally, as proved in \\cite[Prop.~2.4-2.5]{Calaque2005}, we get:\n\\begin{theorem}[Fedosov Resolutions]\n  \\label{thm:FedRes}\n  The maps \n  \\begin{equation}\n    \\lambda_D\n    \\colon \n    \\left(\\Dpoly{E}{}(M),\\partial\\right)\n    \\longrightarrow\n    \\left(\\FormsD{}{M}{},\\partial + D\\right)\n  \\end{equation} \n  and \n  \\begin{equation}\n    \\lambda_T\n    \\colon \n    \\left(\\Tpoly{E}{}(M),0\\right)\n    \\longrightarrow \n    \\left(\\FormsT{}{M}{},D\\right)\n  \\end{equation} \n  both given by $\\tau\\circ \\nu^{-1}$  are DGLA quasi-isomorphisms.\n\\end{theorem}\nLet us sketch the remaining steps necessary to obtain the\n$L_\\infty$-quasi-isomorphisms from $\\Tpoly{E}{}(M)$ to $\\Dpoly{E}{}(M)$.\nAs a second step, one notices that in a trivializing neighborhood\n$U\\subset M$ of\n$E$ the connection $\\nabla\\!\\!\\!\\!\\nabla$ on both $\\FormsT{}{M}{}$ and\n$\\FormsD{}{M}{}$ is given by $\\D_E + [B_U,\\argument]$ for some element $B_U\\in \\FormsT{1}{M}{0}\\hookrightarrow\\FormsD{1}{M}{0}$. \nThus, in this neighborhood, we have\n$D = \\D_E + [\\Gamma,\\argument]$, where $\\Gamma$ is a Maurer-Cartan element.  \nWe now observe that\nthe map\n\\begin{equation}\n  \\mathscr{U}\n  \\colon \n  \\FormsT{}{U}{}\n  \\longrightarrow \n  \\FormsD{}{U}{}\n\\end{equation} \ngiven by applying the map $\\mathscr{K}$ from\nTheorem~\\ref{thm:kontsevich} fiberwise\ncommutes with $\\D_E$. \nThe next step consists in twisting this map by\n$\\Gamma$ to obtain $L_\\infty$-quasi-isomorphisms\n\\begin{equation}\n  \\mathscr{U}^\\Gamma\\circ\\lambda_T\n  \\colon \\Tpoly{E}{}(U)\n  \\longrightarrow\n  \\FormsD{}{U}{}.\n\\end{equation}\nThe twisting procedure is essential in our paper and will be discussed\nin full detail in Section~\\ref{sec:Twisting}.  By using the properties\nof Kontsevich's quasi-isomorphism \\eqref{eq:kontsevich} and the fact\nthat $\\nabla\\!\\!\\!\\!\\nabla$ is a $\\mathfrak{gl}(d,\\mathbb{R})$ connection we find that these\nquasi-isomorphisms coincide on intersections and thus we obtain\n\\begin{equation}\n  \\mathscr{U}^\\Gamma\\circ\\lambda_T\n  \\colon \n  \\Tpoly{E}{}(M)\\longrightarrow\\FormsD{}{M}{}.\n\\end{equation}\n\n\\begin{remark} \n  Although it may seem that we are being sloppy with notation by\n  writing $\\mathscr{U}^\\Gamma$, since it is not a twist a priori, it\n  is still possible to consider it as a twist in the context of curved\n  $L_\\infty$-algebras. This construction will be discussed in the\n  upcoming paper \\cite{sisters:2017b}.\n\\end{remark}\nFinally we would like to define the $L_\\infty$-quasi-isomorphism $\n\\lambda_D^{-1}\\circ\\mathscr{U}^\\Gamma\\circ\\lambda_T\\colon\n\\Tpoly{E}{}\\longrightarrow \\Dpoly{E}{}.  $ One problem remains and it\nis that, although $\\lambda_D$ is obviously injective, we cannot be\nassured that $\\mathscr{U}^\\Gamma\\circ\\lambda_T$ maps $\\Tpoly{E}{}$\ninto the image of $\\lambda_D$. However, Dolgushev\n\\cite[Prop.~5]{Dolgushev2005a} shows that we can always \nmodify $\\mathscr{U}^\\Gamma\\circ\\lambda_T$ using a so-called\n\\emph{partial homotopy} to obtain a new quasi-isomorphism\n$\\overline{\\mathscr{U}}$ which maps into the image of\n$\\lambda_D$. Thus we obtain the $L_\\infty$-quasi-isomorphism\n\\begin{equation}\n  \\label{eq:Q}\n  F_E\n  :=\n  \\lambda_D^{-1}\\circ\\overline{\\mathscr{U}}\\colon \\Tpoly{E}{}\\longrightarrow \\Dpoly{E}{}.\n\\end{equation} \nAs a consequence, we obtain the formality theorem for a generic\nmanifold $M$ by considering the case $E = TM$ and formality for Lie\nalgebras by considering the case $E = \\lie g$ over a point.\n\\begin{remark}\n  \\label{rem:choice}\n  Note that the constructions of $D$, $\\tau$ and so on are not unique,\n  but they depend only on the choice of the torsion-free\n  $E$-connection $\\nabla\\!\\!\\!\\!\\nabla$.\n\\end{remark}\n\n\n\n\n\n\n\\subsection{Twisting procedure}\n\\label{sec:Twisting}\n\nIn the following we recall the notions of twisting DGLA's and\n$L_\\infty$-morphisms by Maurer--Cartan elements.  The idea of such\ntwisting procedures comes from Quillen's seminal work\n\\cite{Quillen69}. Here we follow Dolgushev's approach as\nlaid out in \\cite{Dolgushev2005}. As an example we show how one\nobtains the local $L_\\infty$-quasi-isomorphisms $\\mathscr{U}^\\Gamma$\nmentioned above.  \n\\begin{lemma}\n\\label{lem:exppi}\n  Suppose $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$, then the element\n  \\begin{equation}\n    \\exp(\\pi):=\\sum_{n=0}^\\infty\\frac{\\pi^k}{k!}\n  \\end{equation}\n  is well-defined, invertible and group-like. \n\\end{lemma}\n\\begin{proof} \n  $\\exp(\\pi)$ is well-defined, since the\n  partial sums converge by virtue of $\\pi$ being in the first\n  filtration (the filtration is respected by $\\wedge$).\n  Invertibility follows from the usual direct computations showing\n  that $\\exp(-\\pi)\\exp(\\pi)=1=\\exp(\\pi)\\exp(-\\pi)$.  The fact that\n  $\\exp(\\pi)$ is group-like can similarly be deduced from a direct\n  computation using the definition of $\\Delta$ given in Section~\\ref{sec:Linfty}.\n\\end{proof}\nGiven $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$ we define the\n$\\pi$-twist of the $L_\\infty$-algebra $(\\mathfrak{L},Q)$ as the\n$L_\\infty$-algebra $\\mathfrak{L}^\\pi$ given by the pair\n$(\\mathfrak{L}, Q^\\pi)$ with\n\\begin{equation}\n  Q^\\pi(a)\n  :=\n  \\exp(-\\pi)\\wedge Q(\\exp(\\pi)\\wedge a).\n \\end{equation}\n %\n\\begin{corollary}\nSuppose $(\\mathfrak{L},Q)$ is an $L_\\infty$-algebra and $\\pi\\in \\mathcal{F}^1\\mathfrak{L}[1]^0$, then the $\\pi$-twist $(\\mathfrak{L}, Q^\\pi)$ is an $L_\\infty$-algebra.\n\\end{corollary}\n\\begin{example} \n  Given a curved Lie algebra $(\\mathfrak{L}, R,\\D,\n  [\\argument,\\argument])$ we find the twisted curved Lie algebra\n  $(\\mathfrak{L}, R^\\pi,\n  \\D + [\\pi,\\argument],[\\argument,\\argument])$, where\n  \\begin{equation}\n  \\label{eq:Rpi}\n    R^\\pi\n    :=\n    R+\\D\\pi + \\frac{1}{2}[\\pi,\\pi].\n  \\end{equation}\n  Note in particular that the $\\pi$-twist is flat exactly when $\\pi$ satisfies the Maurer-Cartan equation.\n\\end{example}\n\n\n\\begin{proposition}\n  \\label{prop:MCtwist} \n  Suppose $\\mathfrak{L}$ is an $L_\\infty$-algebra and $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$, then the $\\pi$-twist $\\mathfrak{L}$ is flat if and only if $\\pi$ is a Maurer-Cartan element. \n\\end{proposition}\n\\begin{proof} \n  We have \n  \\begin{equation} \n    Q^\\pi(1)\n    =\n    \\exp(-\\pi)\\wedge Q(\\exp(\\pi))\n    =\n    \\sum_{n=0}^\\infty \\frac{Q_n(\\pi^n)}{n!}\n    ,\n  \\end{equation}\n  since all terms in $\\bigoplus_{k=2}^\\infty\\Anti^k(\\mathfrak{L}[1])$\n  cancel out by virtue of the fact that\n  $Q^\\pi(1)=Q^\\pi_0(1)\\in\\mathfrak{L}[1]$.\n\\end{proof}\n\\begin{example} \n  For a DGLA $(\\mathfrak{L}, \\D,[\\argument,\\argument])$ Eq.~\\eqref{eq:Rpi}\n  boils down to the usual Maurer--Cartan equation\n  \\begin{equation}\n    \\D\\pi+\\frac{1}{2}[\\pi,\\pi]\n    =\n    0.\n  \\end{equation}\n  If we have similarly a curved Lie algebra with curvature $-R$ it\n  comes down to the non-homogeneous equation\n  \\begin{equation}\n    \\D\\pi+\\frac{1}{2}[\\pi,\\pi]\n    =\n    R.\n  \\end{equation}\n\\end{example}\n\\begin{lemma}\n  \\label{lem:MCcondition}\n  Suppose $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$, then $\\pi$ is an MC\n  element if and only if $Q(\\exp(\\pi))=0$.\n\\end{lemma}\n\\begin{proof}\n  The proof follows from the following equation\n  \\begin{align*}\n    Q(\\exp(\\pi))\n    &=\n    \\sum_{n=0}^\\infty\\sum_{k=0}^n\\sum_{\\sigma\\in \\mbox{\\tiny Sh($k$,$n-k$)}}\\epsilon(\\sigma)\\frac{1}{n!}Q_k(\\pi^k)\\wedge\n    \\pi^{n-k}\n    \\\\\n    &=\n    \\sum_{n=0}^\\infty\\sum_{k=0}^n\\frac{1}{k!(n-k)!}Q_k(\\pi^k)\n    \\wedge\\pi^{n-k}\n    \\\\\n    &=\n    \\left(\\sum_{n=0}^\\infty \\frac{Q_n(\\pi^n)}{n!}\\right)\n    \\wedge\\exp(\\pi).\n  \\end{align*}\n\\end{proof}\n\n\\begin{lemma}\n  \\label{lem:Fep=epF}\n  Given an L$_\\infty$-morphism $F$ from $\\mathfrak{L}$ to\n  $\\mathfrak{L}'$ and an element\n  $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$, we define the $F$-associated\n  element $\\pi_F\\in\\mathcal{F}^1\\mathfrak{L}'[1]^0$ by the formula\n  \\begin{equation}\n    \\pi_F\n    :=\n    \\sum_{n=1}^\\infty\\frac{F_n(\\pi^n)}{n!}.\n  \\end{equation}\n  We have\n  \\begin{equation}\n    F(\\exp(\\pi))=\\exp(\\pi_F)\n  \\end{equation}\n\\end{lemma}\n\\begin{proof} \n  It follows from explicit computation using the formula\n  \\eqref{eq:coalgebramorphism}.\n\\end{proof}\nLemmas \\ref{lem:Fep=epF} and \\ref{lem:MCcondition} imply the following corollary.\n\\begin{corollary}\n   If $\\pi$ is an MC element, then $\\pi_F$ is also an MC element.\n\\end{corollary}\nLet $F\\colon (\\mathfrak{L},Q)\\longrightarrow (\\widetilde{\\mathfrak{L}},\\widetilde{Q})$\nbe an $L_\\infty$-morphism and $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$. \n\\begin{definition}[$\\pi$-twist morphism]\n  The $\\pi$-twist of $F$ is a map\n  \\begin{equation}\n    F^\\pi\n    \\colon \n    (\\mathfrak{L},Q^\\pi)\\longrightarrow (\\widetilde{\\mathfrak{L}},\\widetilde{Q}^\\pi)\n  \\end{equation}\n  defined by \n  \\begin{equation}\n    F^\\pi(a)\n    :=\n    \\exp(\\pi_F)\\wedge F(\\exp(\\pi)\\wedge a).\n  \\end{equation}\n\\end{definition}\n\\begin{corollary}\n  The $\\pi$-twist of an $L_\\infty$-morphism $F$ is an\n  $L_\\infty$-morphism.\n\\end{corollary}\n\\begin{proof}\n  Note that, by Lemma~\\ref{lem:exppi} and Remark~\\ref{rem:bialgebra},\n  the operators of multiplication by $\\exp(\\pi)$ and $\\exp(\\pi_F)$ are\n  co-algebra morphisms. Thus $F^\\pi$ is a co-algebra morphism. The\n  relation $F^\\pi Q^\\pi=\\widetilde{Q}^\\pi F^\\pi$ follows from\n  the definitions.\n\\end{proof}\n\\begin{remark}\n  \\label{comptwist}\n  Given two $L_\\infty$-morphisms $F$ and $G$ from $\\mathfrak{L}$\n  to $\\mathfrak{L}'$ and $\\mathfrak{L}'$ to $\\mathfrak{L}''$,\n  respectively, and the elements $\\pi, B\\in\n  \\mathcal{F}^1\\mathfrak{L}[1]^0$, we have that\n  \\begin{align} \n    (Q^{\\pi})^B\n    &=\n    Q^{\\pi+B}=(Q^B)^\\pi,\n    \\\\ \n    (F^\\pi)^B\n    &=\n    F^{\\pi+B}=(F^B)^\\pi,\n    \\\\\n    \\pi_F+B_{F^\\pi}\n    &=(\\pi+B)_F\n    =\n    B_F+\\pi_{F^B},\n    \\\\\n    (\\pi_F)_G\n    &=\n    \\pi_{G\\circ F}.\n  \\end{align}\n\\end{remark}\nFor the proof of the following proposition we refer to\n\\cite[Prop. 1]{Dolgushev2005}.\n\\begin{proposition}\n  \\label{prop:twistquis}\n  Let $F\\colon\\mathfrak{L}\\rightarrow\\widetilde{\\mathfrak{L}}$ be\n  an $L_\\infty$-quasi-isomorphism such that the induced morphisms\n  \\begin{equation}\n    F|_{\\mathcal{F}^k\\mathfrak{L}}\n    \\colon \n    \\mathcal{F}^k\\mathfrak{L}\\longrightarrow \\mathcal{F}^k\\widetilde{\\mathfrak{L}}\n  \\end{equation}\n  are also $L_\\infty$-quasi-isomorphisms for all $k$. Suppose further\n  that $\\pi\\in\\mathcal{F}^1\\mathfrak{L}[1]^0$ is an MC element. Then\n  the $\\pi$-twist\n  \\begin{equation}\n    F^\\pi\\colon \\mathfrak{L}^\\pi\\longrightarrow \\widetilde{\\mathfrak{L}}^{\\pi_F}\n  \\end{equation}\n  of $F$ is also a quasi-isomorphism. \n\\end{proposition}\n\\begin{remark}\n  \\label{rem:thepoint}\n  The proposition above says that the class of\n  $L_\\infty$-quasi-isomorphisms is closed under the operation of\n  twisting by a Maurer--Cartan element. This provides the method of\n showing that an $L_\\infty$-morphism is an $L_\\infty$-quasi-isomorphism by showing\n that it is the twist of a known $L_\\infty$-quasi-isomorphism.\n\\end{remark}\n\\begin{example}[Formality for $\\mathbb{R}^d$]\n  \\label{ex:formalityRd}\n  Here we generalize the result of Theorem \\ref{thm:kontsevich} from\n  $\\Rf$ to $\\mathbb{R}^d$ by providing an example of the claim in\n  Remark~\\ref{rem:thepoint}. From now on we set $E = TM$ and drop the\n  $E$ for notational convenience. Proposition \\ref{prop:twistquis}\n  allows us to obtain an $L_\\infty$-quasi-isomorphism witnessing the\n  formality of $\\Dpoly{}{}(M)$ for any manifold by twisting the formal\n  quasi-isomorphism of Theorem~\\ref{thm:kontsevich}.\n  We set $M=\\mathbb{R}^d$\n \n \n  and recall that we are looking for an $L_\\infty$-quasi-isomorphism\n  \\begin{equation}\n    \\mathscr{U}^\\delta\n    \\colon\n    (\\Omega(\\mathbb{R}^d;\\FibT{}),D)\n    \\longrightarrow \n    (\\Omega(\\mathbb{R}^d,\\FibD{}),\\partial+D),\n  \\end{equation}\n  since this would complete \n  the diagram \n  \\begin{equation}\n  \\label{eq:seqquisRd}\n    (\\Tpoly{}{}(\\mathbb{R}^d),0)\n    \\stackrel{\\lambda_T}{\\longrightarrow} \n    (\\Omega(\\mathbb{R}^d;\\FibT{}),D)\n    \\stackrel{\\mathscr{U}^\\delta}{\\longrightarrow} \n    (\\Omega(\\mathbb{R}^d,\\FibD{}),\\partial+D)\n    \\stackrel{\\lambda_D}{\\longleftarrow} (\\Dpoly{}{}(\\mathbb{R}^d),\\partial)\n  \\end{equation}\n  of $L_\\infty$-quasi-isomorphisms. \n \n \n \n  Also, recall that $D:=-\\delta+\\D$. We obtain this map\n  $\\mathscr{U}^\\delta$ as follows.\n  First we note that, by applying the map $\\mathscr{K}$ from \n  Theorem~\\ref{thm:kontsevich} fiberwise, we obtain the $L_\\infty$-morphism\n  \\begin{equation}\n    \\mathscr{U}\n    \\colon \n    (\\Omega(\\mathbb{R}^d;\\FibT{}),\\D)\n    \\longrightarrow \n    (\\Omega(\\mathbb{R}^d;\\FibD{}),\\partial+\\D).\n  \\end{equation}\n  By considering the filtrations by exterior degree on both these\n  algebras we construct spectral sequences which show that\n  $\\mathscr{U}$ is a quasi-isomorphism. Using this same filtration we\n  may consider the MC element\n  $-A_{-1}\\in\\mathcal{F}^1\\Omega(\\mathbb{R}^d;\\FibT{})$. Now note that\n  $\\Omega(\\mathbb{R}^d;\\FibT{})^{-A_{-1}}$ is exactly\n  $(\\Omega(\\mathbb{R}^d;\\FibT{}),D)$ and\n  $\\Omega(\\mathbb{R}^d;\\FibD{})^{(-A_{-1})_\\mathscr{U}}$ is exactly\n  $(\\Omega(\\mathbb{R}^d;\\FibD{}),D)$, since $(-A_{-1})_{\\mathscr{U}}=-A_{-1}$\n  by point \\refitem{thm:kontsevich2} of Theorem~\\ref{thm:kontsevich}. So we obtain the\n  diagram \\eqref{eq:seqquisRd} by setting $\\mathscr{U}^\\delta:=\n  \\mathscr{U}^{-A_{-1}}$. This concludes the example of the claim in\n  Remark~\\ref{rem:thepoint}.\n  In order to obtain the quasi-isomorphism \n  \\begin{equation}\n    \\Tpoly{}{}(\\mathbb{R}^d)\\longrightarrow \\Dpoly{}{}(\\mathbb{R}^d)\n  \\end{equation} \n  we need to invert the final arrow of diagram \\eqref{eq:seqquisRd}.\n  This arrow is actually an identification (by\n  DGLA-morphism) with the kernel of $D$ in exterior degree $0$.  Thus\n  it can be inverted without problems if we can guarantee that the map\n  $\\mathscr{U}^\\delta\\circ\\lambda_T$ maps $\\Tpoly{}{}(\\mathbb{R}^d)$ into this\n  kernel.  We refer to \\cite[Sect.~4.2]{Dolgushev2005a} for an\n  explanation of a way to correct $\\mathscr{U}^\\delta$ to have this\n  property.\n\\end{example}\n\n\\begin{example}[Formality for Lie algebras]\n  Let us conclude this section by providing the equivalent of the\n  proof of formality for the case where $M=\\{\\mbox{pt}\\}$ is the\n  connected $0$-dimensional manifold and $E$ is a $d$-dimensional Lie\n  algebra $\\mathfrak{g}$. The DGLA of polyvector fields is given by\n  $\\mathrm{CE}_\\bullet(\\mathfrak{g})$, the Chevalley-Eilenberg complex\n  with the trivial differential. The complex of $E$-differential forms\n  with values in the fiberwise polyvector fields is thus given by\n  \\begin{equation}\n    \\mathrm{CE}^\\bullet(\\mathfrak{g};\\mathrm{CE}_\\bullet(\\mathfrak{g};\n    \\widehat{\\mathcal{S}}(\\mathfrak{g}^*))),\n  \\end{equation}\n  where we have denoted $\\widehat{\\mathcal{S}}(\\mathfrak{g}^*)=\n  \\prod_{k\\geq 0}S^k\\mathfrak{g}^*$ and the differential\n  $\\delta-d_E$ coincides with the usual Chevalley-Eilenberg\n  differential. A linear $E$-connection $\\nabla\\!\\!\\!\\!\\nabla$ is simply given by a\n  linear map\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla\\colon \\mathfrak{g}\\otimes\\mathfrak{g}\\longrightarrow \\mathfrak{g}.\n  \\end{equation}\n  The corresponding map\n  $\\nabla\\!\\!\\!\\!\\nabla\\colon\\Anti^\\bullet\\mathfrak{g}^*\\rightarrow \\Anti^{\\bullet\n    +1}\\mathfrak{g}^*$ is given by extending the formula\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla\\alpha(X, Y)=-\\alpha(\\nabla\\!\\!\\!\\!\\nabla(\\frac{1}{2}(X\\otimes Y-Y\\otimes X)))\n  \\end{equation}\n  from one-forms as a $\\wedge$-derivation. Note that the\n  $\\mathfrak{g}$-differential $\\D_\\mathfrak{g}$ is simply given by\n  $X\\otimes Y\\mapsto [X,Y]$. Suppose $\\{e_i\\}_{i=1}^d$ is a basis for\n  $\\mathfrak{g}$ with dual basis $\\{e^i\\}_{i=1}^d$. Then the\n  torsion-freeness of the connection $\\nabla\\!\\!\\!\\!\\nabla$ can be expressed  as\n  $\\tilde{\\Gamma}_{ij}^k=\\tilde{\\Gamma}_{ji}^k$ in terms\n  of the Christoffel symbols $\\tilde{\\Gamma}_{ij}^k\\in \\mathbb{R}$ defined by\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla(e_i\\otimes e_j)=\\tilde{\\Gamma}_{ij}^ke_k,\n  \\end{equation}\n  where we have used the Einstein summation convention. Let us consider\n  also the \\emph{relative} Christoffel symbols\n  $\\Gamma_{ij}^k$ defined by\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla(e_i\\otimes e_j)-[e_i,e_j]=\\Gamma_{ij}^ke_k,\n  \\end{equation}\n  i.e. $\\Gamma_{ij}^k=\\tilde{\\Gamma}_{ij}^k-\\frac{1}{2}c_{ij}^k$ where\n  $c_{ij}^k$ are the structure constants. In terms of these torsion-freeness is\n  equivalent to the equation\n  \\begin{equation}\n    \\Gamma_{ij}^k-\\Gamma_{ji}^k-c_{ij}^k=0.\n  \\end{equation}\n  Note in particular that the connection $\\D_\\mathfrak{g}$ is\n  \\underline{not} torsion-free. The most obvious choice of\n  torsion-free connection is given by\n  $\\Gamma_{ij}^k=\\frac{1}{2}c_{ij}^k$, but we leave the choice of\n  symmetric part open. Given any connection $\\nabla\\!\\!\\!\\!\\nabla$ it is given on\n  $\\mathrm{CE}^\\bullet(\\mathfrak{g};\\mathrm{CE}_\\bullet(\\mathfrak{g};\n  \\widehat{\\mathcal{S}}(\\mathfrak{g}^*)))$ by the formula\n  \\begin{equation}\n    \\nabla\\!\\!\\!\\!\\nabla=\\D_\\mathfrak{g}+[\\Gamma_{ij}^k e^i\\hat{e}^je_k,\\argument]\n  \\end{equation}\n  where we have used the hat to signify that we consider $\\hat{e}^j\\in\n  \\widehat{S}(\\mathfrak{g}^*)$. Similar statements hold for\n  $\\Dpoly{\\mathfrak{g}}{}$.\n  Now the example proceeds identically to the previous one. \n\\end{example}\n\n\n\n\n\n\\section{Formality and Deformation Symmetries}\n\\label{sec:FormalityActions}\n\nIn this section we prove the main result of this paper, which \nleads to a new perspective on Drinfeld's approach to deformation\nquantization. First we construct certain $L_\\infty$-algebras related\nto a Hopf algebra or more generally a unital bialgebra and show how\none obtains deformations from Drinfeld twists and maps into a\nHochschild cochain complex. Then we briefly recall the basic notions\nof Poisson action and triangular Lie algebra.  We consider the\nparticular case of a Poisson action of a triangular Lie algebra $(\\lie\ng, r)$ on a manifold $M$ and we show that we can construct a\ncorresponding $L_\\infty$-morphism between polydifferential operators\n$\\Dpoly{\\mathfrak{g}}{}$ and $\\Dpoly{}{}(M)$.  This morphism induces a\nDGLA morphism between a quantum group associated to our Lie algebra\nand a deformed algebra of smooth functions on $M$.\n\n\\subsection{Deformation Symmetries}\n\nIn the following we define the concept of a deformation\nsymmetry. This notion is inspired by Drinfeld's work on deformation\nthrough quantum actions and Drinfeld twists. Let us start by\nrecalling the definition of Drinfeld twist. In this section we shall\nfix the Hopf algebra $(H,\\Delta, \\epsilon, S)$ over the PID $\\ring R$\ncontaining $\\mathbb{Q}$.\n\n\\begin{definition}[Drinfeld twist, \\cite{drinfeld:1983a, drinfeld:1988a}]\n  \\label{def:TwistUEA}%\n  An element $J\\in H\\otimes H$ is said to be a twist on $H$ if the\n  following three conditions are satisfied.\n  \\begin{definitionlist}\n  \\item\\label{def:twist1} $J$ is invertible;\n  \\item \\label{def:twist2} \n    $(\\Delta\\tensor 1)(J)(J\\tensor 1) = (1\\tensor\\Delta)(J)(1\\tensor J)$\n    and\n  \\item \\label{def:twist3}\n    $(\\epsilon\\tensor 1)J =\n    (1\\tensor\\epsilon)J = 1$.\n  \\end{definitionlist}\n\\end{definition}\nIn the following we consider formal deformations. If we consider\ntwists in $H[\\![\\hbar]\\!]$, the condition of invertibility and\n``co-invertibility'' (condition \\refitem{def:twist3} in the above definition) \nmay be replaced by a stronger condition which is\neasier to check. In fact this condition may be formulated for any Hopf\nalgebra equipped with a complete filtration $H = \\mathcal{F}^0 H\n\\supset \\mathcal{F}^1H\\supset\\ldots $.\n\\begin{definition}[Formal Drinfeld twist]\n  \\label{def:FTwistUEA}%\n  Let $H$ be equipped with the complete filtration\n  \\\\$H = \\mathcal{F}^0 H \\supset \\mathcal{F}^1 H \\supset \\ldots$. Then an\n  element $J \\in H \\otimes H$ is said to be a formal twist on $H$\n  if $J$ satisfies \\refitem{def:twist2} of Definition \\ref{def:TwistUEA} and\n  $J-1\\otimes 1\\in \\mathcal{F}^1(H\\otimes H)$.\n\\end{definition}\n\\begin{corollary}\n  \\label{cor:ft=t}\n  A formal twist on $H$ is a twist on $H$.\n\\end{corollary}\n\\begin{proof} \n  This follows immediately from the compatibility of the Hopf\n  algebra structure with the complete filtration.\n\\end{proof}\nIt turns out that the definition of formal twist coincides exactly\nwith the definition of Maurer-Cartan element on a certain DGLA that we\nshall now define. The main observation is that the formulas\n\\eqref{eq:Pre-Lie} and \\eqref{eq:Lie} for the Gerstenhaber bracket on\n$\\Dpoly{E}{}$ only involve the structure of a unital bialgebra.\nFrom now on we denote \n\\begin{equation}\n  TH = \\bigoplus_{k=0}^\\infty T^k H\n  \\qquad\n  \\mbox{with}\n  \\qquad\n  T^k H := H^{\\otimes k}.\n\\end{equation}\nFor $P_1\\in T^{k_1+1}H$ and $P_2\\in\nT^{k_2+1}H$ set\n\\begin{equation}\n  \\label{H-Pre-Lie}\n  P_1\\bullet P_2\n  :=\n  \\sum_{i=0}^{k_1}\n  (-1)^{ik_2}(\\id^{\\otimes i}\\otimes\\Delta^{(k_2)}\\otimes \n  \\id^{\\otimes k_1-i})(P_1)\\cdot (1^{\\otimes i}\\otimes P_2\\otimes 1^{\\otimes k_1-i})\n\\end{equation}\nand \n\\begin{equation}\n  \\label{H-Lie}\n  [P_1,P_2]_H\n  :=\n  P_1\\bullet P_2-(-1)^{k_1k_2}P_2\\bullet P_1\n\\end{equation}\n\\begin{proposition} \n  The graded vector space $TH[1]$ equipped with the bracket\n  $[\\argument,\\argument]_H$ is a graded Lie algebra.\n\\end{proposition}\n\\begin{proof}\n\n  \\leavevmode\n\n  \\noindent We can immediately extend $[\\argument,\\argument]_H$ to\n  non-homogeneous elements, since $\\bullet$ can be extended by bilinearity.  Thus\n  the bilinearity and anti-symmetry of $[\\argument,\\argument]_H$\n  follow immediately from the bilinearity of $\\bullet$, which follows\n  in turn from the linearity of the coproduct and the bilinearity of\n  the product. Finally denote the associator of $\\bullet$ by $\\alpha$,\n  i.e.\n  \\begin{equation} \n    \\alpha(A,B,C)=A\\bullet(B\\bullet C)- (A\\bullet B)\\bullet C.\n  \\end{equation}\n  Then the average of $\\alpha$ over the symmetric group $S_3$ is $0$, i.e \n  \\begin{equation}\\label{real-pre-lie}\n    \\sum_{\\sigma\\in S_3}\\sigma^*\\alpha =0.\n  \\end{equation}\n  Here $S_3$ acts on $(TH[1])^{\\otimes 3}$ through the usual signed\n  permutation of tensor legs. The last equation is obviously\n  equivalent to the Jacobi identity for $[\\argument,\\argument]_H$.\n\\end{proof}\n\\begin{remark}\n  \\label{rem:braces}\n  The structure $\\bullet$ on $TH[1]$ is\n  actually the pre-Lie structure coming from a brace algebra\n  structure.  As such the identity \\eqref{real-pre-lie} can actually\n  be proved by showing the pre-Lie identity\n  \\begin{equation}\n    \\alpha(A,B,C)\n    =\n    (-1)^{|A||B|}\\alpha(B,A,C).\n  \\end{equation}\n  The braces underlying the brace algebra structure are given by \n  \\begin{align*}\n    &P \\langle Q_1,\\ldots, Q_r\\rangle \n    =\n    \\sum_{0\\leq i_1< i_2<\\ldots<i_r\\leq k}(-1)^{i_1k_1+i_2k_2+\\ldots+i_rk_r}\n    \\left(\n    \\id^{\\otimes i_1}\\otimes\\Delta^{(k_1)}\\otimes \\id^{\\otimes (i_2-i_1)}\\otimes\\ldots \\right.\n    \\\\\n    & \\left. \\ldots \\otimes\\Delta^{(k_r)}\\otimes \\id^{\\otimes (k-i_r)}\n    \\right)(P)\\cdot \n    \\left(1^{\\otimes i_1}\\otimes Q_1\\otimes 1^{\\otimes (i_2-i_1)}\\otimes\\ldots\\otimes 1^{\\otimes (i_{r}-i_{r-1})}\\otimes Q_r\\otimes 1^{\\otimes (k-i_r)}\\right),\n  \\end{align*}\n  where $P\\in T^{k+1}H$ and $Q_j\\in T^{k_j+1}H$ for all $j$. Note that\n  restricting this brace algebra structure to $H=T^1H=(TH[1])^0$ we\n  find the brace algebra given in \\cite[Section~6]{Aguiar}.\n\\end{remark}\n \n \nLet us denote the twist of $TH[1]$ by the ``Maurer-Cartan\" element $1\\otimes 1$ as\n\\begin{equation}\n  \\left(H^\\bullet_{poly},[\\argument,\\argument],\\partial\\right)\n  :=\n  \\left(T^{\\bullet +1}H,[\\argument,\\argument]_H,[1\\otimes1,\\argument]_H\\right).\n\\end{equation}\n\\begin{lemma}\n  An element $J\\in T^2H$ is a formal twist on $H$ if and only if\n  $J-1\\otimes 1$ is a Maurer-Cartan element in $H_{poly}$.\n\\end{lemma}\n\\begin{proof} \n\n  \\leavevmode\n\n  \\noindent Suppose first that $J\\in T^2H$ is a formal twist on\n  $H$. Then, by definition, $F := J-1\\otimes 1$ is an element of\n  $\\mathcal{F}^1H_{poly}^1$ and\n  \\begin{equation}\n    \\partial F+\\frac{1}{2}[F,F]_H\n    =\n    \\frac{1}{2}[J,J]_H\n    =\n    (\\Delta\\otimes \\id)(J)(J\\otimes 1)-(\\id\\otimes \\Delta)(J)(1\\otimes J)=0\n  \\end{equation}\n  by condition \\refitem{def:twist2} of Definition \\ref{def:TwistUEA}.  Conversely,\n  suppose $F$ is a Maurer-Cartan element in $H_{poly}$, then, for $J =\n  1\\otimes 1 + F$,\n  \\begin{equation} \n    (\\Delta\\otimes \\id)(J)(J\\otimes 1)-(\\id\\otimes \\Delta)(J)(1\\otimes J)\n    =\n    \\frac{1}{2}[J,J]_H\n    =\n    \\partial F +\\frac{1}{2}[F,F]_H\n    =\n    0\n  \\end{equation}\n  by the Maurer-Cartan equation. So $J$ satisfies \\refitem{def:twist2} of\n  \\ref{def:TwistUEA}, while clearly $J-1\\otimes\n  1=F\\in\\mathcal{F}^1(H\\otimes H)$.\n \\end{proof}\nDrinfeld discovered \\cite{drinfeld:1988a,drinfeld:1983b} that one can\ntwist the Hopf algebra structure on $H$ by any (formal) twist $J$. \nMore explicitely, one obtains the twisted Hopf algebra $H_J$\nby changing only the coproduct $\\Delta$ to $\\Delta_J$ given by\n\\begin{equation} \n  \\Delta_J(X)\n  =\n  J^{-1}\\Delta(X)J\n\\end{equation} \nLet us fix the (formal) twist $J$ on $H$. It\nis convenient to introduce the following notation\n\\begin{align*}\n  J_k\n  &:=\n  \\prod_{i=1}^{k}(\\Delta^{(k-i)}\\otimes \\id^{\\otimes i})(J\\otimes 1^{\\otimes i-1})\n  \\\\\n  &=\n  (\\Delta^{(k-1)}\\otimes \\id)(J)\\cdot(\\Delta^{(k-2)}\\otimes \\id\\otimes \\id)(J\\otimes 1)\n  \\cdot\\ldots\\cdot\n  (\\Delta\\otimes \\id^{\\otimes k-1})(J\\otimes 1^{\\otimes k-2})\\cdot (J\\otimes 1^{\\otimes k-1})\n\\end{align*}\nand we set $J_0=1$.  Note that $J_k\\in H^{\\otimes k+1}$ is invertible\nwith $J_k^{-1}$ given by reversing\nthe order of terms above and replacing $J$ by $J^{-1}$.\n\\begin{lemma}\n  \\label{lem:DeltaJk}\n  The iterates of $\\Delta_J$ are given by the formula \n  \\begin{equation} \n    \\Delta_J^{(k)}(X)\n    =\n    J_k^{-1}\\Delta^{(k)}(X)J_k.\n  \\end{equation}\n\\end{lemma}\n\\begin{proof} \n  The proof is given by straightforward computation.  \n\\end{proof}\nWe also need the following (slightly technical) lemma.\n\\begin{lemma}\n  \\label{lem:tech}\n  The $J_k$'s satisfy the relation \n  \\begin{equation} \n  \\label{eq:tech}\n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes \\id^{\\otimes k-i})(J_k)\n    \\cdot\n    (1^{\\otimes i}\\otimes J_l\\otimes 1^{\\otimes k-i})\n    =\n    J_{k+l}\n  \\end{equation}\n  for all $0\\leq i\\leq k\\in\\mathbb{Z}$ and $l\\in \\mathbb{Z}_{\\geq0}$. \n\\end{lemma}\n\\begin{proof}\n  For $k=i=0$, the formula \\eqref{eq:tech} reads\n  \\begin{equation} \n    \\Delta^{(l)}(1)\\cdot J_l\n    =\n    J_l\n  \\end{equation}\n  which is obviously satisfied. For $k=1$ and $i=0$ we get\n  \\begin{equation} \n    (\\Delta^{(l)}\\otimes \\id)(J)\\cdot(J_l\\otimes 1)\n    =\n    J_{l+1},\n  \\end{equation}\n  which follows immediately from the definition of $J_{l+1}$. For $k=1$, $i=1$ and $l=0$ we find \n  \\begin{equation}\n  J\\cdot(1\\otimes 1)=J.\n  \\end{equation}\n  We will fully establish the $k=1$ case by induction now. So suppose the formula \\ref{eq:tech} holds for $k=1$, $i=1$, $l-1\\in \\mathbb{Z}_{\\geq0}$. Then \n  \\begin{align*}\n    ((\\id\\otimes \\Delta^{(l)})(J)\\cdot(1\\otimes J_l)\n    &= (\\id\\otimes \\Delta^{(l-1)}\\otimes\\id)\\left((\\id\\otimes \\Delta)(J)\\cdot(1\\otimes J)\\right)\\cdot(1\\otimes J_{l-1}\\otimes 1)\n    \\\\\n    &=\n    (\\id\\otimes\\Delta^{(l-1)}\\otimes \\id)\\left((\\Delta\\otimes \\id)(J)\\cdot\n    (J\\otimes 1)\\right)\\cdot(1\\otimes J_{l-1}\\otimes 1)\n    \\\\\n    &=\n    (\\Delta^{(l)}\\otimes \\id)(J)\\cdot(\\id\\otimes \\Delta^{(l-1)}\\otimes \\id)(J\\otimes 1)\\cdot (1\\otimes J_{l-1}\\otimes 1)\n    \\\\\n    &=\n    (\\Delta^{(l)}\\otimes \\id)(J)\\cdot\\left(((\\id\\otimes \\Delta^{(l-1)})(J)\\cdot(1\\otimes J_{l-1}))\\otimes 1\\right)\n    \\\\\n    &=(\\Delta^{(l)}\\otimes \\id)(J)\\cdot (J_l\\otimes 1)=J_{l+1}.\n  \\end{align*}\n  Thus we have established the $k=0$ and $k=1$ cases completely. \n \n \n To establish the $k\\geq 2$ cases, we proceed by induction. Suppose that the formula \\eqref{eq:tech} is\n  satisfied for all triples $(\\kappa,i,l)\\in(\\mathbb{Z}_{\\geq 0})^3$ where $i\\leq \\kappa$ and $\\kappa<k$. Then we\n  have\n  \\begin{align*} \n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes \\id^{\\otimes k-i})(J_k)&\\cdot (1^{\\otimes i}\\otimes J_l\\otimes 1^{\\otimes k-i})\n    =\n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes \\id^{k-i})\\left((\\id\\otimes \\Delta^{(k-1)})(J)\\cdot(1\\otimes J_{k-1})\\right)\\cdot\\\\ \n    &\\hspace{0.4cm}\\cdot(1^{\\otimes i}\\otimes J_l\\otimes 1^{\\otimes k-i})\n    \\\\\n    &=\n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes\\id^{\\otimes k-i})((\\id\\otimes \\Delta^{(k-1)})(J))\\cdot\n    \\\\\n    &\\hspace{0.4cm}\\cdot(1\\otimes \\left((\\id^{\\otimes (i-1)}\\otimes \\Delta^{(l)}\\otimes \\id^{\\otimes k-i})(J_{k-1})\\right)\\cdot(1^{\\otimes i-1}\\otimes J_l\\otimes 1^{k-i}))\n    \\\\\n    &=\n    (\\id^{\\otimes i}\\otimes \\Delta^{(l)}\\otimes \\id^{\\otimes k-i})((\\id\\otimes \\Delta^{(k-1)})(J))\\cdot(1\\otimes J_{k+l-1})\n    \\\\\n    &=(\\id\\otimes \\Delta^{(k+l-1)})(J)\\cdot(1\\otimes J_{k+l-1})\n    \\\\\n    &=\n    J_{k+l}.\n  \\end{align*}\n  Thus, Eq.~\\eqref{eq:tech} is satisfied for all triples $(k,i,l)$. \n\\end{proof}\n\\begin{proposition}\n  \\label{prop:twistJ}\n  The map $\\mathscr{J}\\colon (H_J)_{poly}\\longrightarrow\n  (H_{poly})^{J-1\\otimes 1}$ given by $\\mathscr{J}(P)=J_k\\cdot P$ for\n  $P\\in (H_J)_{poly}^k$ is a DGLA isomorphism.\n\\end{proposition}\n\\begin{proof} \n  First we note that $\\mathscr{J}^{-1}(P)=J_k^{-1}P$ for $P\\in\n  H_{poly}^k$ is obviously an inverse of $\\mathscr{J}$. Furthermore we\n  observe that $\\mathscr{J}(1\\otimes 1) = J$, which shows that we only need\n  to check that $\\mathscr{J}$ preserves the brackets. This follows by\n  direct computation from Lemmas \\ref{lem:DeltaJk} and \\ref{lem:tech}\n\\end{proof}\n\\begin{definition}[Deformation Symmetry]\n  \\label{Defsym}\n  Let $\\algebra A$ be a unital associative algebra over $\\ring R$\n  equipped with a complete filtration $\\algebra A =\\mathcal{F}^0\n  \\algebra A\\supset\\mathcal{F}^1 \\algebra A\\supset\\ldots$ and consider\n  the Hochschild DGLA structure on the complex $C^\\bullet(\\algebra\n  A) = C^\\bullet(\\algebra A; \\algebra A)[1]$. A deformation symmetry of\n  $H$ in $\\algebra A$ is a map\n  \\begin{equation} \n    \\Phi\n    \\colon \n    H_{poly}\\longrightarrow C(\\algebra A)\n  \\end{equation}\n  of $L_\\infty$-algebras. \n\\end{definition}\nThe previous proposition implies the following claim. \n\\begin{corollary}\n  Any formal twist on $H$ produces\n  deformations of all algebras $\\algebra A$ equipped with a\n  deformation symmetry of $H$.\n\\end{corollary}\nThe notion of deformation symmetry is a generalization of the\nstandard notion of universal deformation via Drinfeld twist, see e.g.\n\\cite{bieliavsky.gayral:2015a,ESW2016,giaquinto.zhang:1998a}. Universal deformation formula \nrelies on the notion of Hopf\nalgebra action that we recall in the following definition.\n\\begin{definition}[Hopf algebra action]\n  Let $\\algebra A$ be as in Definition \\ref{Defsym}. Then an action of\n  the Hopf algebra $H$ on $\\algebra A$ is defined as a map\n  \\begin{equation}\n    \\phi\\colon H\\otimes \\algebra A\\longrightarrow \\algebra A\n  \\end{equation}\n  such that \n  \\begin{align}\n    \\phi\\circ(\\id_H\\otimes \\mu) \n    &= \n    \\mu\\circ (\\phi\\otimes\\phi)\n    \\circ\n    (\\id_H\\otimes \\tau_{H\\otimes \\algebra A}\\otimes\\id_{\\algebra A})\n    \\circ \n    (\\Delta\\otimes\\id_{\\algebra A\\otimes \\algebra A}), \n    \\\\\n    \\phi\\circ(\\mu_H\\otimes \\id_{\\algebra A})\n    &=\n    \\phi\\circ(\\id_H\\otimes \\phi)\n    \\\\\n    \\phi\\circ(\\eta_H\\otimes \\id_{\\algebra A})\n    &=\n    \\mu_{\\algebra A}\\circ(\\eta_{\\algebra A}\\otimes \\id_{\\algebra A})\n    \\\\\n    \\phi\\circ(\\id_H\\otimes \\eta_{\\algebra A}) &= \\eta_{\\algebra A}\\circ\\mu_{\\ring R}\\circ(\\epsilon\\otimes \\id_{\\ring R}).\n  \\end{align}\n  Here $\\mu$ denotes the multiplication of $\\algebra A$, $\\mu_{\\ring R}$\n  denotes the multiplication of $\\ring R$, $\\mu_H$ denotes the\n  multiplication of $H$, $\\eta_{\\algebra A}\\colon \\ring R\\rightarrow\n  A$ denotes the unit of $A$, $\\eta_H\\colon \\ring R\\rightarrow H$\n  denotes the unit of $H$ and $\\tau_{H\\otimes A}$ denotes the flip\n  $h\\otimes a\\mapsto a\\otimes h$.\n\\end{definition}\nNotice that $\\phi$ can be regarded as a map $H\\rightarrow \\End_{\\ring\n  R}(A) = C^1(A;A)$ satisfying certain conditions.\n\\begin{proposition}\n  \\label{actionimpliesdefsym}\n  Given a Hopf algebra action $\\phi$ of $H$ on $A$, the map $\\Phi$\n  defined by\n  \\begin{equation}\n    h_0\\otimes \\ldots \\otimes h_k\\mapsto \\mu_{\\algebra A}^{(k)}\\circ(\\phi(h_0)\\otimes\n    \\phi(h_1)\\otimes \\ldots\\otimes \\phi(h_k))\n  \\end{equation}\n  is a deformation symmetry. Here $\\mu_{\\algebra A}^{(k)}$ denotes the\n  $k$-th iteration $\\mu_{\\algebra A}\\circ(\\id\\otimes \\mu_{\\algebra\n    A})\\circ\\ldots\\circ(\\id^{\\otimes k-1}\\otimes \\mu_{\\algebra A})$ of\n  $\\mu_{\\algebra A}$.\n\\end{proposition}\n\\begin{proof} \n\n  \\leavevmode\n\n  \\noindent We prove this proposition by showing that $\\Phi$ is a\n  map of DGLA's.  Note that for $P=P_0\\otimes \\ldots\\otimes\n  P_k$, $Q=Q_0\\otimes \\ldots\\otimes Q_l$ and $a=a_0\\otimes\n  \\ldots\\otimes a_{k+l}$ we have\n  \\begin{align*}\n    \\Phi(P\\bullet Q)(a)\n    &=\n    \\Phi\\left(\\sum_{i=0}^k(-1)^{il}P_0\\otimes \\ldots\\otimes P_{i-1}\\otimes \\Delta^{(l)}(P_i)Q\\otimes P_{i+1}\\otimes \\ldots\\otimes P_k\\right)(a)\n    \\\\\n    &=\n    \\sum_{i=0}^k(-1)^{il}\n    \\phi(P_0)(a_0)\\cdot\\ldots\\cdot\n    \\phi(P^{(0)}_iQ_0)(a_i)\\cdot\\ldots\\cdot\\phi(P^{(l)}_iQ_l)(a_{i+l})\\cdot\n    \\\\\n    &\\qquad\\cdot\n    \\phi(P_{i+1})(a_{i+1+l})\\cdot\\ldots\\cdot\\phi(P_k)(a_{k+l})\n    \\\\\n    &=\n    \\sum_{i=0}^k(-1)^{il}\\phi(P_0)(a_0)\n    \\cdot\\ldots\\cdot\n    \\phi(P_i)(\\phi(Q_0)(a_i)\\cdot\\ldots\\cdot\\phi(Q_l)(a_{l+i}))\\cdot\n    \\\\\n    &\\qquad\\cdot \\phi(P_{i+1}(a_{l+i+1})\\cdot\\ldots\\cdot\\phi(P_k)(a_{l+k})\n    \\\\\n    &=\n    \\sum_{i=0}^k(-1)^{il}\\phi(P_0)(a_0)\n    \\cdot\\ldots\\cdot\n    \\phi(P_{i-1})(a_{i-1})\\cdot\\phi(P_i)(\\Phi(Q)(a_{i}\\otimes\\ldots\\otimes a_{i+l})\\cdot\n    \\\\\n    &\\qquad\\cdot \\phi(P_{i+1})(a_{l+i+1})\\cdot \\ldots\\cdot\\phi(P_{k})(a_{k+l})\n    \\\\\n    &=\n    \\sum_{i=0}^k(-1)^{il}\\Phi(P)(a_0\\otimes \n    \\ldots\\otimes a_{i-1}\\otimes \\Phi(Q)(a_i\\otimes\\ldots\\otimes a_{i+l})\\otimes a_{i+l+1}\\otimes \\ldots\\otimes a_{k+l})\n    \\\\\n    &=\n    \\Phi(P)\\{\\Phi(Q)\\}(a)\n  \\end{align*}\n  where we have used the \\emph{brace} notation, see e.g.\n  \\cite{GerstVoron1995}. Note that\n  $[A,B]_G=A\\{B\\}-(-1)^{|A||B|}B\\{A\\}$. In short the above computation \n  boils down to\n  \\begin{equation}\n    \\Phi(P\\bullet Q)\n    =\n    \\Phi(P)\\{\\Phi(Q)\\}.\n  \\end{equation} \n  Thus $\\Phi$ respects the brackets and observing that $\\Phi(1\\otimes\n  1)=\\mu_{\\algebra A}$ the proposition is proved.\n\\end{proof}\n\\begin{remark}\n  Recall that $TH[1]$ is endowed with a brace algebra structure, as described in\n  Remark~\\ref{rem:braces}.  The above proposition actually follows\n  from the fact that a Hopf algebra action $\\phi$ of $H$ on\n  $\\mathscr{A}$ actually induces a brace algebra morphism\n  \\begin{equation}\n    \\Phi\\colon TH[1]\\longrightarrow C^\\bullet(\\mathscr{A}),\n  \\end{equation}\n  where the braces on $C^\\bullet(\\mathscr{A})$ are defined as usual by \n  \\begin{align*}\n    P\\{Q_1,\\ldots, Q_r\\}&(a_0\\otimes \\ldots\\otimes a_{k+k_1+\\ldots+k_r})\n    =\n    \\sum_{0\\leq i_1<i_2<\\ldots<i_r\\leq k}(-1)^{i_1k_1+i_2k_2+\\ldots+i_rk_r}\n    P(a_0\\otimes \\ldots\\otimes \n    \\\\\n    &\\otimes\n    Q_1(a_{i_1}\\otimes \n    \\ldots \\otimes a_{i_1+k_1})\\otimes a_{i_1+k_1+1}\\otimes\\ldots \n    \\otimes a_{i_2+k_1-1}\\otimes \n    Q_2(a_{i_2+k_1}\\otimes \\ldots\\otimes a_{i_2+k_1+k_2})\\otimes \n    \\\\\n    &\\ldots\\otimes \n    Q_r(a_{i_r+k_1+\\ldots+k_{r-1}}\\otimes a_{i_r+k_1+\\ldots+k_r})\\otimes \\ldots \\otimes a_{k+k_1+\\ldots+k_r}),\n  \\end{align*}\n  for $P\\in C^{k+1}(\\mathscr{A};\\mathscr{A})$, $Q_j\\in\n  C^{k_j+1}(\\mathscr{A};\\mathscr{A})$ for all $j$ and $a_i\\in\n  \\mathscr{A}$ for all $i$. So we have\n  \\begin{equation} \n    \\Phi(P\\langle Q_1,\\ldots, Q_r\\rangle)\n    =\n    \\Phi(P)\\{\\Phi(Q_1),\\ldots, \\Phi(Q_r)\\}\n  \\end{equation}\n  for all $P, Q_1,\\ldots, Q_r\\in TH[1]$. \n\\end{remark}\n\n\n\n\n\n\n\\subsection{Twisting Poisson actions}\n\n\nA Lie bialgebra is a pair $(\\lie g, \\gamma)$ where $\\lie g$ is a Lie\nalgebra and $\\gamma$ is a $1$-cocycle, $ \\gamma \\colon \\lie g \\to \\lie\ng\\wedge \\lie g$.  In this paper we consider a particular class of Lie\nbialgebras.  Recall that an element $r \\in \\lie g\\wedge \\lie g$ is\ncalled $r$-matrix if it satisfies the Maurer--Cartan equation\n$\\Schouten{r, r}= 0$.  It can be proved that $r$-matrices always\ninduce a Lie bialgebra structure on $\\lie g$, by setting $\\gamma =\n\\Schouten{r, \\argument }$. We refer to the pair $(\\lie g, r)$ as\n\\emph{triangular Lie algebra}. For further details on Lie bialgebras\nwe refer to \\cite{Kosmann-Schwarzbach2004}.\n\\vspace{0.3cm}\n\nLet us consider a Lie algebra action $\\varphi : \\lie{g} \\to\n\\Secinfty(TM)$.\n\\begin{definition}[Poisson action]\n  The action $\\varphi$ is Poisson if it satisfies\n  \\begin{equation}\n    \\D_\\pi \\varphi(X)\n    =\n    \\varphi \\wedge \\varphi \\circ \\gamma (X)\n  \\end{equation}\n  where $\\D_\\pi = \\Schouten{\\pi , \\argument}$. \n\\end{definition}\n\\begin{proposition}\n  \\label{prop:PoissonR}\n  Let $\\lie{g}$ be a finite dimensional triangular Lie algebra and\n  $\\varphi \\colon \\lie{g} \\to \\Secinfty(TM)$ a Lie algebra action.\n  \\begin{propositionlist}\n  \\item The bitensor $\\pi$ defined as the image of $r$ via $\\varphi$\n    is a Poisson tensor.\n  \\item $\\varphi$ is a Poisson action wrt $\\pi = \\varphi \\wedge\n    \\varphi (r)$.\n  \\end{propositionlist}\n\\end{proposition}\n\\begin{proof}\n  Let us consider the $r$-matrix $r = \\frac{1}{2} \\sum_{i,j} r^{ij}\n  e_i \\wedge e_j$ and define\n  \\begin{equation}\n    \\label{eq:PoissonR}\n    \\pi\n    :=\n    \\frac{1}{2} \\sum_{i,j} r^{ij} \\varphi(e_i) \\wedge \\varphi(e_j).\n  \\end{equation}\n  From $\\Schouten{r, r} = 0$, using the fact that $\\varphi$ is a Lie\n  algebra morphism, it follows that $\\Schouten{\\pi, \\pi} = 0$. The second\n  claim is a straightforward computation.\n\\end{proof}\nIt is easy to see that \nthe notion of Poisson action can be extended to a morphism of\nDGLA's\n\\begin{equation}\n  \\label{eq:PoissonActionExt}\n  \\wedge^{\\bullet}\\varphi\n  : \n  (\\wedge^{\\bullet}\\lie g ,\\delta, [\\argument ,\\argument])\n  \\longrightarrow \n  (\\Secinfty (\\wedge^{\\bullet} TM), \\D_\\pi, \\Schouten{\\argument , \\argument}).\n\\end{equation}\n\nWe now show how one may use the formality of $\\Dpoly{\\mathfrak{g}}{}$\nand $\\Dpoly{}{}(M)$ to obtain a deformation symmetry of\n$\\mathscr{U}(\\mathfrak{g})$ in $\\Cinfty(M)$ given an infinitesimal\naction $\\varphi$ of $\\mathfrak{g}$ on $M$. Thus, given an r-matrix of\n$\\mathfrak{g}$, we obtain deformations of all relevant structures and\nwe comment on these. In Section~\\ref{sec:formality} we have obtained\nthe \\emph{formal} DGLA's $\\Dpoly{\\mathfrak{g}}{}$ and $\\Dpoly{}{}(M)$.\nThe classical $r$-matrix $r\\in\\Tpoly{\\mathfrak{g}}{1}$ yields a\nMaurer-Cartan element. Although it satisfies the Maurer-Cartan\nequation $\\Schouten{r,r}=0$ it is in fact not an MC element according\nto our Definition~\\ref{MCdef} as we have neglected to consider any\nfiltration. The filtration is needed since we go through formality,\nwhich means we encounter infinite sums. More precisely, we obtain a\nfiltration by considering the formal power series ring $\\mathbb{R}[\\![\\hbar]\\!]$. \nGiven a DGLA $\\mathfrak{L}$ we denote the DGLA obtained by\nextending scalars to the formal power series ring by\n$\\mathfrak{L}[\\![\\hbar]\\!]$.  We consider these DGLA's as filtered by the\ndegree in $\\hbar$. Note that, given $L_\\infty$-morphisms of DGLA's\n$\\mathfrak{L}\\rightarrow \\mathfrak{L}'$ we obtain also\n$L_\\infty$-morphisms of the extended DGLA's\n$\\mathfrak{L}[\\![\\hbar]\\!]\\rightarrow\\mathfrak{L}'[\\![\\hbar]\\!]$.  In this way\n$L_\\infty$-quasi-isomorphism go to $L_\\infty$-quasi-isomorphisms.\n\n\nFrom Section~\\ref{sec:formality} it follows that we\nhave a \\emph{horse-shoe} diagram in the category of\n$L_\\infty$-algebras:\n\\begin{equation}\n  \\label{horse-shoe}\n  \\begin{tikzcd}[row sep=3em,column sep=4em]\n    & \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]\n    \\arrow[r, \"\\varphi\"]\\arrow[d, swap, \"F_{\\mathfrak{g}}\"] \n    & \\Tpoly{}{}(M)[\\![\\hbar]\\!] \\arrow[d,  \"F_M\"]\n    \\\\\n    & \\Dpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]\n    & \\Dpoly{}{}(M)[\\![\\hbar]\\!].\n  \\end{tikzcd}\n\\end{equation}\nHere the vertical maps $F_\\mathfrak{g}$ and $F_M$ are\n$L_\\infty$-quasi-isomorphisms constructed as discussed in\nSection~\\ref{sec:formality} and the horizontal arrow is induced by\n$\\varphi$ as in \\eqref{eq:PoissonActionExt}.\nThus we obtain, given an r-matrix $r\\in\\mathfrak{g}\\wedge\\mathfrak{g}$, the MC elements:\n\\begin{equation}\n  \\begin{array}{l}\n    \\hbar r\\in \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!];\n    \\\\\n    \\hbar\\pi\n    =\n    (\\hbar r)_\\varphi\\in \\Tpoly{}{}(M)[\\![\\hbar]\\!];\n    \\\\\n    \\rho_\\hbar\n    :=\n    (\\hbar r)_{F_\\mathfrak{g}}\\in \\Dpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]\n    \\\\\n    B_\\hbar\n    :=\n    (\\hbar \\pi)_{F_M}\\in \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!].\n  \\end{array}\n\\end{equation}\n\n\\begin{remark} \n  \\label{rem:WrongCandidate}\n  Notice that we obtain the deformed versions of $\\Cinfty(M)$ and\n  $\\mathscr{U}(\\mathfrak{g})$ by twisting by the Maurer-Cartan\n  elements listed above. We would like then to obtain a deformation\n  symmetry completing the square in the horse-shoe diagram above, such\n  that it commutes.  Commutativity ensures that the two formal\n  deformations of $\\Cinfty(M)$ induced by $\\hbar r$ by transporting it\n  along the bottom or the top to $\\Dpoly{}{}(M)[\\![\\hbar]\\!]$ coincide. An\n  obvious candidate for such a map would be\n  \\begin{equation}\n    X_1\\otimes\\ldots\\otimes X_k\n    \\mapsto \n    \\varphi(X_1)\\otimes\\ldots\\otimes\\varphi(X_k),\n  \\end{equation}\n  i.e. the map induced by the map of Lie-Rinehart pairs\n  $(\\mathfrak{g}, \\mathbb{R})\\rightarrow (\\Secinfty(TM),\\Cinfty(M))$.  In\n  other terms the deformation symmetry induced by the obvious Hopf\n  algebra action of $\\mathscr{U} (\\mathfrak{g})$ on $\\Cinfty(M)$\n  through proposition \\ref{actionimpliesdefsym}. However this map may\n  not make the diagram commute. In fact it is the opinion of the\n  authors that such commutation would involve some condition of\n  compatibility of the connections used in defining these maps, see\n  Remark~\\ref{rem:choice}.\n\\end{remark}\nWe complete the horse-shoe diagram \\eqref{horse-shoe} to a commuting\nsquare by observing that $L_\\infty$-quasi-isomorphisms are\n\\emph{invertible}. The following lemma is essentially contained in\n\\cite[Chapt.~10.4]{AO}. We remind the reader that the field underlying\n$L_\\infty$-algebras is of characteristic $0$.\n\\begin{lemma}\n  \\label{invert}\n  Suppose $\\mathfrak{L}$ and $\\mathfrak{L}'$ are formal\n  $L_\\infty$-algebras and $f\\colon H(\\mathfrak{L})\\rightarrow\n  H(\\mathfrak{L'})$ is an $L_\\infty$-morphism, then there exists a\n  lift $\\tilde{f}\\colon \\mathfrak{L}\\rightarrow \\mathfrak{L}'$ of\n  $f$. In other words there exists a commuting diagram\n  \\begin{equation}\n    \\begin{tikzcd}[row sep=3em,column sep=4em]\n      & H(\\mathfrak{L})\n      \\arrow[r, \"f\"]\\arrow[d, swap, \"i_l\"] \n      & H(\\mathfrak{L}') \\arrow[d,  \"i_r\"]\n      \\\\\n      & \\mathfrak{L} \\arrow[r, swap, \"\\tilde{f}\"] \n      & \\mathfrak{L}'.\n    \\end{tikzcd}\n  \\end{equation}\n  in the category of $L_\\infty$-algebras such that the vertical arrows\n  are quasi-isomorphisms.\n\\end{lemma}\n\\begin{proof}\n  Note that we start by hypothesis with the horse-shoe\n  \\begin{equation}\n    \\begin{tikzcd}[row sep=3em,column sep=4em]\n      & H(\\mathfrak{L})\n      \\arrow[r, \"f\"]\\arrow[d, swap, \"i_l\"] \n      & H(\\mathfrak{L}') \\arrow[d,  \"i_r\"]\n      \\\\\n      & \\mathfrak{L}\n      & \\mathfrak{L'}.\n    \\end{tikzcd}\n  \\end{equation}\n  in the category of $L_\\infty$-algebras such that the vertical arrows\n  are quasi-isomorphisms. As shown in \\cite[Sect.~10.4.4]{AO} we can\n  always find a quasi-inverse $l_i\\colon\\mathfrak{L}\\rightarrow\n  H(\\mathfrak{L})$ of $i_l$ such that the induced maps in cohomology\n  are inverse to each other. \\ We define $\\tilde{f}:=i_r\\circ f\\circ\n  l_i$. Note that this already proves that $\\tilde{f}\\circ\n  i_l=i_r\\circ f$ in cohomology (and thus ``up to homotopy''). To get\n  the stronger statement in the lemma we will need to consider the\n  construction of the map $l_i$. This construction involves the notion\n  of a so-called $\\infty$-isomorphism. An $\\infty$-isomorphism is a\n  morphism of $L_\\infty$-algebras $F$ such that $F_1$ is an\n  \\emph{isomorphism}. The main observations for the construction of\n  $l_i$ are two-fold. First, the homotopy transfer theorem\n  \\cite[Sect.~10.3]{AO} yields an $L_\\infty$-structure on\n  $H(\\mathfrak{L})$ which is unique up to $\\infty$-isomorphism. It is\n  obtained by picking a retraction of $\\mathfrak{L}$ onto\n  $H(\\mathfrak{L})$, we may pick the retraction given by\n  $i_l$. Secondly in section 10.4.2 of \\cite{AO} it is shown that any\n  $L_\\infty$-algebra $\\mathfrak{L}$ is $\\infty$-isomorphic to the sum\n  $H(\\mathfrak{L})\\oplus K$ where $K$ is an acyclic chain complex\n  (with trivial $Q_0$, $Q_2$, $Q_3$ and so on). Now the construction\n  of $l_i$ follows by the fact that $\\infty$-isomorphisms are\n  invertible (shown in section 10.4.1 of \\cite{AO}). Our lemma follows\n  from the fact that, for a formal $L_\\infty$-algebra the\n  $L_\\infty$-structure induced on $H(\\mathfrak{L})$ by homotopy\n  transfer equals the canonically induced structure up to\n  $\\infty$-isomorphism. Thus we may simply consider the splittings\n  $\\mathfrak{L}\\simeq H(\\mathfrak{L})\\oplus K_{\\mathfrak{L}}$ and\n  $\\mathfrak{L}'\\simeq H(\\mathfrak{L}')\\oplus K_{\\mathfrak{L}'}$,\n  where $\\simeq$ means $\\infty$-isomorphism, given by the retractions\n  induced by $i_l$ and $i_r$. Then the map $\\tilde{f}$ simply maps\n  $H(\\mathfrak{L})$ to $H(\\mathfrak{L}')$ by $f$.\n\\end{proof}\n\\begin{corollary}\n\\label{twistaction}  The diagrams \n  \\begin{equation}\n    \\label{square}\n    \\begin{tikzcd}[row sep=3em,column sep=4em]\n      & \\Tpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!]\n      \\arrow[r, \"\\varphi\"]\\arrow[d, swap, \"F_{\\mathfrak{g}}\"] \n      & \\Tpoly{}{}(M)[\\![\\hbar]\\!] \\arrow[d,  \"F_M\"]\n      \\\\\n      & \\Dpoly{\\mathfrak{g}}{}[\\![\\hbar]\\!] \\arrow[r, \"\\tilde{\\varphi}\"]\n      & \\Dpoly{}{}(M)[\\![\\hbar]\\!]\n    \\end{tikzcd}\n  \\end{equation}\n  and\n  \\begin{equation}\n    \\label{twistsquare}\n    \\begin{tikzcd}[row sep=3em,column sep=4em]\n      & \\Tpoly{\\mathfrak{g}}{\\hbar r}[\\![\\hbar]\\!]\n      \\arrow[r, \"\\varphi^{\\hbar r}\"]\\arrow[d, swap, \"F^{\\hbar r}_{\\mathfrak{g}}\"] \n      & \\Tpoly{}{\\hbar\\pi}(M)[\\![\\hbar]\\!] \\arrow[d,  \"F^{\\hbar\\pi}_M\"]\n      \\\\\n      & \\Dpoly{\\mathfrak{g}}{\\rho_\\hbar}[\\![\\hbar]\\!] \\arrow[r, \"\\tilde{\\varphi}^{\\rho_\\hbar}\"]\n      & \\Dpoly{}{B_\\hbar}(M)[\\![\\hbar]\\!]\n    \\end{tikzcd}\n  \\end{equation}\n  commute.\n\\end{corollary}\n\\begin{proof}\n  Applying the above lemma to our situation we immediately find that\n  the diagram \\eqref{square} commutes. Also, by applying the results\n  of Section~\\ref{sec:Twisting} we obtain diagram \\eqref{twistsquare},\n  which commutes thanks to Remark~\\ref{comptwist}.\n\\end{proof}\n\n\\subsection{Twisted structures}\n\nIn the following we show that the twisted complexes obtained above are\ncoming from a formal deformation quantization of $\\Cinfty(M)$ (in the\ncase of $\\Dpoly{}{B_\\hbar}(M)$) and a deformation of\n$\\mathscr{U}(\\mathfrak{g})$ into a quantum group (in the case of\n$\\Dpoly{\\mathfrak{g}}{\\rho_\\hbar}$).\n\\begin{proposition}\n  \\label{defHoch}\n  There is a formal deformation quantization $\\mathscr{A}_\\hbar$ of\n  $(\\Cinfty(M),\\pi)$ such that\n  \\begin{equation}\n    \\Dpoly{}{B_\\hbar}(M)\n    \\hookrightarrow \n    C^\\bullet(\\algebra{A}_\\hbar,\\algebra{A}_\\hbar),\n  \\end{equation}\n  i.e. $\\Dpoly{}{B_\\hbar}(M)$ is a subcomplex of the Hochschild\n  cochain complex of $\\algebra{A}_\\hbar$.\n\\end{proposition}\n\\begin{proof}\n  Note that the Maurer--Cartan equation \\eqref{eq:MC} matches exactly\n  the associativity condition of $m + B_\\hbar$, where $m$ denotes the\n  pointwise multiplication in $\\Cinfty(M)[\\![\\hbar]\\!]$. Since\n  \\begin{equation}\n    B_\\hbar\n    =\n    \\sum_{n\\geq 1}\\frac{1}{n!}(F_M)_n(\\hbar\\pi,\\ldots,\\hbar\\pi)\n    =\n    \\sum_{n\\geq 1}\\frac{\\hbar^n}{n!}(F_M)_n(\\pi,\\ldots,\\pi),\n  \\end{equation}\n  we see that $m+B_\\hbar$ defines a deformation quantization\n  $\\algebra{A}_\\hbar$. The differential on the Hochschild complex\n  $C^\\bullet(\\algebra A, \\algebra A)$ is given by taking the\n  Gerstenhaber bracket with the multiplication for any associative\n  algebra $\\algebra A$.  Thus the twisted differential on\n  $\\Dpoly{}{B_\\hbar}(M)$ coincides with the differential of\n  $C^\\bullet(\\algebra{A}_\\hbar,\\algebra{A}_\\hbar)$.  Finally we recall\n  that\n  \\begin{equation}\n    B_\\hbar\n    =\n    \\sum_{n\\geq 1}\\frac{\\hbar^n}{n!}(F_M)_n(\\pi,\\pi,\\ldots,\\pi)\n    =\n    \\hbar(F_M)_1(\\pi)  \n    \\hspace{0.4cm}\\mbox{mod}\\hspace{0.2cm}\\hbar^2.\n  \\end{equation}\n  Since $(F_M)_1$ is a quasi-isomorphism we find that the alternating\n  part of $B_\\hbar$ is $\\hbar\\pi$ modulo $\\hbar^2$ and\n  \\begin{equation}\n    \\frac{[f,g]_{\\star}}{\\hbar}\n    =\n    \\pi(f,g)+O(\\hbar)\n  \\end{equation}\n  for all $f,g\\in \\Cinfty(M)$. Here $[\\argument, \\argument]_\\star$\n  denotes the commutator bracket of $\\algebra{A}_\\hbar$.\n\\end{proof}\nThus we obtain in particular the deformation quantization\n$\\algebra{A}_\\hbar$.  On the other hand we also obtain the MC element\n$\\rho_\\hbar$ in $\\Dpoly{\\mathfrak{g}}{}$. This yields the\ncoproduct $\\Delta_{J_\\hbar}$ (where $J_\\hbar=1\\otimes 1 +\\rho_\\hbar)$\nthus establishing a quantum group, obtained by quantization of the Lie\nbialgebra $(\\mathfrak{g},\\Schouten{r,\\argument})$.\nFinally we obtain the following theorem as a corollary of\nProp.~\\ref{prop:twistJ} and Corollary~\\ref{twistaction}. \n\n\\begin{theorem}\n  Suppose $(\\lie g, r)$ is a triangular Lie algebra and $\\varphi\\colon\n  \\lie g\\rightarrow \\Secinfty(TM)$ is an action on the manifold\n  $M$. Then there exist a formal deformation quantization\n  $\\mathscr{A}_\\hbar$ of $(M,\\pi)$ (where\n  $\\varphi\\wedge\\varphi(r)=\\pi$) and a quantization\n  $\\mathscr{U}_\\hbar(\\lie g)$ of $(\\lie g, r)$ which allow a\n  deformation symmetry\n  \\begin{equation} \n    \\tilde{\\varphi}^{\\rho_\\hbar}\\circ\\mathscr{J}\n    \\colon \n    \\mathscr{U}_\\hbar(\\lie g)_{poly}\\longrightarrow C(\\mathscr{A}_\\hbar;\\mathscr{A}_\\hbar)\n  \\end{equation}\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Comparison with Drinfeld's construction}\n\n\n\n\n\n \n \n \n\nFirst, let us briefly recall the original construction of Drinfeld\n(see\n\\cite{aschieri.schenkel:2014a,drinfeld:1989a,giaquinto.zhang:1998a}). Consider\na formal twist $J$ on $\\mathscr{U}(\\mathfrak{g})[\\![\\hbar]\\!]$ and a generic\n$\\mathscr{U}(\\lie{g})$-module algebra $\\algebra A$. Drinfeld proved\nthat we can then always define an associative star product on\n$\\algebra A$. In particular, consider $\\algebra A = \\Cinfty(M)$ with\npointwise multiplication $m$.  Given a Lie algebra action $\\varphi:\n\\lie g \\to \\Secinfty(TM)$ we obtain a Hopf algebra action\n\\begin{equation}\n  \\label{eq:Action}\n  \\acts\n  \\colon\n  \\mathscr{U}(\\lie{g}) \\times \\Cinfty(M)\n  \\longrightarrow\n  \\Cinfty(M),\n\\end{equation}\nwhich makes $\\Cinfty(M)$ into a left $\\mathscr{U}(\\lie{g})$-module\nalgebra. More precisely, $X \\acts f = \\Lie_{\\varphi(X)} f$ where\n$\\Lie$ denotes the Lie derivative. The action $\\acts$ extends to\nformal power series \n\\begin{equation}\n  \\acts \\colon\n  \\mathscr{U}(\\lie{g})[\\![\\hbar]\\!] \\times \\Cinfty(M)[\\![\\hbar]\\!]\\longrightarrow\n  \\Cinfty(M)[\\![\\hbar]\\!].\n\\end{equation}\nThus the product defined by\n\\begin{align}\n  \\label{eq:StarTwist}\n  f\\star_J g\n  =\n  m(J\\acts(f\\tensor g))\n  =\n  m (\\Lie_{\\varphi \\tensor \\varphi(J)}(f \\tensor g))\n\\end{align}\nfor $f, g \\in \\Cinfty(M)[\\![\\hbar]\\!]$ is a star product.  The classical\nlimit of \\eqref{eq:StarTwist} is given by\n\\begin{equation}\n  \\label{eq:ThePoissonStructure}\n  \\{f, g\\}\n  =\n  m (r \\acts (f \\tensor g)),\n\\end{equation}\nwhere $r:=\\frac{J-\\tau(J)}{\\hbar}|_{\\hbar=0}$ is the $r$-matrix\nassociated to the twist $J$, here $\\tau$ denotes the flip $X\\otimes\nY\\mapsto Y\\otimes X$.  It is important to underline that the deformed\nalgebra $(\\Cinfty (M)[\\![\\hbar]\\!], \\star_J)$ is then a module-algebra for\nthe quantum group:\n\\begin{equation}\n  \\algebra U_J(\\lie{g}) \n  := (\\algebra U(\\lie{g})[\\![\\hbar]\\!], \\Delta_J).\n\\end{equation}\nIn other words, $\\acts$ is a Hopf agebra action of the twisted Hopf\nalgebra $\\algebra U_J(\\lie{g})$ on $(\\Cinfty (M)[\\![\\hbar]\\!], \\star_J)$.\n\n\n\nThus, the construction takes a formal twist $J\\in\n\\mathscr{U}(\\mathfrak{g})[\\![\\hbar]\\!]$ and an infinitesimal action of $\\lie\ng$ on $M$ as input and produces a deformation quantization $\\algebra\nA_\\hbar$ together with an action of the quantum group\n$\\mathscr{U}_J(\\lie g)$ on it. \nIn our approach one starts with\nan $r$-matrix $r\\in\\lie g\\wedge\\lie g$ and an infinitesimal action of\n$\\lie g$ on $M$ and obtains a formal twist $J$ and a deformation\nsymmetry $\\tilde{\\varphi}$. These then also yield a deformation\nquantization given by\n\\begin{equation}\n  f\\star_r g=\\tilde{\\varphi}(J)(f\\otimes g),\n\\end{equation}\nand another deformation symmetry $\\tilde{\\phi}^{\\rho_\\hbar}\\circ\\mathscr{J}$ \nof the quantum group $\\algebra U_J(\\lie g)$ in the deformed algebra $\\algebra A_\\hbar$.\n %\n \n \n \n \nThe main difference of the two approaches is that the formal twist is taken as\ngiven in Drinfeld's approach while we obtain it through quantization of an $r$-matrix. The trade-off\nis however that we do not obviously obtain an action of a quantum\ngroup anymore, instead we obtain the deformation symmetry. A direct comparison of the two approaches will be\nnontrivial and it implies a study of the compatibility condition\nbetween connections mentioned in Remark \\ref{rem:WrongCandidate}. In\nparticular it is of interest whether the process of quantization so\nobtained can be made functorial for equivariant maps between the\nmanifolds. We will come back to this in a future project.\n\n\n\n\n\n\n\n{\n  \\footnotesize\n  \\renewcommand{\\arraystretch}{0.5}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nWithin the past two years, three neutron star binaries, PSR\nJ1824$-$2452I in M28 \\citep{Pap13}, PSR J1023+0038 \\citep{Pat14}, and\nXSS J12270$-$4859 \\citep{Bassa14}, have been observed to\nswitch between accreting and rotation-powered pulsar states. These\ndiscoveries have confirmed the standard model for the formation of\nrotation-powered millisecond pulsars (MSPs) via spin-up by accretion\nin a low-mass X-ray binary (LMXB).\n\nThe two confirmed transitional MSPs in the field of the Galaxy, PSR\nJ1023+0038 and XSS J12270$-$4859, exhibit strong $\\gamma$-ray emission\nin \\textit{Fermi} Large Area Telescope data even during their\naccreting states. This implies that other similar objects may be\nhiding among the unidentified \\textit{Fermi} LAT sources \\citep[see,\n  e.g.,][for the case of 1FGL J1417.7$-$4407]{Stra15}.  While targeted\nradio pulsation searches of \\textit{Fermi} LAT sources have proven to\nbe particularly fruitful for discovering new rotation-powered MSPs,\nyielding at least 38 published discoveries to date\\footnote{See\n  https:\/\/confluence.slac.stanford.edu\/display\/GLAMCOG\/\n  \\\\ Public+List+of+LAT-Detected+Gamma-Ray+Pulsars for an up-to-date\n  list.}, transitional objects may elude detection at radio\nfrequencies since the pulsar emission mechanism appears to be\nextinguished when an accretion disk is present in the system\n\\citep[e.g.,][]{Stap14}.\n\nIn their low-luminosity ($\\sim$$10^{33}$\\,erg s$^{-1}$ in the soft\nX-ray band) accreting states, PSR J1023+0038, XSS\nJ12270$-$4859, and PSR J1824--2452I exhibit sudden, unpredictable\ndrops in flux and occasional intense flares\n\\citep{deM10,deM13,Lin14,Ten14,Bog14b}. This characteristic X-ray\nbehavior is a unique signature that can be used to identify additional\ntransition systems and distinguish them from other varieties of\naccreting objects such as cataclysmic variables (CVs).\n\nWe have identified a particularly promising candidate for an LMXB\/MSP\ntransition object, namely, 3FGL J1544.6$-$1125, previously known as\n2FGL J1544.5$-$1126. The only X-ray-bright object within the 95\\%\nconfidence error ellipse of this 3FGL source is the \\textit{ROSAT}\nsource 1RXS J154439.4$-$112820 \\citep{Step10}.  Optical identification of\n\\textit{ROSAT}-selected candidates for unassociated \\textit{Fermi} LAT\nsources by \\citet{Mas13} revealed this object to have an optical\nspectrum in 2012 consistent with those of accretion-disk-dominated\nsystems, with prominent emission lines of H and He. On this basis, it\nwas classified as a CV and was dismissed as the counterpart of 3FGL\nJ1544.6$-$1125 because no CVs are known to exhibit persistent\n$\\gamma$-ray emission.  However, given the fact that both PSR\nJ1023+0038 and XSS J12270--4859 were mis-classified as CVs upon their\ndiscovery suggests that similar objects may also be masquerading as\nCVs.  Based on this, we suspected that 1RXS J154439.4$-$112820 is\nindeed the counterpart of the \\textit{Fermi} LAT source. Here, we\npresent an analysis of X-ray and UV data from \\textit{XMM-Newton} and\noptical time-series photometry obtained at MDM Observatory of 1RXS\nJ154439.4$-$112820, which reveal the true nature of this object.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.42\\textwidth]{finding_chart.ps}\n\\end{center}\n\\caption{Finding chart for \\rxs\\ (tic marks) at position \\pos\\ from\n  MDM 1.3m images in the GG420 filter.  The size is\n  $4^{\\prime}\\!.3\\times4^{\\prime}\\!.3$.  North is up and east is to\n  the left.  The star marked ``C'' has magnitude $R=14.25$ in the USNO\n  B1.0 catalog, and is used for appoximate calibration in GG420.\n  Faint fringing is a detector artifact due to the filter transmission\n  at long wavelengths.  }\n\\label{fig:finder}\n\\end{figure} \n\n\n\\begin{deluxetable*}{cccccl}\n\\tabletypesize{\\small}\n\\tablewidth{0pt}\n\\tablecaption{Log of MDM Observatory Time Series of \\rxs\\tablenotemark{a}}\n\\tablehead{\n\\colhead{Telescope} & \\colhead{Date (UT)} & \\colhead{Time (UT)} &\n\\colhead{Exp. time (s)} &  \\colhead{$N_{\\rm exp}$} & Conditions\n}\n\\startdata\n  2.4-m  &  2014 Mar 22  & 09:53--12:42  &  10  &  677  & Clear \\\\\n  2.4-m  &  2014 Mar 23  & 07:15--12:40  &  10  & 1331  & Photometric \\\\\n  1.3-m  &  2015 Feb 13  & 09:44--13:14  &  30  &  379  & Cirrus, near moon \\\\\n  1.3-m  &  2015 Feb 14  & 09:45--13:13  &  30  &  376  & Thin clouds \\\\\n  1.3-m  &  2015 Feb 17  & 09:19--13:13  &  30  &  425  & Partly cloudy \\\\\n  1.3-m  &  2015 Feb 18  & 09:14--11:52  &  30  &  288  & Photometric \n\\enddata\n\\label{tab:log}\n\\tablenotetext{a}{All observations in the GG420 filter.}\n\\end{deluxetable*}\n\n\n\n\n\\section{OBSERVATIONS AND DATA REDUCTION}\n\n\\subsection{XMM-Newton}\nThe field around 1RXS J154439.4$-$112820 was observed with\n\\textit{XMM-Newton} on 2014 February 16 (ObsID 072080101). The EPIC MOS1\ninstrument was configured in ``full frame'' mode The MOS2 and PN\ndetectors were used in ``timing'' mode, which provide time resolution\nof 1.75 ms and 30 $\\mu$s, respectively. In this mode, the fine time\nresolution is enabled by imaging in only one spatial dimension (RAWX),\nwith the other dimension (RAWY\/TIME) used for fast read out.  The\nmedium optical blocking filter was used for all three instruments.\nThe effective exposures for the MOS1, MOS2, and pn instruments were\n40.9 ks, 40.6 ks, and 38.9 ks, respectively. All three data sets were\nprocessed with the SAS\\footnote{The XMM-Newton SAS is developed and\n  maintained by the Science Operations Centre at the European Space\n  Astronomy Centre and the Survey Science Centre at the University of\n  Leicester.} version {\\tt xmmsas\\_20141104\\_1833}.  The MOS1 source\nevents were extracted from a circular region of radius 36$''$. The\nMOS2 and pn data were extracted from regions of width 60 and 7\ndetector pixels, respectively, in the imaging (RAWX) direction and the\nfull detector width in the readout (RAWY\/TIME) direction. The standard\nflag and pattern event screening criteria were applied to all three\ndata sets.\n\nThe \\textit{XMM-Newton} Optical Monitor (OM) was used with the $U$\nfilter (centered on 3440 {\\AA} with a band pass of 840 {\\AA}) in place\nand in \"fast\" mode, which permits rapid readout and photon counting\ncapabilities.  A photometric light curve was obtained using the {\\tt\n  omfchain} processing pipeline with the default parameter settings.\n\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[height=5.8cm]{j1544_lc_new.ps}\n\\includegraphics[height=5.8cm]{j1544_histo_final.ps}\n\\end{center}\n\\caption{Top left: Exposure-corrected, background-subtracted 0.3--10\n  keV X-ray light curve of 1RXS J154439.4$-$112820 from the\n  \\textit{XMM-Newton} EPIC observation obtained by co-adding the\n  MOS1\/2 and pn light curves binned at a resolution of 20\n  seconds. Note the sudden jumps in count rate between $\\approx$0.2\n  and $\\approx$2 counts s$^{-1}$. Top right: Histogram of\n    number of 20 second time bins ($N$) as a function of X-ray count\n    rate, showing strong bimodality in flux.  Bottom:\n  \\textit{XMM-Newton} OM $U$ filter photometric light curve binned at\n  20 minute resolution. The dotted line shows the best-fit sinusoid\n  with period 5.2 hours.}\n\\end{figure*}\n\n\n\n\\subsection{MDM Observatory}\nWe obtained time-series optical photometry of \\rxs\\ using the MDM\nObservatory's 2.4-m Hiltner telescope and 1.3-m McGraw-Hill telescope\non Kitt Peak during the 2014 and 2015 observing seasons.\nTable~\\ref{tab:log} is a summary of the data obtained on six nights.\nThe detector was the thinned, backside illuminated SITe CCD\n``Templeton''.  It has $1024\\times1024$ pixels, with a scale of\n$0.\\!^{\\prime\\prime}275$ pixel$^{-1}$ on the 2.4-m and\n$0.\\!^{\\prime\\prime}509$ pixel$^{-1}$ on the 1.3-m.  The filter used\nwas a GG420, to maximize throughput.  In order to reduce the readout\ntime, the detector was windowed down and the pixels were binned\n$2\\times2$.  Exposure times were 10~s with a read\/prep time of\n5\\,s, or 30~s with a read\/prep time of 3\\,s.\nFigure~\\ref{fig:finder} is a finding chart from the MDM GG420 filter\nimages, showing the object at position \\pos\\ from the USNO B1.0\ncatalog. \n\n\n\n\\section{X-ray Analysis}\nTo maximize the photon statistics of the data for the variability\nanalysis, we extracted a total exposure-corrected and\nbackground-subtracted X-ray light curve of 1RXS J154439.4$-$112820 by\ncombining the individual MOS1\/2 and pn time series over the period\nwhen the three instruments are collecting data simultaneously. This\nwas done using the {\\tt epiclccorr} task in SAS. For this relatively\nbright source, removal of periods contaminated by background flares\nwas not neccessary. We verified that the observed variability in the\nbackground light curve is not present in the background-subtracted\nsource light curves.\n\nThe total 0.3--10 keV light curve, binned at a resolution of\n20 s to look for rapid variability, is shown in Figure 2.  The X-ray\nlight curve exhibits obvious short-timescale, large-amplitude\nvariability. The X-ray flux alternates irregularly between two\nclearly distict modes that differ in flux by a factor of\n$\\approx$10. Indeed, we find clear bimodality in the X-ray count rate\nwith two peaks at $\\approx$0.2 and $\\approx$2 counts s$^{-1}$.  The\ntransition between the two flux levels is very rapid, with a typical\nduration of order 10 s. There is no evidence for periodicity in the\nobserved variability pattern.\n\nArchival \\textit{Swift} XRT data obtained in 2006, 2007, and 2012 also\nshow X-ray variability comparable to that observed with\n\\textit{XMM-Newton}. However, due to the brief \\textit{Swift}\nexposures it is not possible to determine whether the variability\npattern is the same.\n\n\nWe carried out a spectroscopic analysis of the \\textit{XMM-Newton}\nEPIC data in XSPEC\\footnote{Available at\n  \\url{http:\/\/heasarc.nasa.gov\/docs\/xanadu\/xspec\/index.html}.}  12.7.1\n\\citep{Arnaud96}. Due to known issues with the spectral calibration of\nthe pn instrument in timing mode, we only considered the MOS1\/2\ndata. The time-averaged spectrum of 1RXS J154439.4$-$112820 is\nwell-described by a simple absorbed power-law. The best fit parameters\nare $\\Gamma=1.68\\pm0.04$ for the spectral photon index, $N_{\\rm H}=(1.4\n\\pm 0.1)\\times 10^{21}$ cm$^{-2}$ for the hydrogen column density\nalong the line of sight, and $\\chi^2_{\\nu}=0.95$ for 399 degrees of\nfreedom. The unabsorbed flux in the 0.3--10 keV band is $F_X=(3.51 \\pm\n0.07) \\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$. All uncertainties\nquoted are at a 90\\% confidence level.  The derived $N_{\\rm H}$ is in\nfull agreement with the value measured through the Galaxy of $1.3\n\\times 10^{21}$ cm$^{-2}$ in the direction of this source\n\\citep{Kalb05}.\nFitting the high and low emission separately (by making a cut\n  at 0.9 counts s$^{-1}$ in the light curve from Figure 2) produces\n  $\\Gamma=1.67\\pm0.04$ and $\\Gamma=1.97\\pm0.28$, implying only\n  marginally significant evidence for spectral variations as a\n  function of source brightness. As the distance to 1RXS\nJ154439.4$-$112820 is unconstrained, it is not possible to determine\nthe X-ray luminosity levels of the high and low modes.\n\nAlthough the 30 $\\mu$s time resolution of the pn permits the search for\nX-ray pulsations at millisecond periods, the lack of an orbital period\nand ephemeris, combined with the relatively low source count rate,\nrender a blind periodicity search infeasible at this time.\n\n\n\n\\section{UV and Optical Variability}\nThe bottom panel of Figure 2 shows the \\textit{XMM-Newton} OM $U$\nfilter photometric data at a 20 minute resolution, which shows\nvariability by up to $\\approx$0.5 mag.  A fit with a sinusoid\n  indicates a periodicity of 5.2 hours. However, it should be noted\n  that the low points in the U filter data may not correspond to a\n  binary modulation as they could be related to the intervals\n  dominated by low mode X-ray emission that occur at the same time.\nDue to the faint nature of 1RXS J154439.4$-$112820, it is not possible\nto probe variability on timescales of tens of seconds with the OM\ndata.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{time_series.ps}\n\\end{center}\n\\caption{MDM light curves of \\rxs.\n  Magnitude is based on differential photometry\n  in the GG420 filter with respect to the comparison star indicated\n  in Figure~\\ref{fig:finder}.\n}\n\\label{fig:phot}\n\\end{figure} \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{psrj1023.ps}\n\\end{center}\n\\caption{MDM 1.3m $R$-band light curve of \\psr\\ with 13~s cadence.\n  The orbital period of the binary is 0.198~d, which is evident\n  as the long wave in the light curve. The magnitude\n  is based on differential photometry with respect to a\n  calibrated comparison star as described in \\citet{Bog14b}.\n}\n\\label{fig:psrj1023}\n\\end{figure} \n\n\nDifferential photometry on the much more sensitive MDM time series was\nperformed with respect to field stars, and the results are shown in\nFigure~\\ref{fig:phot}.  Although not calibrated in the GG420 filter,\nwe adopt the $R$ magnitudes from stars in the USNO B1.0 catalog as an\napproximate reference, in particular, $R=14.25$ for the bright star\nmarked ``C'' on the finding chart.  On this scale, the magnitude of\n\\rxs\\ varies from about 18.3 to 19.1, similar to $R=18.53$ as listed\nin the USNO B1.0 catalog.\n\nThe nature of the optical variability of \\rxs\\ observed at MDM is\nsubtly different from that of most cataclysmic binaries.  Rather than\na random walk or flickering, the brightness appears to be confined\nbetween minimum and maximum values separated by $\\approx$$0.5$\nmagnitudes. This is especially obvious in the higher-quality 2.4m\ndata. In this regard, it resembles the limit-cycle behavior\noccasionally seen from \\psr, an example of which is shown in\nFigure~\\ref{fig:psrj1023}.  There is no obvious candidate for an\norbital or any other period in these light curves of \\rxs. A\n  periodicity search of the longest light curve on 2014 March 23 shows\n  a peak in the power spectrum at $\\sim$5.4 hours, but this is unreliable\n  because it is comparable to the length of the observation, and it\n  cannot be confirmed by any of the shorter time series. In contrast,\nin all light curves of \\psr, e.g., Figure~\\ref{fig:psrj1023}, its\n4.75~hr orbital period is clearly manifest as a modulation due to\nheating of the side of the companion facing the neutron star.\n\n\nThe Catalina Real-time Transient Survey \\citep{Dra09} contains 324\nphotometric measurements of \\rxs\\ over a span of 7 years.  We searched\nthese for periodicity in the range 1--24 hours, such as may arise from\norbital modulation, but did not find a significant detection.\n\n\\section{DISCUSSION AND CONCLUSIONS} \nWe have presented an X-ray, UV, and optical investigation of 1RXS\nJ154439.4$-$112820, the most likely counterpart of the \\textit{Fermi}\nLAT source 3FGL J1544.6$-$1125.  When combined with previous studies,\nwe can establish the following observational properties of the system:\ni) strong emission lines of H and He, suggestive of an accretion disk\n\\citep[see][]{Mas13}; ii) fast optical variaiblity within limits; iii)\npower-law X-ray emission with $\\Gamma \\approx 1.7$ that exhibits rapid\nmoding between two distinct X-ray flux levels; iv) persistent\n$\\gamma$-ray emission with no evidence for a spectral cut-off\n\\citep[see][]{Fermi15}.  These characteristics are strikingly similar\nto what is observed for the transitional MSPs PSR J1824$-$2452I\n\\citep{Pap13,Pal13,Lin14,Fer14}, PSR J1023+0038\n\\citep{Stap14,Cot14,Tak14,Bog14b} and XSS J12270--4859\n\\citep{deM10,deM13,Pap14,Bogd14a,Roy15,Pap15} in their low-luminosity accreting states.\nTherefore, 1RXS J154439.4$-$11282 is the counterpart of 3FGL\nJ1544.6$-$1125 and is almost certainly a binary that hosts an MSP and\nis presently in a low-luminosity LMXB state.\n\nAs argued by \\citet{Del14} for the case of PSR J1023+0038, the\n$\\gamma$-ray emission seen by \\textit{Fermi} LAT is likely generated\nin a jet\/wind outflow, perhaps driven by propeller-mode accretion.\nThe peculiar X-ray moding appears to be caused by rapid transitions\nbetween channelled accretion onto the neutron star surface (the high\nmode) and no accretion (the low mode).  As with PSR J1023+0038\n\\citep{Arch14} and XSS J12270--4859 \\citep{Pap14}, we predict that\n1RXS J154439.4$-$112820 should exhibit coherent X-ray pulsations at\nthe neutron star spin period during the high flux mode. However, due\nto the lack of a precise orbital ephemeris, at present, a blind\nperiodicity search of the X-ray data is not practical.  In the absence\nof evidence for the orbit of \\rxs\\ in either X-ray or optical light\ncurves, a spectroscopic period should be obtained from its optical\nemission lines.\n\nThis system can serve as an additional laboratory for studying\nthe peculiar phenomenology of transitional MSP systems, especially the\nhighly unusual X-ray flux switching, and the relation, if any, with\nthe optical moding we see in 1RXS J154439.4$-$112820. In turn, this\nmay shed light on the little-understood low-luminosity (``quiescent'')\nregime in neutron star X-ray binaries.  In addition, extending the\npresently small sample of transitional MSPs has important implication\nfor understanding key aspects of the evolution of compact neutron star\nbinaries.\n\nBased on the recent history of the three confirmed LMXB\/RMSP\ntransition objects, we expect 1RXS J154439.4$-$112820 to transform\ninto a disk-free state in which the rotation-powered millisecond\npulsar activates. Therefore, this object warrants intensive\nmulti-wavelength monitoring in order to catch such an occurence.\n\n\n\\acknowledgements\nA portion of the results presented were based on observations obtained\nwith \\textit{XMM-Newton}, an ESA science mission with instruments and\ncontributions directly funded by ESA Member States and NASA.  This\nwork used observations obtained at the MDM Observatory, operated by\nDartmouth College, Columbia University, Ohio State University, Ohio\nUniversity, and the University of Michigan. We have made\nextensive use of the NASA Astrophysics Data System (ADS) and the arXiv.\n\nFacilities: \\textit{XMM}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn Random Matrix Theory (RMT) many statistical quantities can be\ndescribed as determinants. This is especially true in the cases of\nwhat are known as the classical ensembles, for example, the Gaussian\nUnitary Ensemble (GUE) or the Laguerre Ensembles. Since the\ngroundbreaking work of Tracy and Widom \\cite{TW1994}, which\ncharacterized the largest eigenvalue distribution for the GUE, one\ncommon approach is to write some statistical quantity as a\ndeterminant and then express the determinant as something involving\na Painlev\\'e transcendent, a solution to one of the classical\nPainlev\\'e second order non-linear differential equations.\n\nTechniques to find the determinant and then the resulting\ndifferential equation are quite complicated. Tracy and Widom used\nmore of an operator theory approach, others have used an integrable\nsystem approach, whilst others still have used a stochastic equation\napproach.\n\nThis paper highlights a technique known as the ``ladder operator''\napproach and is, in some sense, one long example illustrating the\ntechnique and describing the application of interest. The specific\napplication relates to the performance of multiple-input\nmultiple-output (MIMO) wireless communication systems, in which both\nthe transmitter and receiver devices are equipped with multiple\nantennas. Such systems have been the subject of intense interest\nsince the key papers \\cite{Telatar} and \\cite{Foschini}, and now\nform the cornerstone of most modern day wireless systems (Wi-Fi,\ncellular networks, etc). Here, a key fundamental performance measure\nis considered, the so-called ``outage probability'',  and this is\nshown to involve the probability distribution of a certain ``linear\nstatistic'' in a Laguerre random matrix ensemble. The problem thus\nfalls naturally within the realm of the ladder operator framework.\n\nHere is the idea of the approach. For the example at hand, the\nlinear statistic of interest can be described through its moment\ngenerating function as a Hankel determinant, which using the theory\nof orthogonal polynomials (for a certain nonstandard weight\ngenerally) can be computed via a product of norms of monic\northogonal polynomials. Now it is well-known that orthogonal\npolynomials satisfy three term recurrence equations. In fact, the\ntwo coefficients in the recurrence combined with initial conditions,\ncompletely determine the polynomials. Thus information about the\ncoefficients in the recurrence equations should yield information\nabout the Hankel determinants.\n\nThe path to this information is from the ladder operators, two\nformulas that connect the polynomials one index apart to their\nderivatives. These yield, using basic complex analysis, a set of\nequations in the coefficients along with two additional auxiliary\nquantities that arise.  The story would end here, except for the\nfact that often there is a ``time'' parameter implicit in the\noriginal weight and thus in the polynomials themselves. Using\n``time'' evolution one can then, using only elementary means, find a\npair of coupled Ricatti equations in the two auxiliary quantities.\nThese then lead directly to a Painlev\\'e equation. It should be\npointed out that this method works at least in principle, if the\n``time'' parameter is present and if the derivative of the logarithm\nof the weight is a rational function. Then one can in many cases\nfollow the steps illustrated in this paper.\n\n\nHere is an outline of the paper. The next section contains the\npreliminaries of the theory including the ladder operator equations.\nSection III shows how the application of interest, the outage\nprobability performance measure which arises in the application of\nMIMO wireless communication, can be described using the RMT\nframework. Section IV provides the details of the path to the\ndifferential equation solutions, which are presented Theorems 1 and\n2.\n\n\\section{Preliminaries}\n\n\n\\subsection{Linear Statistics of Hermitian Random Matrices and Hankel Determinants}\n\nFor the MIMO capacity application, it will be seen that the problem\nof interest falls within the general theory of linear statistics of\nHermitian random matrices, with a close connection to the theory of\northogonal polynomials. Here a brief introduction to the general\ntheory is given, and preliminaries are established for later use.\n\nWe will require the distribution of a certain linear statistic\n\\begin{align} \\label{eq:LinStat}\n\\sum_{k=1}^N f(x_k)\n\\end{align}\nin the eigenvalues $\\{ x_k \\}$ of a $N \\times N$ Hermitian random\nmatrix, with joint eigenvalue density of the form\n\\begin{align} \\label{eq:EigDensGeneric}\np(x_1, \\ldots, x_N) \\propto \\prod_{k = 1}^N  w_0(x_k) \\prod_{1 \\leq\ni <  j \\leq N} ( x_j - x_i )^2  ,\n \\quad  x_k \\in (a, b)\n\\end{align}\nfor some weight function $w_0(\\cdot)$. It is convenient to attempt\nto characterize the distribution of the linear statistic\n(\\ref{eq:LinStat}) through its moment generating\nfunction\\footnote{The parameter $\\lambda$ is an indeterminate which\ngenerates the random variable\n$\\sum_{k=1}^N f(x_k).$\\\\\n},\n\\begin{align} \\label{eq:MGFFirst}\n\\mathcal{M}(\\lambda) = E \\left[ \\exp \\left( \\lambda \\sum_{k=1}^N f (x_k)\n\\right) \\right] \\; = \\; E \\left[ \\prod_{k=1}^N e^{\\lambda f(x_k)}\n\\right]\n\\end{align}\nwhich upon substituting for (\\ref{eq:EigDensGeneric}) gives\n\\begin{align} \\label{eq:DnGeneric}\n\\mathcal{M}(\\lambda) &= \\frac{ \\frac{1}{N!} \\int_{(a,b)^N}\\prod_{1\\leq\ni<j\\leq N}(x_j-x_i)^2\\prod_{k=1}^{N} w(x_k) dx_k} {\\frac{1}{N!}\n\\int_{(a,b)^N}\\prod_{1\\leq i<j\\leq\nN}(x_j-x_i)^2\\prod_{k=1}^{N}w_0(x_k) dx_k}\n\\end{align}\nwhere\n$$\nw(x) := w_0(x) e^{\\lambda f(x) }\n$$\ndenotes the \\emph{deformed} version of the reference weight\n$w_0(\\cdot)$. Application of the Andreief-Heine identity \\cite{Sze}\nnow directly leads to\n\\begin{align} \\label{eq:MGFRatio}\n\\mathcal{M}(\\lambda) = \\frac{D_N[ w ]}{D_N[w_0 ]} \\; = \\; \\frac{ \\det\n\\left( \\int_a^b x^{i+j-2} w(x) dx \\right)_{i,j=1}^{N} } {\\det \\left(\n\\int_a^b x^{i+j-2}w_0(x) dx \\right)_{i,j=1}^{N}}\\; \\; ,\n\\end{align}\nwhich is a ratio of Hankel determinants. By virtue of the Selberg\nintegral, for most ``classical'' weight functions of interest, the\nHankel determinant in the denominator of (\\ref{eq:DnGeneric}) admits\nan explicit closed-form (non determinantal) representation. The\nnumerator, on the other hand, is much more difficult to\ncharacterize, since it involves the more complicated deformed weight\n$w$. To proceed, methods based on orthogonal polynomials will be\nintroduced in the sequel.\n\n\\subsection{Orthogonal Polynomials and their Ladder Operators}\n\\label{sec:LadderOperators}\n\nWe start by noting that\n\\begin{align}\n\\prod_{1\\leq i < j \\leq N}(x_j-x_i) = \\det \\left( x_j^{i-1}\n\\right)_{i,j=1}^N = \\det \\left( P_{i-1}(x_j) \\right)_{i,j=1}^N\n\\end{align}\nwhere $P_j (\\cdot)$ represents any monic polynomial of degree $j$,\n {\\small \\begin{eqnarray} \\label{eq:MonicDefn}\nP_j(z)=z^{j}+\\textsf{p}_1(j)\\:z^{j-1}+... \\end{eqnarray}} Applying this in\n(\\ref{eq:DnGeneric}) and once again integrating via the\nAndreief-Heine identity, the numerator evaluates to\n\\begin{align} \\label{eq:DnIntermed}\nD_N[w] = \\det \\left( \\int_a^b P_{i-1}(x) P_{j-1}(x) w(x) dx\n\\right)_{i,j=1}^{N} \\; .\n\\end{align}\nIf we orthogonalize the polynomial sequence $\\{P_{n}(x)\\}$ with\nrespect to $w(x)$ over the interval $[a,b],$ i.e.,\n\\begin{align}\n\\label{eq:OrthRel} \\int_{a}^{b}\\:P_{j}(x)P_k(x)\nw(x)dx=h_{j}\\delta_{j,k},\\;\\; j,k=0,1,2,...\n\\end{align}\nwith $h_j$ denoting the square of the $L^2$ norm of $P_j$ over\n$[a,b],$ then (\\ref{eq:DnIntermed}) reduces to\n\\begin{align} \\label{eq:DetRelation}\nD_N[ w ] = \\prod_{k=0}^{N-1} h_k \\; .\n\\end{align}\n\nThe key challenge is how to characterize the class of polynomials\nwhich obey the orthogonality constraints in (\\ref{eq:OrthRel}) or,\nmore importantly, the norms of such polynomials required to evaluate\n(\\ref{eq:DetRelation}).\n\n\nIf all the moments of the weight $w$ exist, then the theory of\northogonal polynomials states that the $P_n(z)$ for $n=0,1,2,...$\nsatisfy the three term recurrence relations,\n\\begin{align}\n\\label{eq:RecRel}\nzP_n(z)=P_{n+1}(z)+\\alpha_n\\:P_n(z)+\\beta_n\\:P_{n-1}(z).\n\\end{align}\nThe above sequence of polynomials can be generated from the\northogonality conditions (\\ref{eq:OrthRel}), the recurrence\nrelations (\\ref{eq:RecRel}), and the initial conditions,\n\\begin{align}\nP_0(z)=1,\\;\\;\\beta_0P_{-1}(z)=0.\n\\end{align}\nSubstituting (\\ref{eq:MonicDefn}) into the recurrence relations, an\neasy computation shows that \\begin{eqnarray} \\label{eq:p1Defn}\n\\textsf{p}_1(n)-\\textsf{p}_1(n+1)=\\alpha_n, \\end{eqnarray} with\n$\\textsf{p}_1(0):=0.$ A telescopic sum of (\\ref{eq:p1Defn}) gives\n\\begin{align}\n\\textsf{p}_1(n) =-\\sum_{j=0}^{n-1} \\alpha_j .\n\\end{align}\n>From the recurrence relation (\\ref{eq:RecRel}) and the orthogonality\nrelations (\\ref{eq:OrthRel}), we find {\\small \\begin{eqnarray}\\label{eq:betah}\n \\beta_n=\\frac{h_n}{h_{n-1}}. \\end{eqnarray}}\n\nWe shall see that $\\textsf{p}_1(n)$ plays an important role in later\ndevelopments. For more information on orthogonal polynomials, we\nrefer the reader to Szeg\\\"o's treatise \\cite{Sze}.\n\nNext, we present three Lemmas which are concerned with the ``ladder\noperators'' associated with orthogonal polynomials, as well as\ncertain supplementary conditions. Note that these have been known\nfor quite sometime; we reproduce them here for the convenience of\nthe reader using the notation of \\cite{chen+its}, where one can also\nfind a list of references to the literature. We also mention that\nMagnus \\cite{Magnus} was perhaps the first to apply these\nlemmas---albeit in a slightly different form---to random matrix\ntheory and the derivation of Painlev\\'e equations. Tracy and Widom\nalso made use of the compatibility conditions in their systematic\nstudy of finite $n$ matrix models \\cite{twdet}. See also\n\\cite{for1}.\n\\\\\n\\noindent {\\bf Lemma 1} \\emph{Suppose $\\v(x)=-\\log w(x)$ has a\nderivative in some Lipshitz class with positive exponent. The\nlowering and raising operators satisfy the differential-difference\nformulas:} {\\small \\begin{eqnarray}\nP_n'(z)&=&-B_n(z)P_n(z)+\\beta_n\\:A_n(z)P_{n-1}(z)\\\\\nP_{n-1}'(z)&=&[B_n(z)+{\\mathsf{v}^{\\prime}}(z)]P_{n-1}(z)-A_{n-1}(z)P_n(z), \\end{eqnarray}} where\n{\\small \\begin{eqnarray}\nA_n(z)&:=&\\frac{1}{h_n}\\int_{a}^{b}\\frac{{\\mathsf{v}^{\\prime}}(z)-{\\mathsf{v}^{\\prime}}(y)}{z-y}\\:P_n^2(y)w(y)dy \\label{eq:AnNewDefn}\\\\\nB_n(z)&:=&\\frac{1}{h_{n-1}}\\int_{a}^{b}\\frac{{\\mathsf{v}^{\\prime}}(z)-{\\mathsf{v}^{\\prime}}(y)}{z-y}P_n(y)P_{n-1}(y)w(y)dy.\n\\label{eq:BnNewDefn} \\end{eqnarray}} A direct computation produces two\nfundamental supplementary or compatibility conditions valid for all\n$z\\in \\mathbb{C}\\cup\\{\\infty\\}$. These are stated in the next Lemma.\n\\\\\n{\\bf Lemma 2} \\emph{The functions $A_n(z)$ and $B_n(z)$ satisfy the\nsupplementary conditions:} {\\small $$\nB_{n+1}(z)+B_n(z)=(z-\\alpha_n)A_n(z)-{\\mathsf{v}^{\\prime}}(z)\\eqno(S_1)\n$$\n$$\n1+(z-\\alpha_n)[B_{n+1}(z)-B_n(z)]=\\beta_{n+1}A_{n+1}-\\beta_nA_{n-1}(z).\\eqno(S_2)$$}\n\\\\\nIt turns out that there is an equation which gives better insight\ninto the coefficients $\\alpha_n$ and $\\beta_n$, if $(S_1)$ and $(S_2)$\nare suitably combined to produce a ``sum rule'' on $A_n(z).$ We\nstate this in the next lemma. The sum rule, we shall see later,\nprovides important information about the logarithmic derivative of\nthe Hankel determinant.\n\\\\\n{\\bf Lemma 3} \\emph{The functions $A_n(z),$ $B_n(z)$, and the sum}\n{\\small $$ \\sum_{j=0}^{n-1}A_j(z),\n$$}\n\\emph{satisfy the condition:} {\\small $$\nB_n^2(z)+{\\mathsf{v}^{\\prime}}(z)B_n(z)+\\sum_{j=0}^{n-1}A_j(z)=\\beta_n\\:A_n(z)\\:A_{n-1}(z).\\eqno(S_2')\n$$}\n\n\n\\section{Information Theory of MIMO Wireless Systems}\n\nIn this section we introduce the wireless communication problem of\ninterest, and connect it with the general linear statistics\nframework introduced previously.\n\nWe consider a MIMO communication system in which a transmitter\nequipped with $n_t$ antennas communicates with a receiver equipped\nwith $n_r$ antennas. Denoting the transmitted signal vector as ${\\bf x}\n\\in \\mathbb{C}^{n_t}$ and the received signal vector as ${\\bf y} \\in\n\\mathbb{C}^{n_r}$, under a certain assumption on the channel (known\nas ``flat fading''), these signals are related via the linear model\n{\\small \\begin{eqnarray} \\label{eq:LinearModel} {\\bf y} = {\\bf H} {\\bf x} + {\\bf n} \\; , \\end{eqnarray}}\nwhere ${\\bf n}  \\in \\mathbb{C}^{n_r}$ the receiver noise vector, is\ncomplex Gaussian with zero mean and covariance $E( \\mathbf{n}\n\\mathbf{n}^\\dagger ) = \\mathbf{I}_{n_r}$. The matrix \\emph{channel\nmatrix}, ${\\bf H} \\in \\mathbb{C}^{n_r \\times n_t}$,\n represents the wireless fading coefficients\nbetween each transmit and receive antenna. The channel is modeled\nstochastically, with distribution depending on the specific wireless\nenvironment. Under the realistic assumption that there are\nsufficient scatterers surrounding the transmit and receive\nterminals, the channel matrix ${\\bf H}$ is well modeled by a\ncomplex Gaussian distribution with independent and identically\ndistributed (i.i.d.) elements having zero mean and unit variance.\nThis matrix is assumed to be known at the receiver\\footnote{In\npractice, this information can be obtained using standard estimation\ntechniques.}, but the transmitter only has access to its\ndistribution. The transmitted signal ${\\bf x}$ is designed to meet a\npower constraint:\n{\\small \\begin{align}\n \\label{eq:PowerConstraint} E({\\bf x}^{\\dag}{\\bf x})\\leq P \\; .\n\\end{align}}\n\nOur objective is to study the fundamental capacity limits of a MIMO\ncommunication system. Such limits are described by the field of\ninformation theory, founded by Claude Shannon in 1948\n\\cite{Shannon}. Specifically, information-theoretic measures allow\none to precisely determine the highest data rate that can be\ncommunicated with negligible errors by any transmission scheme.\nConsequently, information theory offers a benchmark for the design\nof practical transmission technologies, and has become an\nindispensable tool for modern communication system design.\n\nThe capacity of a communication link is determined by the so-called\n``mutual information'' between the input and output signals. For the\nMIMO model (\\ref{eq:LinearModel}) it is given by: {\\small \\begin{eqnarray}\n\\label{eq:MIDefn}\nI({\\bf x};{\\bf y} | {\\bf H} \n={\\cal H}({\\bf y} | {\\bf H})-{\\cal H}({\\bf n}) \\end{eqnarray} } with ${\\cal H}({\\bf y} | {\\bf H})$\ndenoting the conditional entropy of ${\\bf y}$, {\\small \\begin{eqnarray} {\\cal H}({\\bf y}\n| {\\bf H})= -\\int_{\\mathbb{C}^{n_r} }p({\\bf y} | {\\bf H})\\log p({\\bf y} | {\\bf H})d {\\bf y}\n, \\end{eqnarray}} where $p({\\bf y} | {\\bf H})$ denotes the conditional density of ${\\bf y}$\ngiven ${\\bf H}$.  This formula represents the maximum amount of\ninformation that can be reliably transported between the transmitter\nand receiver (i.e., it represents the rate which is ``supportable''\nby a given realization of the MIMO channel). It was proved in\n\\cite{Telatar} that the conditional mutual information $I({\\bf x};{\\bf y} |\n{\\bf H} )$ is maximized by choosing the input signal vector ${\\bf x}$\naccording to a zero-mean circularly-symmetric complex Gaussian\ndistribution with covariance ${\\bf Q}_{\\rm x} = E \\left( {\\bf x} {\\bf x}^{\\dag}\n\\right)$ satisfying ${\\rm tr} \\left({\\bf Q}_{\\rm x} \\right) \\leq P$.\nIn this case, the mutual information (\\ref{eq:MIDefn}) was shown to\nbe {\\small \\begin{align}\n \\label{eq:MIDefn2} I({\\bf x};{\\bf y} | {\\bf H} ) = \\log \\det \\left( {\\bf I}_{n_r}  + {\\bf H} {\\bf Q}_{\\rm x} {\\bf H}^{\\dag}\n \\right) \\; .\n\\end{align}}\n\nIn this paper, we will consider a scenario in which the channel is\nselected randomly at the beginning of a transmission, and remains\n\\emph{fixed} during the transmission.  In this scenario, it is\nimpossible to guarantee that the communication will be completely\nreliable, since no matter what transmission rate $R$ we choose\n(which is assumed fixed) there is always a non-zero probability that\nthe rate may not be supportable by the channel. In other words,\nthere is always a chance that the mutual information $I({\\bf x};{\\bf y} |\n{\\bf H} )$ falls below $R$, and thus communicating at rate $R$ becomes\nimpossible. This is referred to as an ``outage event'', and the\nprobability of this occurring is called the \\emph{outage\nprobability}, {\\small \\begin{align} \\label{eq:outProb} P_{\\rm out} (\nR ) &:= {\\rm Pr} \\left( I({\\bf x};{\\bf y} | {\\bf H} ) < R  \\right)\n.\n\\end{align}}\n\n\nHere, we will make the common assumption that {\\small \\begin{align}\n\\label{eq:xCov} \\mathbf{Q}_{\\rm x} = \\frac{P}{n_t} {\\bf I}_{n_t} \\; ,\n\\end{align}}\ncorresponding to sending independent complex Gaussian signals from\neach transmit antenna, each with power $P\/n_t$. Hence, with this\ninput signal covariance, the quantity $P \\, (>0)$ will also\nrepresent the signal-to-noise ratio (SNR).  With $\\mathbf{Q}_{\\rm\nx}$ given by (\\ref{eq:xCov}), the mutual information $I({\\bf x};{\\bf y} |\n{\\bf H} )$ becomes\n{\\small \\begin{align} I({\\bf x};{\\bf y} | {\\bf H} ) = \\log\\det \\left( {\\bf I}_{n_r}\n+ \\frac{1}{t} {\\bf H} {\\bf H}^{\\dag} \\right) , \\; \\quad t := \\frac{n_t}{P} .\n\\end{align}}\nTo fix notation, let $M := \\textsf{max}\\{n_{r},n_{t}\\}, N :=\n\\textsf{min}\\{n_{r},n_{t}\\} ,\\alpha := M - N$ and define {\\small \\begin{eqnarray}\n{\\bf W} &:=& \\left\\{\n\\begin{array}{lr}\n {\\bf H}{\\bf H}^{\\dag},  &  n_{r}<n_{t}\\nonumber\\\\\n{\\bf H}^{\\dag}{\\bf H}, &    n_{r} \\geq n_{t} \\nonumber\n\\end{array}\n\\right. \\end{eqnarray}} with positive eigenvalues $\\{x_{k}\\}_{k=1}^{N}$. With\nthese definitions, and with $\\det({\\bf I}+ {\\bf A} {\\bf\nB})=\\det({\\bf I}+{\\bf B}{\\bf A})$, we can further evaluate {\\small\n\\begin{align} \\label{eq:linStat}\nI({\\bf x};{\\bf y} | {\\bf H} ) &= \\sum_{k=1}^N \\log \\left( 1 + \\frac{1}{t} x_k\n\\right)  \\;  \\nonumber \\\\\n&= - N \\log t + \\sum_{k=1}^N \\log \\left( t + x_k \\right) .\n\\end{align}}\n\nComputation of the outage probability (\\ref{eq:outProb}) requires\nthe probability distribution of $I({\\bf x};{\\bf y} | {\\bf H} )$. From\n(\\ref{eq:linStat}), this is clearly a linear statistic in the\neigenvalues of the Hermitian random matrix ${\\bf W}$ (with a constant\nshift of $-N \\log t$). Moreover, ${\\bf W}$ is complex Wishart\ndistributed \\cite{Muirhead}, thus the eigenvalues\n$\\{x_{k}\\}_{k=1}^{N}$ are well-known to admit the joint density\n{\\small\n\\begin{align}\n \\label{eq:EigPDF_Wishart} p(x_1,x_2,...,x_N)\\propto\n\\prod_{l=1}^{N} w_{{\\rm Lag}} (x_l) \\prod_{1\\leq j<k\\leq\nN}(x_k-x_j)^2 , \\quad \\quad  x_l\\in[0,\\infty)\n\\end{align}\n} where {\\small\n\\begin{align} w_{{\\rm Lag}} (x):=x^{\\alpha}\\:{\\rm e}^{-x},\n\\;\\;\\alpha>-1,\\;\\;x\\in[0,\\infty)\n\\end{align}}\nis the classical Laguerre weight.\n\nOur aim will be to compute the moment generating function of the\nlinear statistic, {\\small\n\\begin{align}\n{\\tilde {\\cal M}}(\\lambda) := E_{{\\bf H}} \\left( e^{\\lambda I({\\bf x};{\\bf y} |\n{\\bf H}) } \\right)  \\; = \\; t^{-N \\lambda} {\\cal M}(\\lambda)\n\\end{align}\n} where ${\\cal M}(\\lambda)$ is identified by (\\ref{eq:MGFFirst}) but\nwith the following particularizations:\n\\begin{align}\n\\left( f(x), w_0(x), w(x), a, b \\right) \\Longrightarrow \\left(\n\\log(t+x), w_{\\rm Lag}(x), w_{\\rm dLag}(x, t), 0, \\infty \\right) \\;\n\\end{align}\nwhere $w_{\\rm dLag}(x,t)$ is a deformed Laguerre weight, {\\small\n\\begin{align}\n\\label{eq:DefLagWeight} w_{\\rm dLag}(x,t):= (x+t)^{\\lambda} w_{{\\rm\nLag}} (x) ,\\;\\;\\; t>0 \\;\\;\\;x>0 \\; .\n\\end{align}\n}\nThus, (\\ref{eq:MGFRatio}) immediately gives {\\small\n\\begin{align}\n \\label{eq:MGF_MainResultsSec} {\\cal M}(\\lambda) = \\frac{D_N(t,\\lambda)}{D_N[w_{\\rm Lag}]} \\;\n\\end{align}}\nwhere \\begin{eqnarray} D_N(t,\\lambda)=\\det\\left( \\mu_{i+j-2}(t, \\lambda)\n\\right)_{i,j=1}^{N} \\label{eq:HankelDefin1} \\end{eqnarray} is the Hankel\ndeterminant generated from $w_{\\rm dLag}(x,t)$ with moments {\\small\n\\begin{align} \\label{eq:MomentDefn}\n \\mu_{k}(t, \\lambda) := \\int_{0}^{\\infty}x^{k} w_{\\rm dLag}(x) dx \\;\n, \\hspace*{1cm} k = 0, 1, 2, \\ldots .\n\\end{align}\n}\n\n\nThe quantity $D_N[w_{{\\rm Lag}}]$ in the denominator of\n(\\ref{eq:MGF_MainResultsSec}) is the Hankel determinant generated\nfrom the classical Laguerre weight, $w_{{\\rm Lag}} (x)$, and can be\ncomputed in terms of the Barnes $G$--function as {\\small\n\\begin{align} \\label{eq:DnLaguerre_LambdaZero} D_N[w_{{\\rm\nLag}}]=\\frac{G(N+1)G(N+\\alpha+1)}{G(\\alpha+1)},\\quad G(1)=1.\n\\end{align}}\n\nOur next objective will be to compute a non-determinantal\nrepresentation for the (scaled) moment generating function\n(\\ref{eq:MGF_MainResultsSec}).  This, in turn, will require\nevaluation of the Hankel determinant $D_N(t,\\lambda)$ in\n(\\ref{eq:HankelDefin1}).  We will address this problem in the sequel\nby appealing to the orthogonal polynomial framework introduced in\nSection \\ref{sec:LadderOperators}.  We should like to mention here\nthat, unlike the classical ladder operators, the ``coefficients'' in\nour ladder operators are $``x\"$ dependent, as we shall see later.\n\n\n\\setcounter{equation}{0}\n\n\n\n\n\\section{Painlev\\'e V and the Continuous and Discrete $\\sigma$-Form}\n\n\\subsection{Main Results}\n\nThe following two theorems present the main results of the paper:\n\n{\\bf Theorem 1\\;\\;} \\emph{The Hankel determinant $D_N(t,\\lambda)$\nadmits the following integral representation:}\n\\begin{align}\n\\label{eq:Intrep} \\frac{D_N(t,\\lambda)}{D_N[w_{\\rm Lag}]} =\nt^{N\\lambda} \\exp \\left( \\int^{\\infty}_{t}f(y(s),y'(s),s) ds \\right)\n\\end{align}\n\\emph{where}\n\\begin{align} &f(y(s),y'(s),s):=\n\\frac{\\lambda^2+2(s+\\alpha-\\lambda)y+(4 N\ns+(s+\\alpha)^2-2(s+2\\alpha)\\lambda+\\lambda^2)y^2\n}{4y(y-1)^2} \\\\\n& \\hspace*{4cm} + \\frac{-2(2 N\ns+\\alpha(s+\\alpha-\\lambda))y^3+\\alpha^2y^4-[s\\;y'(s)]^2}{4y(y-1)^2}\n\\; .\n\\end{align}\n\n\n\\noindent {\\bf Theorem 2\\;\\;} \\emph{Equivalently, the Hankel\ndeterminant $D_N(t,\\lambda)$ also admits the following integral\nrepresentation:} \\begin{eqnarray} \\label{eq:HankelDet_ContSigma}\n\\frac{D_N(t,\\lambda)}{D_N[w_{\\rm Lag}]}= t^{N\\lambda} \\exp\n\\left(\\int_{\\infty}^t \\frac{H_N(x)-N\\lambda}{x} dx \\right)\n\\label{eq:tmp} \\end{eqnarray} \\emph{where} $H_N(t)$ \\emph{ satisfies the\nJimbo-Miwa-Okamoto $\\sigma-$form of Painlev\\'e V:} {\\small\n\\begin{align}\n & (t H_N'')^2=\\left[t H_N'- H_N + H_N'(2N+\\alpha+\\lambda) + N \\lambda \\right]^2 \\nonumber \\\\ & \\hspace*{2cm} - 4(t\nH_N'- H_N + {\\delta_N} )\\left[( H_N')^2+\\lambda H_N'\\right] \\label{eq:JimboPV}\n\\end{align}}\n\\emph{with} ${\\delta_N} := N(N+\\alpha+\\lambda)$.\n\nBefore presenting the proof of these results, we would like to point\nout that Painlev\\'e equations first appeared in the early 1900's\nthrough the work of Painlev\\'e and his collaborators \\cite{book}. In\nthe mid 1970's, Painlev\\'e equations first appeared in\ncharacterizing the correlation function of an Ising model through\nthe pioneering work of Barouch, McCoy, Tracy and Wu, see\n\\cite{Tracy}. The 1-particle reduced density matrix was shown in\n1980 to satisfy a particular Painlev\\'e V, see \\cite{Jimbo}.  For a\nrecent review on this and other related problems in matrix\nensembles, see \\cite{TW2011}. Another Painlev\\'e V appeared in the\nHankel determinant associated with the ``time evolved\" Jacobi\npolynomials, see \\cite{Basor}.\n\n\\subsection{Proof of Theorems 1 and 2}\n\n\\subsubsection{Compatibility Conditions, Recurrence Coefficients and\nDiscrete Equations}\n\n\n\n\nFor the purpose of applying the ladder operator framework introduced\nin Lemmas 1--3, an easy computation shows that {\\small \\begin{eqnarray}\n\\v(z,t)&:=&-\\log w_{\\rm dLag}(z,t)=-\\alpha \\log z-\\lambda \\log(z+t)+z , \\nonumber\\\\\n\\v'(z,t)&=&-\\frac{\\alpha}{z}-\\frac{\\lambda}{z+t}+1\\nonumber \\end{eqnarray} and therefore \\begin{eqnarray}\n\\frac{\\v'(z,t)-\\v'(y,t)}{z-y}&=&\\frac{\\alpha}{zy}+\\frac{\\lambda}{(z+t)(y+t)}.\\nonumber\n\\end{eqnarray}}\nSubstituting the above into (\\ref{eq:AnNewDefn}) and\n(\\ref{eq:BnNewDefn}), followed by integration by parts, we obtain\n{\\small \\begin{align}\n\\label{eq:AnDefn}\nA_n(z)&=\\frac{1-R_n(t)}{z}+\\frac{R_n(t)}{z+t}\\\\\nB_n(z)&=-\\frac{n+r_n(t)}{z}+\\frac{r_n(t)}{z+t}\n\\end{align}}\nwhere we have introduced the auxiliary quantities:\n{\\small \\begin{align}\nR_n(t)&:=\\frac{\\lambda}{h_n}\\int_{0}^{\\infty}\\frac{[P_n(x)]^2}{x+t}w_{{\\rm dLag}}(x,t)dx\\\\\nr_n(t)&:=\\frac{\\lambda}{h_{n-1}}\\int_{0}^{\\infty}\\frac{P_n(x)P_{n-1}(x)}{x+t}w_{{\\rm dLag}}(x,t)dx.\n\\label{eq:rnDefn}\n\\end{align}}\nThe next Lemma gives a representation of the recurrence coefficients\n$\\alpha_n,$ $\\beta_n,$ $\\sum_{j}\\:R_j,$ and $\\textsf{p}_1(n)$---the\ncoefficient of $z^{n-1}$ of $P_n(z)$--- in terms of the auxiliary\nvariables $r_n$ and $R_n.$ Note that $\\textsf{p}_1(n)$ also depends\non $t$ but we do not display this if there is no confusion.\n\\\\\n{\\bf Lemma 4} \\emph{The recurrence coefficients $\\alpha_n$ and $\\beta_n$\nare expressed in terms of the auxiliary quantities $r_n$ and $R_n$\nas:}\n{\\small \\begin{eqnarray} \\alpha_n&=&2n+1+\\alpha+\\lambda-tR_n\\\\ \\label{eq:AlphanRelation}\n\\beta_n&=&\\frac{1}{1-R_n}\\left[r_n(2n+\\alpha+\\lambda)+\\frac{r_n^2-\\lambda\nr_n}{R_n}+n(n+\\alpha)\\right]. \\label{eq:BetanRelation}\n\\end{eqnarray}}\n\\emph{Furthermore,}\n{\\small \\begin{eqnarray}\nt\\sum_{j=0}^{n-1}R_j&=&n(n+\\alpha+\\lambda)-\\beta_n-tr_n, \\label{eq:SumRjRelation} \\\\\n\\textsf{p}_1(n)&=&-\\beta_n-tr_n. \\label{eq:p1Relation}\n\\end{eqnarray}}\n{\\bf Proof\\;\\;} We start from  $(S_1)$. Equating the coefficients of $z^{-1}$ and $(z+t)^{-1}$, we obtain\nthe following difference equations relating $\\alpha_n$ to $r_n$ and $R_n$:\n{\\small \\begin{align}\n\\label{eq:S1Diff1}\n-(2n+1+r_{n+1}+r_n)&=\\alpha-\\alpha_n(1-R_n)\\\\\nr_{n+1}+r_n &= \\lambda-R_n(t+\\alpha_n).\n\\label{eq:S1Diff2}\n\\end{align}}\nTo proceed further, we take note of\n(\\ref{eq:AnDefn})--(\\ref{eq:rnDefn}), and derive\nidentities based on the supplementary condition $(S_2')$, which will be of particular interest.\nA straightforward (but long)  computation shows that the r.h.s. of $(S_2')$ becomes\n{\\small \\begin{align}\n& B_n^2(z)+\\v'(z)B_n(z)+\\sum_{j=0}^{n-1}A_j(z) \\nonumber \\\\\n& \\hspace*{2cm} = z^{-2}[(n+r_n)^2+\\alpha(n+r_n)]\\nonumber\\\\\n& \\hspace*{2cm} +z^{-1}\\Big\\{n-\\sum_{j=0}^{n-1}R_j+r_n[\\lambda-\\alpha-t-2(n+r_n)]\/t+(n-\\lambda)\/t\\Big\\}\\nonumber\\\\\n& \\hspace*{2cm} +(z+t)^{-1}\\Big\\{\\sum_{j=0}^{n-1}R_j+r_n[t+\\alpha-\\lambda+2(n+r_n)]\/t-n\\lambda\/t\\Big\\}\\nonumber\\\\\n& \\hspace*{2cm} +(z+t)^{-2}[r_n^2-\\lambda r_n] . \\nonumber\n\\end{align}}\nNow focusing on $(S_2')$ as presented above and equating the coefficients\nof $z^{-2},$ $z^{-1},$ $(z+t)^{-1},$ $(z+t)^{-2}$,\ngive rise to the following difference equations involving $\\beta_n$, $r_n$, $R_n$ and $\\sum_j\\:R_j$ :\n{\\small \\begin{align}\n\\label{eq:S2Diff1} (n+r_n)^2+\\alpha(n+r_n) =\\beta_n(1-R_n)(1-R_{n-1})\n\\end{align}}\n{\\small \\begin{align}\n\\small n-\\sum_{j=0}^{n-1}R_j+\\frac{r_n}{t}[\\lambda-\\alpha-t-2(n+r_n)]+\\frac{n(\\lambda-t)}{t} =\n\\frac{\\beta_n}{t}\\left[(1-R_{n-1})R_n+(1-R_{n-1})R_n\\right]\n\\label{eq:S2Diff2a}\n\\end{align}}\n{\\small \\begin{align}\n\\sum_{j=0}^{n-1}R_{j}+\\frac{r_n}{t}[t+\\alpha-\\lambda+2(n+r_n)]-\\frac{n\\lambda}{t}\n =\n-\\frac{\\beta_n}{t}\\left[(1-R_n)R_{n-1}+(1-R_{n-1})R_n\\right]\n\\label{eq:S2Diff2b}\n\\end{align}}\n{\\small \\begin{align}\nr_n^2-\\lambda r_n =\\beta_nR_nR_{n-1}.\\quad \\label{eq:S2Diff3}\n\\end{align}}\nObserve that (\\ref{eq:S2Diff2a}) and\n(\\ref{eq:S2Diff2b}) are equivalent. We shall see later that (\\ref{eq:S2Diff2a}), when combined with\ncertain identities, performs the sum {\\large $\\sum_{j=0}^{n-1}R_j$}\nautomatically in closed-form. This sum will provide an important\nlink between the logarithmic derivative of the Hankel determinant\nwith respect to $t$, and $\\beta_n$, $r_n$, which is an essential step\nin establishing the Painlev\\'e equation. Whilst the difference relations\n(\\ref{eq:S1Diff1})--(\\ref{eq:S2Diff1}) and (\\ref{eq:S2Diff3}) look\nrather complicated, these can be manipulated to give us insight into\nthe recurrence coefficients $\\alpha_n$ and $\\beta_n.$ To this end,\nsumming (\\ref{eq:S1Diff1}) and (\\ref{eq:S1Diff2}) gives us a simple\nexpression for the recurrence coefficient $\\alpha_n$ in terms of $R_n$:\n{\\small \\begin{eqnarray}\n\\label{eq:alrn} \\alpha_n=2n+1+\\alpha+\\lambda-tR_n.\n\\end{eqnarray}}\nFrom\n(\\ref{eq:S2Diff1}) and (\\ref{eq:S2Diff3}) we find after a minor re-arrangement\n{\\small \\begin{eqnarray} \\label{eq:RnBnCond}\n\\beta_n(R_{n}+R_{n-1})=\\beta_n-n(n+\\alpha)-r_n(\\alpha+\\lambda+2n).\n\\end{eqnarray}}\nNow substituting (\\ref{eq:S2Diff3}) and (\\ref{eq:RnBnCond}) into either\n(\\ref{eq:S2Diff2a}) or (\\ref{eq:S2Diff2b}) to eliminate $R_n$ and\n$R_{n-1}$ leaves us the following very simple form for\n$\\sum_{j=0}^{n-1}R_j,$ which will play a crucial role later,\n{\\small \\begin{eqnarray}\n\\label{eq:RSum} t\\sum_{j=0}^{n-1}R_j=n(n+\\alpha+\\lambda)-\\beta_n-tr_n.\n\\end{eqnarray}}\nBut in view of (\\ref{eq:alrn}), we also have an alternative representation of $\\sum_{j=0}^{n-1}R_j,$ namely,\n{\\small \\begin{eqnarray}\nt\\sum_{j=0}^{n-1}R_j&=&n(n+\\alpha+\\lambda)-\\sum_{j=0}^{n-1}\\alpha_j\n=n(n+\\alpha+\\lambda)+\\textsf{p}_1(n) \\label{eq:RSumb}.\n\\end{eqnarray}}\nIn summary, we have obtained two different ways to express $\\sum_{j}R_j,$ and comparing\n(\\ref{eq:RSum}) with (\\ref{eq:RSumb}) gives us the important relation (\\ref{eq:p1Relation}).\nlinking $\\textsf{p}_1(n)$ to $\\beta_n$  and $r_n.$\nWe are now in the position to find an expression for $\\beta_n$ in terms\nof $r_n$ and $R_n$. This is found by eliminating $R_{n-1}$ from\n(\\ref{eq:RnBnCond}) and (\\ref{eq:S2Diff3}) resulting in (\\ref{eq:BetanRelation}).\n{\\bf End of Proof.}\n\n\\subsubsection{$t$ Evolution and Painlev\\'e V}\nIn the next stage of the development, we vary $t$ and $n.$ The\ndifferential-difference relations generated here when combined with the\ndifference relations obtained previously will give us the desired\nPainlev\\'e equation. A straightforward computation shows that\n{\\small \\begin{eqnarray} \\frac{d}{dt}\\log\nh_n=R_n. \\end{eqnarray}}\n But, from (\\ref{eq:betah}), it follows that\n {\\small \\begin{eqnarray}\n\\frac{d\\beta_n}{dt}&=&\\beta_n(R_n-R_{n-1})\\\\\n&=&\\beta_n R_n-\\frac{r_n^2-\\lambda r_n}{R_n}, \\label{eq:BetaDiff} \\end{eqnarray}}\nwhere the last equality follows from (\\ref{eq:S2Diff3}).\n\nDifferentiating\n{\\small $$\n0=\\int_{0}^{\\infty}x^{\\alpha}(x+t)^{\\lambda}{\\rm e}^{-x}P_{n}(x)P_{n-1}(x)dx\n$$}\nwith respect to $t$ produces\n{\\small \\begin{align}\n0&=\\lambda\\int_{0}^{\\infty}\\:x^{\\alpha}\\:(x+t)^{\\lambda-1}{\\rm e}^{-x}P_n(x)P_{n-1}(x)dx+\n\\int_{0}^{\\infty}x^{\\alpha}(x+t)^{\\lambda}{\\rm e}^{-x}\\left[\\frac{d}{dt}\\textsf{p}_1(n)\\:x^{n-1}+...\\right]P_{n-1}(x)dx\\nonumber\\\\\n&=\\lambda\\int_{0}^{\\infty}\\frac{P_{n-1}(x)P_{n}(x)}{x+t}w_{{\\rm dLag}}(x,t)dx+h_{n-1}\\:\\frac{d}{dt}\\textsf{p}_1(n),\\nonumber\n\\end{align}}\nresulting in\n{\\small \\begin{eqnarray} \\label{eq:p1Diff}\n\\frac{d}{dt}\\textsf{p}_1(n)&=&-r_n .\n\\end{eqnarray}}\nUpon noting\n(\\ref{eq:p1Defn}), this implies\n\\begin{eqnarray} \\label{eq:AlphaDiff}\n\\frac{d\\alpha_n}{dt}&=&r_{n+1}-r_{n} .\n\\end{eqnarray}\nNow differentiating\n(\\ref{eq:p1Relation}) with respect to $t$ and noting\n(\\ref{eq:p1Diff}), we find\n{\\small \\begin{eqnarray}\n\\frac{d}{dt}\\textsf{p}_1(n)&=&-\\frac{d\\beta_n}{dt}-\\frac{d}{dt}(tr_n)\n=-\\frac{d\\beta_n}{dt}-r_n-t\\frac{dr_n}{dt}\n=-r_n.\\nonumber\n\\end{eqnarray}}\nThe above result combined with (\\ref{eq:BetaDiff})\ngives\n{\\small \\begin{eqnarray}\n\\frac{d\\beta_n}{dt}=-t\\frac{dr_n}{dt}=\\beta_nR_n-\\frac{r_n^2-\\lambda\nr_n}{R_n}.\n\\end{eqnarray}}\nWe now come to a key Lemma which gives the first order derivative of $r_n(t)$ and $R_n(t)$ with respect to $t,$ and where $n$\nappears as a parameter.\n\\\\\n\\\\\n{\\bf Lemma 5} \\emph{The auxiliary variables} $r_n$ \\emph{and} $R_n$\n\\emph{satisfy the following coupled Riccatti equations,}\n{\\small \\begin{align}\n\\label{eq:RicEqr} t\\frac{dr_n}{dt}=\\frac{r_n^2-\\lambda\nr_n}{R_n}-\\frac{R_n}{1-R_n}\\left[r_n(2n+\\alpha+\\lambda)+\\frac{r_n^2-\\lambda\nr_n}{R_n}+n(n+\\alpha)\\right],\n\\end{align}}\n\\emph{and}\n{\\small \\begin{align}\n\\label{eq:RicEqR} 2r_n=t\\frac{dR_n}{dt}+\\lambda-R_n\\:(t+2n+\\alpha+\\lambda-t\\:R_n).\n\\end{align}}\n\\emph{Furthermore,}\n{\\small $$\ny(t)=y(t, n):=1-\\frac{1}{1-R_n(t)},\n$$}\n\\emph{satisfies the following second order non-linear ode,}\n\\begin{align}\ny''&=\\frac{3y-1}{2y(y-1)}\\:(y')^2- \\frac{y'}{t}+\\frac{(y-1)^2}{t^2}\n\\: \\left(\\frac{\\alpha^2}{2}y-\\frac{\\lambda^2}{2y}\\right)+\\frac{(2n+1+\\alpha+\\lambda)\\:y}{t}-\\frac{y(y+1)}{2(y-1)},\n\\label{eq:PVyt}\n\\end{align}\n\\emph{which is recognized to be a} {\\small\n$$P_V\\left(\\frac{\\alpha^2}{2},-\\frac{\\lambda^2}{2},2n+1+\\alpha+\\lambda,-1\/2\\right).$$}\n\n{\\bf Proof\\;}Because (\\ref{eq:BetanRelation}) expresses $\\beta_n$ as a\nquadratic in $r_n,$ we see that $r_n$ satisfies the Riccatti\nequation (\\ref{eq:RicEqr}). Eliminating $r_{n+1}$ from\n(\\ref{eq:S1Diff2}) and (\\ref{eq:AlphaDiff}), and upon referring to\n(\\ref{eq:AlphanRelation}), we obtain (\\ref{eq:RicEqR}). Next, we\nsimply substitute  $r_n(t)$ from (\\ref{eq:RicEqR}) into\n(\\ref{eq:RicEqr}), to see that $R_n(t)$ satisfies a second order\nnon-linear ode in $t$, in which $n$, $\\alpha$, and $\\lambda$ appear as\nparameters. A further linear fractional change of variable {\\small\n$$ R_n(t)=1-\\frac{1}{1-y(t)}\\quad{\\rm or}\\quad\ny(t) =1-\\frac{1}{1-R_n(t)},\n$$}\nestablishes that $y(t)$ satisfies the Painlev\\'e V displayed in the Lemma.{\\;\\bf End of Proof.}\n\nWe begin here a series of computations which ultimately give rise\ntwo integral representations of the Hankel determinant of interest,\ni.e., {\\small\n$$ D_N(t,\\lambda)=\n\\det\\left(\\int_{0}^{\\infty}x^{j+k-2}(x+t)^{\\lambda}x^{\\alpha}{\\rm\ne}^{-x}dx\\right)_{1\\leq j,k\\leq N} \\; ,$$} given in Theorems 1 and\n2. To this end, an easy computation shows that {\\small \\begin{eqnarray}\nH_N(t)&:=&t\\frac{d}{dt}\\log D_N(t,\\lambda)=t\\frac{d}{dt}\\sum_{j=0}^{N-1}\\log h_j=t\\sum_{j=0}^{N-1}R_j\\nonumber\\\\\n&=& N(N+\\alpha+\\lambda)-\\beta_N-tr_N \\label{eq:Hna} \\\\\n&=& N(N+\\alpha+\\lambda)+\\textsf{p}_1(N) \\label{eq:Hnb}, \\end{eqnarray}} where the last\ntwo equations follow from (\\ref{eq:SumRjRelation}) and\n(\\ref{eq:p1Relation}) of Lemma 4. Integrating (\\ref{eq:Hna}) with\nrespect to $t$, while noting (\\ref{eq:BetanRelation}),\n(\\ref{eq:RicEqR}) and $R_N(t)=1-1\/(1-y(t)),$ we obtain the result\nstated\nin Theorem 1.\n\n\nTo obtain the second integral representation for $D_n(t,\\lambda)$\nstated in Theorem 2 (i.e., in terms of $H_n(t)$), we note that from\n(\\ref{eq:p1Diff}), (\\ref{eq:Hna}), and (\\ref{eq:Hnb}), we obtain\nexpressions for $\\beta_N$ and $r_N$ in terms of $H_N$ and $H_N'$,\n \\begin{eqnarray}\n\\beta_N&=&N(N+\\alpha+\\lambda)+tH_N'-H_N \\label{eq:BetanHn} \\\\\nr_N &=& -H_N'. \\end{eqnarray} What we need to do is to eliminate $R_N$ to find\na functional equation satisfied by $H_N,$ $H_N'$ and $H_N''.$ For\nthis purpose, we examine two quadratic equations satisfied by $R_N$,\none of which is simply a rearrangement of (\\ref{eq:BetanRelation})\nand reads \\begin{eqnarray} \\label{eq:Frac1} \\frac{r_N^2-\\lambda r_N}{R_N}+\\beta_N\nR_N=\\beta_N-r_N(2N+\\alpha+\\lambda)-N(N+\\alpha). \\end{eqnarray} The other follows from a\nderivative of the first equation of (\\ref{eq:BetanHn}) with respect\nto $t$ and (\\ref{eq:BetaDiff}), \\begin{eqnarray} \\label{eq:Frac2} \\beta_N\nR_N-\\frac{r_N^2-\\lambda r_N}{R_N} &=& tH_N''. \\end{eqnarray} Solving for $R_N$ and\n$1\/R_N$ from the linear system (\\ref{eq:Frac1}) and\n(\\ref{eq:Frac2}), we find\n\\begin{eqnarray}\n2R_N&=&1+\\frac{tH_N''-(2N+\\lambda+\\lambda)r_N-N(N+\\alpha)}{tH_N'-H_N+N(N+\\alpha+\\lambda)} \\label{eq:Rna} \\\\\n\\frac{2}{R_N}&=&\\frac{-tH_N''+(t+2N+\\alpha+\\lambda)H_N'-H_N+N\\lambda}{(H_N')^2+\\lambda\\:H_N'}\n\\label{eq:Rnb}, \\end{eqnarray} where we have replaced $\\beta_N$ and $r_N$ in\nterms of $H_N,\\;H_N' {\\rm and\\;} H_N''$ with (\\ref{eq:BetanHn}). The\nproduct (\\ref{eq:Rna}) and (\\ref{eq:Rnb}) gives us the desired\n$\\sigma-$form (\\ref{eq:JimboPV}).\n\nIt is finally worth noting that with\n$D_N(t,\\lambda)=:t^{{\\delta_N}}\\tilde{D}_N,$\nwe find, after a little computation that $\\tilde{D}_N$ satisfies the Toda\nmolecule equation \\cite{Toda} \\begin{eqnarray}\n\\frac{d^2}{dt^2}\\log\\tilde{D}_N=\\frac{\\tilde{D}_{N+1}\\tilde{D}_{N-1}}{\\tilde{D}_N^2} . \\end{eqnarray}\n\n\n\n\\subsection{The Discrete $\\sigma-$Form}\n\nAs an alternative to the Jimbo-Miwa-Okamoto $\\sigma-$form satisfied\nby $H_N$ in Theorem 2, we state (without proof) a non-linear\ndifference equation satisfied by $H_N$. We believe this to be new.\nSpecifically, we find that the logarithmic derivative of the Hankel\ndeterminant in (\\ref{eq:HankelDefin1}), generated by the deformed\nLaguerre weight in (\\ref{eq:DefLagWeight}), satisfies a second order\nnon-linear difference equation, which we call the discrete\n$\\sigma-$form,\n\\begin{align}\n&\\left[\\frac{N(N+\\alpha)t+(\\Delta^2H_N+t)(H_N-\\delta_N)}{\\Delta^2H_N+2N+\\alpha+\\lambda+t}\\right]^2\n-\\lambda\\:\\frac{N(N+\\alpha)t+(\\Delta^2H_N+t)(H_N-\\delta_N)}{\\Delta^2H_N+2N+\\alpha+\\lambda+t}\\nonumber\\\\\n& \\hspace*{0.5cm}= \\left[\\delta_N-H_N\n+\\frac{N(N+\\alpha)t+(\\Delta^2H_N-t)(H_N-\\delta_N)}{\\Delta^2H_N+2N+\\alpha+\\lambda+t}\\right]\n(H_{N+1}-H_N)(H_N-H_{N-1}) \\quad \\nonumber\n\\end{align}\nwhere $\\Delta^2H_N:=H_{N+1}-H_{N-1}$. The initial conditions are\n$H_1(t) = \\frac{d}{dt}\\log D_1(t,\\lambda),\\: H_2(t) = \\frac{d}{dt}\\log\nD_2(t,\\lambda)$ with $ D_1(t,\\lambda) = \\mu_0(t),\\: D_2(t,\\lambda) =\n\\mu_0(t)\\mu_2(t)-\\mu_1^2(t),$ and the moments are defined in\n$(\\ref{eq:MomentDefn}).$\n\n\nWe do not further elaborate on this result, but present it here\nsince we believe that it may provide a useful tool for efficiently\ncomputing $H_N$ in an iterative manner, and give further insight\ninto the mutual information of a single-user MIMO system. A thorough\nanalysis of the implications of this non-linear difference equation\nin regards to wireless communication applications is the subject of\non-going work.\n\n\n\n\n\n \n  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n  \n  \n\n\n  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding Remarks}\n\nThis is a short review is about Hankel determinants that arise in\nthe information-theoretic study of MIMO communication systems.  We\nconsider the capacity of single-user MIMO systems, in which case the\ndeterminant of interest is generated from a certain deformed\nLaguerre weight. We have obtained two exact integral representations\nfor this Hankel determinant, one of which is described in terms of\nthe $\\sigma-$form of a particular Painlev\\'{e} V differential\nequation. We have also stated an alternative representation,\ninvolving a second-order non-linear difference equation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOn a landline telephone, the headset is usually connected with the phone via a helical cord. Such a helical cord, although initially nice and straight [Fig.~\\ref{fig:cords}(a)], often tangles up after use [Fig.~\\ref{fig:cords}(b)]. A careful examination would reveal that a tangled helical cord is sharply different from tangled hair or ropes. The later come from knotted structures, while a tangled helical cord is usually knot free. Instead, it is due to kinks, at which a helical cord changes its chirality and turns sharply. \nWhere do these kinks come from? Why do they turn a nice and straight cord into a tangled mess?  Why are they so hard to get rid of? \nInterestingly,  answers to these questions, as we show below, have roots in the fascinating physics and mathematics of materials and structures at the verge of mechanical instability, to which these helical cords belong.\n\n\n\n\nMechanical stability is concerned with the response of a material to external loads---whether it holds its shape like a solid or flows like a liquid.  At the verge between these two types of behaviors, interesting physics arises, such as critical phenomena~\\cite{Jacobs1995,Liu2010,Ellenbroek2011,Ellenbroek2014,Zhang2017} and topological states~\\cite{Kane2014,Lubensky2015,Paulose2015a,Stenull2016,mao2018maxwell,Zhou2018}.  One peculiar feature of these materials is that they can exhibit a small number of floppy modes, i.e., normal modes of deformation that cost little elastic energy, but at the same time remain stable under other types of loads.  By controlling these floppy modes, which are the low energy excitations of the system, mechanical response of a material can be precisely programmed, which has broad applications such as switchable, actuatable, deployable materials~\\cite{bertoldi2017flexible,Mullin2007,schenk2013geometry,Florijn2014,Paulose2015,filipov2015origami,Chen2015,dudte2016programming,Rocklin2017,tang2017programmable}.  \n\nAmong these fascinating systems right at the verge of mechanical instability, a subcategory, known as Maxwell networks\/lattices~\\cite{Lubensky2015}, has been extensively studied recently, and provided deep insight into many problems in soft matter~\\cite{Mao2010,Mao2011a,Broedersz2011,Mao2013b,Mao2013c,Dennison2013,Mao2015,Zhang2015a,Zhang2016,Zhang2017,Liarte2019}, metamaterials~\\cite{Chen2014,Paulose2015,Rocklin2017,Zhang2018,Ma2018} and mechanobiology~\\cite{Alvarado2013,Feng2015,Feng2016,Sharma2016,Ronceray201514208,feng2019cell}. \nThanks to the discreteness of the degrees of freedom and constraints in these lattices\/networks, \nwithin linear elasticity, a universal theorem, known as the the Maxwell-Calladine index theorem~\\cite{Calladine1978,PellegrinoCal1986,Kapko2009,Sun2012} can be rigorously proved, which reveals a profound relation between floppy modes and states of self stress.\nStates of self stress, as we define rigorously below,  describes modes of distributing stress on  a network leaving all components in force balance and thus characterize the ability of a network to carry stress.  In contrast to floppy modes which describe kinematics of a structure, states of self stress focus on the statics.\nIn addition to demonstrating a universal connection between these two seemingly unrelated quantities, this theorem also played a crucial role in the discovery of topological mechanics in Maxwell lattices and networks~\\cite{Kane2014,Lubensky2015}. Moreover, there is a rigorous duality relation between the geometry of  floppy modes and states of self stress via Maxwell reciprocal diagrams~\\cite{maxwell1864xlv,crapo1994spaces,mitchell2016mechanisms,Zhou2019}.\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.8\\columnwidth]{cords.pdf}\n\t\\caption{Kinks in helical cords. (a) A kink-free helical cord is straight.  (b) Kinks sharply turn the helical cord.  Chirality (L or R) of domains and topological charge of kinks are marked (see discussions in Secs.~\\ref{SEC:MiniSurf} and \\ref{SEC:FracRibb}).\n\t}\n\t\\label{fig:cords}\n\\end{figure}\n\nIn contrast, the verge of mechanical instability in continuous elastic media is much less understood, especially in terms nonlinear elasticity which naturally arises when low dimensional elastic objects are embedded in a higher dimensional space~\\cite{Witten2007,Klein1116,sharon2010mechanics,Efrati2011,EFRATI2009762,Armon1726,Kim1201,chen2012nonlinear,guo2014shape,gladman2016biomimetic}.  It is worth pointing out that here mechanical instability refers to intrinsic instability coming from material properties and geometry, rather than instabilities caused by external stress, such as buckling.  In a nonlinear continuous elastic medium, is there a connection between floppy modes and states of self stress, similar to that in lattices\/networks? If so, does it lead to states with nontrivial topological features?\n\nIn continuum, floppy modes correspond to vanishing elastic moduli for certain types of load,  \nwhereas states of self stress dictate the possibility for an elastic body to carry residual stress. \nStress can be generated in elastic bodies in many different ways, such as frozen in stress from manufacturing, topological defects in crystals, and inhomogeneous growth in biological tissue~\\cite{Alexander1998,Withers2001,goriely2007definition}.  When the elastic body is allowed to relax in absence of external load, these stresses may or may not be fully released, and the stress that remains becomes residual stress.  Physically, the existence of residual stress implies that if we cut the system into two parts, neither of them will retain its original shape. Instead, the two pieces will both deform to reduce elastic energy associated with the residual stress. Residual stress often results in ``weak points'' for mechanical failure, but they are also very useful in toughening brittle materials and can be introduced intentionally.  \nStress-free solid materials, on the contrary, usually requires fine tuning, e.g., through careful annealing. Once achieved, cutting the material into pieces will not affect its shape and\/or spacial configuration, in contrast to stressed materials.\n\nIn this manuscript, we examine floppy modes and states of self stress in continuum elasticity, and demonstrate that Maxwell's counting argument still applies, and surprisingly, even in the nonlinear regime.  Based on Maxwell's counting, we classify elastic bodies at different dimensions and study their ground states and low-energy excitations. In light of the counting argument and insights gained from the study of quantum topological states, we identify two types of  thin elastic objects, where topological floppy modes\/low-energy excitations arise. The first class is called Maxwell plates. Such a plate is stress-free and contains sub-extensive number of holographic floppy modes, in strong analogy to 2D Maxwell\/isostatic lattices. The second class is also stress-free and features only one floppy mode. In addition, low-energy excitations in these system are fractional particles, in strong analogy to a $Z_2$  spin liquid and\/or a dimerized spin chain. The nature of these fractional excitations and their connection with topological degeneracy are also discussed. In addition to common features of fractional excitations, fractional excitations in these elastic systems show holographic feature, i.e., the location and configuration of a bulk fractional excitation are be fully dictated by the edge. Interestingly, these topological and holographic features naturally answer the three questions about kinks in a telephone cord mentioned above.\n\n\n\\section{Maxwell lattices and networks}\\label{SEC:counting}\nIn this section, we provide a brief review about elasticity in discrete systems, e.g., Maxwell lattices and networks. Mechanical properties of discrete networks can be conveniently analyzed via a counting argument due to Maxwell~\\cite{Maxwell1864,Calladine1978,Sun2012}.  \n\nThe idea of Maxwell's counting is based on a simple insight about the interplay between degrees of freedom and constraints. If the number of constraints $N_c$ exceeds the\nnumber of the degrees of freedom $N_{\\textrm{d.o.f.}}$, the system is often ``rigid''. On the contrary,\nif $N_c<N_{\\textrm{d.o.f.}}$, the system is often ``floppy''. In this picture, the verge of rigidity is marked by the Maxwell condition\n\\begin{align}\nN_{\\textrm{d.o.f.}}=N_{c},\n\\label{eq:z2D}\n\\end{align}\nand lattices satisfying this condition are called ``Maxwell lattices''. This Maxwell condition also plays an important role in a \nwide range of elastic systems at the verge of mechanical stability~\\cite{Liu2010,Mao2010,Broedersz2011,During2013,Kane2014}.\n\nBefore applying to elasticity,  here we first demonstrate these ideas by considering a set of linear equations, where $N_{\\textrm{d.o.f.}}$ is just the number of variables and \n$N_{c}$ is the number of equations, because each equation enforces one linear constraint on these variables.\nIt is easy to realize that the number of  ``un-fixed'' degrees of freedom can be obtained as\n\\begin{align}\nN_{\\textrm{un-fixed d.o.f.}} = N_{\\textrm{d.o.f.}}-N_{c}+N_{\\textrm{r.c.}}\n\\label{eq:N0}\n\\end{align}\nHere, because constraints may not all be independent of one another, redundant constraints needs to be removed in this counting \nand  $N_{\\textrm{r.c.}}$ here represent the number of redundant constraints.\nFor linear equations, this counting relation can be rigorously proved. However, it must be emphasized that when nonlinearity are taken into account, such\na counting argument is not generically expected, due to the complicated nature of nonlinear constraints. \n\nFor discrete lattices\/networks, the counting relation Eq.~\\eqref{eq:N0}  can be applied if we focus on linear response around stress-free ground states.\nHere, it is easy to realize that the number of  ``un-fixed'' degrees of freedom $N_{\\textrm{un-fixed d.o.f.}}$ gives the number of zero modes $N_0$, i.e. normal modes that don't\ncost any elastic energy. As for redundant constraints, they have a one-to-one correspondence with states of self stress and thus $N_{\\textrm{r.c.}}$ must \ncoincide with the number of states of self stress $N_s$.\nThis is because  by definition a redundant constraint tries to fix a degree of freedom which is already pinned by other constraints. Thus, in order to keep the network stress-free, this redundant constraint must be ``fine-tuned'' such that it is compatible with other constraints. If not compatible, such a redundant constraint will result in conflict constraints and thus generates residual stress,   giving rise to a state of self-stress. \nAs a result, the counting argument discussed above leads to the Maxwell-Calladine index theorem.\n\\begin{align}\nN_0 -N_{s} = N_{\\textrm{d.o.f.}}-N_{c}\n\\label{eq:N0s}\n\\end{align}\nwhich shows that the number of zero modes $N_0$ and the number of states of self-stress $N_{\\textrm{s}}$ are directly related. \nIn particular, for networks\/lattices at the verge of mechanical instability, i.e., Maxwell lattices\/networks with $N_{\\textrm{d.o.f.}}=N_{c}$, these two numbers must coincide ($N_0 =N_{s}$).\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\columnwidth]{counting.pdf}\n\t\\caption{Counting degrees of freedom and constraints.  (a-c) 2D discrete networks composed of sites (black dots) and bonds (black solid lines), which are \n\t(a) underconstraint, (b) isostatic and (c) over-constraint. In (a), $N_{\\textrm{d.o.f.}}=8, N_{c}=4, N_{s}=0$, so $N_0=4$ and thus the network has one floppy mode (green dashed configuration), i.e., $N_f=1$ according to Eq.~\\eqref{eq:Nm}. (b) Introducing one more constraint makes $N_{c}=5$ and $N_f=0$, and thus the network is isostatic and at the verge of mechanical instability.  (The dashed line is not a constraint.)  (c) The addition of more constraints into the network leads to states of self stress: when the new diagonal bond is not at the prescribed length [black dashed line in (b)] the network has residual stress on bonds (yellow for tension and blue for compression). (d-f) $d$ dimensional elastic bodies embedded in a $D=3$ dimensional space.  (d) A curve of $d=1$, with bending and torsional stiffness ignored, is under constrained, in analogy to (a).  (e) A surface of $d=2$, with bending stiffness ignored, is at the verge of mechanical instability, in analogy to (b).  (f) A body of $d=3$ is always over-constrained, in analogy to (c), and able to host residual stress (blue and yellow colors).}\n\t\\label{fig:counting}\n\\end{figure}\n\nIn Fig.~\\ref{fig:counting}(a-c), we demonstrate the Maxwell-Calladine theorem by considering a network consisting of point-like sites (hinges) connected by central-force bonds (struts) and assume that all bonds are at their rest-lengths in the reference state. Here, each site has $D$ degrees of freedom, where $D$ is the spatial dimension, and each bond provides one constraint. Thus the Maxwell-Calladine theorem takes the form of \n\\begin{align}\nN_0-N_{s}= N D - N_b   = ND - N\\langle z \\rangle\/2  .\n\\label{eq:QC}\n\\end{align}\nHere $N D$ is the total number of degrees of freedom $N_{\\textrm{d.o.f.}}$, with $N$ being the number of sites.\nThe total number of constraint $N_{\\textrm{c}}$ is now given by the number of bonds $N_b$, which can be written as $N \\langle z \\rangle\/2$ where $\\langle z \\rangle$ is the average coordination number (number of bonds connecting to one site) of the network. \nFor a periodic lattice, the Maxwell condition [Eq.~\\eqref{eq:z2D}] here takes the form of $\\langle z \\rangle =2D$.\nIn Fig.~\\ref{fig:counting}(c), we present one example demonstrating the relation between redundant constraints and states of self stress. When the last bond is added to the network, if it is not fine-tuned to the length prescribed by other bonds, residual stress is introduced.  Analogous continuous elastic media (d-f) are discussed in Sec.~\\ref{SEC:ContinuumCounting}.\n\nOne important feature of Maxwell lattices\/networks lies in its mechanical properties under open boundary conditions. An infinitely large Maxwell lattice is perfectly constrained everywhere (unless some constraints are redundant).  When a finite piece of Maxwell lattice is cut off, under open boundary conditions, the lattice will have a subextensive number of zero modes, coming from the cut bonds on the surface, inducing a deficit of constraints.  These zero modes exhibit very interesting properties, and their localization has been shown to be a topologically protected property, dictated by the bulk phonon structure~\\cite{Kane2014,Lubensky2015}.\n\nWe conclude this section by discussing the relation between the Maxwell condition [Eq.~\\eqref{eq:z2D}] and isostaticity. For an elastic system, if is often useful \nto exclude trivial degrees of freedom associated with rigid translations and rotations of the whole system, and thus\nthe number of floppy modes is defined as\n\\begin{align}\nN_f = N_0 - \\frac{D(D+1)}{2} ,\n\\label{eq:Nm}\n\\end{align}\nwhich describes elastic deformations that cost no elastic energy. Networks that satisfy $N_f=N_s=0$ are called ``isostatic'' structures, which are both stable (no floppy modes) and stress-free [See Fig.~\\ref{fig:counting}(b) for an example]. According to the Maxwell-Calladine theorem, isostatic systems should satisfy the relation $N_{\\textrm{d.o.f.}}-D(D+1)\/2=N_{c}$. This relation differs slightly from the \nMaxwell condition [Eq.~\\eqref{eq:z2D}], due to the extra term $D(D+1)\/2$ which comes from excluding trivial degrees of freedom (i.e. rigid body translations\/rotations).\nThis difference often becomes negligible for large systems or in the continuum, where $N_{\\textrm{d.o.f.}}$ and $N_c$ diverge.  \n\n\\section{Continuous elasticity of non-flat media}\nIn this section, we write down the elastic theory for a $d$-dimensional elastic medium embedded in a $D$-dimensional Euclidean space ($d \\le D$). In this convention, the typical 3D elastic theory corresponds to $d=D=3$, while a 2D plate embedded in the 3D space has $d=2$ and $D=3$.\nFor 2D plates, unless stated otherwise, in this manuscript, we focus on plates that are homogenous along the thickness direction \n(e.g. a plate formed by identical 2D layers stacked vertically on top of each other), and thus the two sides of each plate are always fully equivalent, in contrast to elastic shells. \nIt turns out that this symmetry plays an important role in the fractional excitations we discuss below.\n\n\nThe spatial configuration of a $d$-dimensional object in a $D$-dimensional Euclidean space is described by the mapping $\\mathbf{r} \\to \\mathbf{R}$, where $\\mathbf{r}$ and $\\mathbf{R}$ are the $d$- and $D$-dimensional coordinates of the original space and the target space respectively. For $d=D$, the elastic energy depends on the  deformation of the elastic medium, characterized by the metric tensor (defined below). For a homogenous or nearly homogenous system (i.e., elastic moduli remain constants but stress can slowly vary in space), the elastic energy takes the following functional form\n\\begin{align}\nE_s=\\int dr \\sqrt{\\det g_0} \\left\\{ \\frac{B-\\frac{2}{d}G}{2} \\tr(g-g_0)^2+ G \\tr[(g-g_0)^2]\\right\\}\n\\label{eq:Es}\n\\end{align}\nwhere $B$ ($G$) is the bulk (shear) modules  and $g$ is the metric tensor\n$g_{\\alpha,\\beta}=\\partial_\\alpha \\mathbf{R}\\cdot \\partial_\\beta \\mathbf{R}$,\nalso known as the first fundamental form, with $\\partial_\\alpha=\\partial\/\\partial \\mathbf{r}_\\alpha$ ($\\alpha=1$, $2$, $\\ldots$, $d$). Same as $g$, $g_0$ is also a symmetric rank-2 tensor ($d\\times d$), which is usually called the reference metric tensor. As shown in Eq.~\\eqref{eq:Es}, $g_0$ is the target for $g$, if we try to minimize the elastic energy.  In terms of discrete elastic networks defined in Sec.~\\ref{SEC:counting}, $g=g_0$ corresponds to the configuration where all bonds are at their rest length.\n\nWhen $g_0=I$, the conventional $D$-dimensional elastic theory is recovered with the strain tensor $\\epsilon=g-I$. In this manuscript, we consider more generic cases where $g_0$ may deviate from $I$. For $d=D$, such a generic $g_0$ describes prestressed materials, in which $g_0$ can never be reduced to identity in any coordinate systems. \nFrom Eq.~\\eqref{eq:Es}, it is easy to realize that for $g_0 \\ne I$, deforming a system (i.e. making $\\mathbf{R}\\ne \\mathbf{r}$ and thus $g\\ne I$) may actually reduce the elastic energy, which corresponds to fully or partially releasing of build-in stress via deformation.\n\nFor $d<D$ (e.g. $d=2$ and $D=3$), there is one more reason to consider a generic $g_0$ beyond identity: if the ground state of the 2D plate is curved, $g_0$ may not be reduced to identity in any coordinate systems, and thus it is necessary to explore this more generic $g_0 \\ne I$ case. For $d<D$, e.g., a 2D sheet embedded in 3D, the elastic energy can be separated into two parts\n\\begin{align}\nE=E_s+E_b.\n\\label{eq:d=2D=3}\n\\end{align}\nwhere the first term $E_s$ describes in plane stress while the second term $E_b$ describes the energy cost associated with bending.\n(For even lower $d<2$ or higher $D>3$, more terms can arise, such as the torsion term, which can be treated along the same line but will not be considered explicitly here).\nFor $d=2$ and $D=3$, this elastic energy [Eq.~\\eqref{eq:d=2D=3}] can be obtained by considering a thin 3D plate with thickness $h$, whose elastic energy takes the form of Eq.~\\eqref{eq:Es}~\\cite{willmore1993riemannian,Klein1116}. \nWhen the thickness of the plate $h$ is small, the total elastic energy can be written as a power-law expansion of $h$. By keeping the leading order terms, Eq.~\\eqref{eq:d=2D=3} is obtained with $E_s \\propto h$ and $E_b \\propto h^3$, while higher order terms (e.g. $O(h^5)$) will be ignored in this manuscript. For the special case of a flat plate, this construction is demonstrated in Landau's book~\\cite{Landau1986}. Here, we consider more generic 2D plates which are not flat.\n\nThe first term of Eq.~\\eqref{eq:d=2D=3} takes the same form as Eq.~\\eqref{eq:Es} but with two modifications\n\\begin{align}\nE_s=h \\int dr{\\sqrt{\\det g_0}}\\large\\{ \\frac{B_0-G_0}{2} & \\tr(g-g_0)^2\n\\nonumber\\\\\n&+ G_0 \\tr[(g-g_0)^2]\\large\\}\n\\label{eq:Es_2D}\n\\end{align}\nFirst, this $E_s$ has an extra pre-factor $h$, which is the thickness of the plate, and secondly, $g$ and $g_0$ are now $d\\times d$ matrices, instead of $D\\times D$.\nThe elastic moduli here are $B_0=\\frac{9 B G}{4(3B+4G)}$ and $G_0=G\/4$. \n\nFor plates that are homogenous along the thickness direction, deformations that make the plate non-flat will cost bending energy, which is the $E_b$ term here. Mathematically, bending is described by the so-called the second fundamental form, and thus $E_b$ shall be in general a functional of the second fundamental form. \nAs will be shown below, for 2D plates, it is always possible to reach $g=g_0$ at least locally, and thus, for a homogenous or nearly-homogenous 2D plate, \n$E_b$ only depends on two qualities, the mean and Gaussian curvatures ($H$ and $K$)\n\\begin{align}\nE_b=h^3  \\int dr{\\sqrt{\\det g_0}}\\frac{G}{12}\\left[ \\frac{8(3B+G)}{3B+4G} H^2 -2 K \\right]\n\\label{eq:Eb}\n\\end{align}\nwhich scales as $h^3$. For a flat plate, this bending energy recovers the text-book example of the elastic theory of thin plates~\\cite{Landau1986}. \n\n\nIt is worthwhile to emphasize that because although the $E_b$ term prefers a plate to be flat, the $E_s$ term may prefer a non-flat ground state, if $g_0$ gives a non-zero curvature, as we discuss below.\n\nFor thin plates (small $h$), because $E_s\\propto h$ while $E_b\\propto h^3$, $E_b \\ll E_s$. Thus, $E_s$ dominates the elastic energy and $E_b$ can be treated as a small perturbation. \nIn the rest of the manuscript, we will follow this perturbative approach and all the conclusion would be accurate to the first order of $E_b$ (i.e., $~h^3$). Higher order contributions, e.g. $~h^5$ will be ignored.\n\n\n\\section{Generalized Maxwell's counting and Janet-Cartan theorem}\\label{SEC:ContinuumCounting}\n\n\\begin{table*}\\label{tab:class}\n\\centering\n\\begin{tabular}[c]{| c | c | >{\\columncolor[rgb]{0.88,1,1}}c || c | c | c | >{\\columncolor[rgb]{0.88,1,1}}c |}\n\\hline\n  &\\multicolumn{2}{c||}{Maxwell systems}   \t&\\multicolumn{4}{c| }{ Over-constrained systems} \\\\\n  \\hline\n  &Maxwell Lattices \t& Maxwell Plates\t&Typically solids\t& \\multicolumn{3}{ c|}{ Plates with bending stiffness}\\\\\n  & \n  $D=2$ or $3$\t& $d=2$, $D=3$\t& $d=D=2$ or $3$\t& \\multicolumn{3}{ c| }{ $d=2$, $D=3$}\\\\\n\\hline\nStress free & Yes$^1$  & Yes$^2$   & No unless $g_0$ is flat  & No in general & \\multicolumn{2}{ c| }{ Yes if criterion in Sec.~\\ref{SEC:PlatBend} is met}\\\\\n\\hline\nGaussian curvature & - & Any & - & Any & Non-negative  & Negative \\\\\n\\hline\nFloppy modes & Subextensive  & Subextensive & None  & None  & None & 1\\\\\n\\hline\nHolographic & \tYes\t & Yes \t& No    & No  & No & Yes \\\\\n \\hline\nShapes &  Any & Any & Any & Any & Sphere or flat & Minimal surface \\\\\n \\hline\nFractional excitations &  No & No  & No\t& No & No & Yes\\\\\n \\hline\n\\end{tabular}\n\\caption{Summary of results. Two types of continuum media with nontrivial topological features are highlighted in cyan. $^1$ if the lattice doesn't contain redundant constraints.  $^2$ if the plate is under open boundary conditions and does not hit singularity points.}\n\\end{table*}\n\n\nIn this section and the following section, we generalize Maxwell's counting argument to continuum elasticity. Similar to discrete systems, this generalized counting argument provides a simple principle for identifying systems with floppy modes and\/or states of self-stress. In addition, we also show that predictions from this generalized Maxwell's counting is in full alignment with the mathematical theorems on local embedding\n~\\cite{janet1926possibilite,cartan1927possibilite}, global embedding and rigidity.\n\nIn this section, we focus on systems with elastic energy $E=E_s$, while more general elastic energy (e.g. with bending $E_b$) will be studied in Sec.~\\ref{SEC:PlatBend}.  \nSystems with $E=E_s$ include two possible cases. The first one is $d=D$, where $E_s$ is naturally the only term in the elastic energy. The second case is for $d<D$, e.g. a 2D plate in a 3-dimensional space. Here, in general, the elastic energy contains more terms beyond $E_s$, e.g., $E_b$. However, as shown above, when the plate is thin enough, $E_b$ becomes negligible, and thus only $E_s$ needs to considered. \n\n\\subsection{Counting argument and its predictions}\nDue to nonlinearity, the counting argument [Eq.~\\eqref{eq:N0s}] is not generically expected here. In this section, we first perform the counting analysis while ignoring \npossible problems caused by nonlinearity. These predictions will then be checked in the next section. Surprisingly, as shown below, for all generic cases considered here, \nwe found that this counting argument happens to provide correct predictions. Of course, it must be emphasized that because such a coincidence cannot be universally proved,\nthere could exist special cases where such a relation is invalid.\n\n\nIn a $D$-dimensional space, each real-space point has $D$ degrees of freedom. As for constraints, following the same spirit of Maxwell's counting argument, $E_s$ can be treated\nas constraints $g=g_0$ , i.e. the elastic energy vanishes if $g=g_0$, in analogy to the a spring network where all springs are at its rest length. The total number of constraints enforced by this relation is $d(d+1)\/2$ per real-space point, because $g=g_0$ is a $d$-dimensional symmetric rank-2 tensor with $d(d+1)\/2$ independent variables. In mathematical literature, this quantity $d(d+1)\/2$ is often referred to as the Janet dimension.\n\nIf $D>d(d+1)\/2$ (e.g. a 1D chain in 3D space with $E=E_s$), the system is under-constrained, and thus it must be floppy with extensive number of floppy modes. In general, unless some incompatible constraints happen to arise in the system, we expect no state of self-stress, which implies that in general, the ground state is stress-free and can reach $g=g_0$.\n\nIf $D<d(d+1)\/2$ (e.g. $d=D>1$), the system is over-constrained and thus we expect extensive number of states of self-stress. These systems don't have to contain floppy modes, and thus in general we expect the system to be rigid with one unique ground state.\n\nAs for the marginal case where the spatial dimension and the Janet dimension coincides, $D=d(d+1)\/2$, the system is at the Maxwell point, where constraints and degrees of freedom have exactly the same number. One such example is a 2D plate embedded in  the 3D space ($d=2$ and $D=3$). If the bending energy is negligible, such a plate will be called a Maxwell plate. \n\n\n\n\\subsection{Local embedding and the Janet-Cartan theorem}\n\\label{sec:janetcartan}\nThe counting argument discussed above is in full alignment with the Janet-Cartan theorem~\\cite{janet1926possibilite,cartan1927possibilite}.\n\nFor any given metric $g=g_0$ (which minimizes $E_s$ and thus fully releases built-in stress), whether a mapping $\\mathbf{r} \\to \\mathbf{R}$ exist such that $(g_{0})_{ij}=\\partial_i \\mathbf{R}\\cdot \\partial_j \\mathbf{R}$, is known as the isometric embedding problem.  $g=g_0$ may not always be achievable, because actual variables of the system are $\\mathbf{R}(\\mathbf{r})$, and different components of $g$ are related with one another.  \nThe  Janet-Cartan theorem of analytic local embedding states tells us that a \\emph{local embedding} always exists for $D\\ge d(d+1)\/2$, where local embedding means that for any real space point, we can always find a region around this point such that $g=g_0$ for every point in this region for any analytic $g_0$. In other words, as long as our elastic system is small enough and has open boundaries, any under-constrained or Maxwell systems ($D\\ge d(d+1)\/2$) must have a stress-free ground state for any analytic $g_0$, i.e., the system has no states of self stress. In contrast, for over-constrained systems $D>d(d+1)\/2$, an isometric embedding doesn't exist in general. Physically, this means that for a generic $g_0$, it is impossible to reach $g=g_0$ and fully released stress. Thus, the system contains states of self-stress.  \n\nThis is in good agreement with conclusions in the previous section following from Maxwell's counting argument.  It is worth noting that the Janet-Cartan theorem incorporates nonlinearities in the constraint $g=g_0$, and interestingly it supports the linear counting argument.\n\nBelow, we consider two examples: an over-constrained system with $d=D$ and a Maxwell plate with $d=2$ and $D=3$, in order to verify relations between zero modes and states of self stress from Eq.~\\eqref{eq:N0s}].\n\n\n\n\\subsection{Over-constrained systems with $d=D$}\nAccording to the counting argument outlined above, a system with $d=D>1$ is always over-constrained, and thus we expect extensive number of states of self stress. \nAs shown above, this conclusion qualitatively agrees with the Janet-Cartan theorem.\nIn this section, we further quantitatively verify the counting argument by directly counting the number of states of self stress and then comparing \nit with the value predicted by the counting argument [Eq.~\\eqref{eq:N0s}]. Same as in discrete systems, we start from a fine-tuned stress-free ground state \n(with $g=g_0$) and ask how many different ways residual stress can arise in such a system, which is a linear elasticity problem and thus a definite answer can be obtained.\n\nBecause the number of floppy modes per real space point is in general zero without fine tuning in an over-constrained system, \naccording to  Eq.~\\eqref{eq:N0s}, the number of states of self stress should equal to the number of constraints minus the number of degrees of freedom, \nwhich is $D(D-1)\/2$ for $d=D$. This is indeed correct. A $D$-dimensional stress tensor field $\\sigma$  has $D(D+1)\/2$ independent components. \nAt the same time, force equilibrium condition requires the stress field to be divergenceless,\n\\begin{align}\n \\partial_i \\sigma_{ij}=0,\n\\end{align}\nwhich provides $D$ partial differential equations and thus fixes $D$ of the $D(D+1)\/2$ independent components at each real space point. As a result, the number of free components becomes $D(D-1)\/2$, matching exactly the expected number of states of self stress of the counting argument. It is worthwhile to mention that to count the states of self stress, in \nprinciple, force-free boundary conditions also need to be enforced in addition to the divergenceless condition. However, because the boundary is subextensive, \nits impact on an extensive quantity (i.e. the number of states of self stress here) is negligible and thus would not change the counting. \n\nWe conclude this subsection by discussing the sufficient and necessary condition, under which this over-constrained system has a stress free ground state. \nSimilar to over-constrained networks\/lattices, although generically our over-constrained system shall contain residual stress, the ground state could be made stress free if we fine tune the constraints to make them fully compatible with each other. For lattices and networks this fine tuning is about the rest length of each bond, while in continuum, we need to fine tune the reference metric tensor $g_0$. For $d=D$, the sufficient and necessary condition for an elastic body to be stress free is that $g_0$ must be flat, because the $D$ dimensional Euclidean space we try to embed this body into is flat. More precisely, if we treat $g_0$ as a metric tensor and compute its Gaussian curvature (as show in App.~\\ref{app:sec:2Din3D}, for a Riemannian manifold, its Gaussian curvature is uniquely determined by the metric tensor), this Gaussian curvature must vanish at every point, which implies that the system is globally flat. \n\n\\subsection{Maxwell plates and holographic floppy modes}\nIf $E_b$ is ignored, a 2D plate ($d=2$ and $D=3$) satisfies the Maxwell condition [Eq.~\\eqref{eq:z2D}]  with $D=d(d+1)\/2$, and thus is right at the Maxwell point, which is the reason\nwhy they are called Maxwell plates.\n\nIn this section, we consider small plates with open boundary conditions, where the Janet-Cartan theorem applies. More generic cases, e.g., large or infinite plates and closed plates without boundary, will be discussed in Sec.~\\ref{SEC:global}.\n\n\nAs shown above, the number of degrees of freedom and constraints perfectly match in the bulk of a Maxwell plate.  \nThus, edge degrees of freedom become important. Same as Maxwell lattices\/networks, a point on an open edge has the same number of \ndegrees of freedom but fewer constraints, because its constraints only comes from one side of the edge, while the other side is empty and thus enforces \nno constraint. This deficit of constraints on the boundary leads to floppy modes in Maxwell plates when they are under open boundary conditions, and according to Eq.~\\eqref{eq:N0s}, the number of these floppy modes is proportional to the length of the boundary, and is a sub-extensive quantity.\nIn addition, same as Maxwell lattices, the edge origin implies that floppy modes in Maxwell plates are \\emph{holographic}, i.e., for any low-energy deformations, the deformation field at the edge fully dictates the bulk deformation field. The existence of sub-extensive holographic floppy modes is a key property of Maxwell plates, in strong analogy to Maxwell lattices.\n\nThese counting-based conclusions on holographic floppy modes can be rigorously proved using  mathematical tools of isometric embedding. As show in Ref.~\\onlinecite{Han2006isometric}, \nmathematically, the problem of isometric embedding is translated into partial differential equations, known as the Gauss-Codazzi equations (See App.~\\ref{app:sec:2Din3D} for\nmore details). \nFor plates considered here, solutions to these partial differential equations alway exists, as long as suitable boundary conditions are applied and the plate is not too large to hit singularity points. In particular, solutions to these equations have a one-to-one correspondence with the boundary conditions, i.e. boundary conditions fully determine the solution in the bulk, which is the mathematical origin of the holographic floppy modes. This is in strong analogy to static electricity in a cavity, which is also governed by a set of partial differential equations. There, the field configuration is also holographic, i.e. it is fully dictated by boundary conditions, e.g. charge or field configurations on the boundary uniquely determines the charge\/field configuration in the bulk.\n\n\n\n\n\\subsection{Global embedding and systems beyond the Janet-Cartan theorem}\\label{SEC:global}\nIn this section, we explore infinitely large 2D plates and\/or closed 2D plates without open boundaries. Here a global embedding is required to fully release the stress,\nwhich is a much stronger requirement in comparison with a local embedding. A global embedding requires $g=g_0$ even if we extends the system (through analytic continuation) to infinity or until the system forms a closed manifold . A global embedding cannot always be achieved for $D=d(d+1)\/2$. \nIn the counting argument, this means that at $D=d(d+1)\/2$ , although $g=g_0$ introduces no redundant constraints for any small regions, if the elastic system is large enough, or if the system form a closed manifold (e.g. by enforcing periodic boundary conditions), redundant constraints may arise and thus the ground state may not be able to fully release stress unless we fine tune these constraints to make them compatible with one another. \n\nIn this section, we consider two different families of Maxwell plates with (1) positive or (2) negative Gaussian curvature. Remarkably, for both cases, their elastic properties obey the counting predictions.\n\nFirst, we consider a 2D closed plate with positive Gaussian curvature and zero genus, which means that the plate is topologically equivalent to a sphere $S^2$. Because such a plate has no boundary, the number of constraints and degrees of freedom perfectly match with each other at every point, and thus the counting argument requires that the density of floppy modes and states of self stress must coincide.  \nThis is indeed true for such a plate. Mathematically, the embedding of such a plate \n(positive Gaussian curvature and zero genus) is known as the Weyl problem~\\cite{Weyl1916uber}. Based on the Nirenberg-Pogorelov theorem~\\cite{Nirenberg1953,Pogorelov1952}\nsuch an embedding with $g=g_0$ always exists and the embedding is ``rigid'',  i.e., any deformation from the ground state will make $g\\ne g_0$ and thus cost energy.\nThe existence of embedding for any $g_0$ implies that the ground state is always stress free, and thus there is no state of self stress $N_s=0$. At the same time, the rigidity\npart of the Nirenberg-Pogorelov theorem implies the absence of floppy mode ($N_f=0$), and thus indeed $N_f=N_s$ as the counting argument predicts.\nIn addition, because such a plate has $N_f=N_s=0$, this system is not only ``Maxwell'', but also ``isostatic''.\n\nFor plates with negative Gaussian curvature, it cannot form a closed manifold and floppy modes can emerge from edge degrees of freedom as shown in the section above. Here, we consider an infinitely-large plate. If the Gaussian curvature decays to zero fast enough at infinity, a global embedding always exits~\\cite{Hong1993} and thus \n$g=g_0$ can always be reached. Here, conclusions of local embedding remain and we shall have no state of self-stress and sub-extensive number of floppy modes. \nIf the Gaussian curvature doesn't approach zero or approaches zero too slowly at infinity, an isometric embedding may hit singularity and thus make a global embedding\nimpossible (e.g. the Efimov theorem~\\cite{Efimov1966surfaces}). Because $g$ cannot reach $g_0$, the ground state carriers residual stress, i.e., state of self stress emerges.\nAs shown in Ref.~\\cite{marder2006geometry,sharon2010mechanics},  wrinkles will start to develop beyond these singular points. According to Maxwell's counting argument, this emergence of state of self-stress must be accompanied by new floppy modes. For this example, these floppy modes correspond to translations of the wrinkles.\n\n\n\n\n\n\\section{2D plates with bending energy}\\label{SEC:PlatBend}\nIn this section, we consider the impact of bending energy $E_b$ for $d<D$. As mentioned above, we will focus on the thin plate limit (small $h$) and thus $E_b \\ll E_s$.\nIn the counting analysis, $E_b$ can also be treated as constraints. \nSame as $E_s$, generally, the number of constraints enforced by $E_b$ is an extensive quantity. For 2D plates in a 3D space, as shown in the previous section, without $E_b$, the number of degrees of freedom exceeds the number of constraints by a small (sub-extensive) margin, and as a result, after adding extensive number of constraints from $E_b$, the system is over-constrained. Therefore, same as the case of $d=D$, we shall in general expect one unique ground state (no floppy mode) and extensive number of states of self-stress.  Thus, unless we fine tune $g_0$, the system shall in general carry residual stress.\n\nIn this section, we answer two questions:\n\\begin{itemize}\n\\item Is it possible to make a 2D plate with bending stiffness stress free? \n\\item Is it possible to make a 2D plate with bending stiffness flexible, i.e. introduce one or more floppy mode(s) into this over-constrained system? \n\\end{itemize}\nIt turns out that the answer to the second question relies on the first one, and thus we will start by exploring the first question. \n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{sphere.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{Enneper.pdf}}\n\t\\caption{Two types of stress-free 2D plates with (a) positive or (b) negative Gaussian curvature. (a) With a positive Gaussian curvature, a sphere is the only configuration for 2D stress-free plates. (b) With a negative Gaussian curvature, a 2D plate must form a minimal surface to be stress-free. Here, we present one such example, an Enneper surface.}\n\t\\label{fig:}\n\\end{figure}\n\n\nHere we show that a 2D plate becomes stress free, if and only if $g_0$ satisfies either one of the following two criteria:\n\\begin{itemize}\n\\item the sphere criterion: the Gaussian curvature $K_0$ calculated from the metric tensor $g_0$ is a non-negative constant over the entire plate.\n\\item the minimal surface criterion: if $K_0<0$, we treat the product $\\sqrt{|K_0|}g_0$ as a new metric tensor and compute its Gaussian curvature $K_1$. $K_1$ must vanish at every point on the plate.\n\\end{itemize}\nIf the first criterion is met, the ground state of such a 2D plate is (part of) a sphere with radius $\\rho_0=1\/\\sqrt{K_0}$. Here, the system is stress free and it has a unique ground state (no floppy mode). \nAs proved by Ricci in 1894, the second criterion is the sufficient and necessary condition for the existence of a minimal-surface with $g=g_0$~\\cite{Ricci1894}. \nIf $g_0$ satisfies the second criterion, the ground state is a minimal surface. Remarkably, such a 2D plate is not only stress free but also has one floppy mode [see Sec.~\\ref{SEC:MiniSurf} below]. One interesting example of minimal-surface plates \ncan be found in Ref.~\\onlinecite{Efrati2011}, which applies to any narrow ribbon with negative Gaussian curvature.\nIf none of these two criteria holds, the corresponding 2D plate must have residual stress, which cannot be released.\n\nHere, we just provide a brief outline of the proof, while details can be found in App.~\\ref{app:sec:stress:free:2D}. Within the perturbative approach described above (keeping terms up to $O(h^3)$), in order to find the ground state, we first minimize the leading-order term $E_s$, which enforces the constraint $g=g_0$, and then, under the constraint of $g=g_0$, $E_b$ is then minimized. The constraint $g=g_0$ not only fixes $g$ but also the Gaussian curvature $K$ of the 2D plate setting $K=K_0$, because the Gaussian curvature  is uniquely determined by the metric tensor and its spatial derivatives (known as Gauss's  \\emph{theorema egregium}, see App.~\\ref{app:sec:2Din3D} for more details). For $E_s$, this means that the value of $K$ in Eq.~\\eqref{eq:Eb} is already pinned as we set $g=g_0$. As a result, the $K$ term in $E_b$ does not participle in the minimization of $E_b$ and thus can be ignored. After dropping $K$,  it is straightforward that $E_s$ prefers to minimize the square of the mean curvature $H=(k_1+k_2)\/2$, where $k_1$ and $k_2$ are the two principle curvatures. This minimization needs to obey the constraint on the Gaussian curvature $K=k_1 \\times k_2=K_0$. With a fixed $k_1\\times k_2$, the minimization of $H^2$ has two possible scenarios: (1) $k_1=k_2=\\sqrt{\\kappa_0}$ for $K \\ge 0$ or (2) $k_1=-k_2$ for $K<0$. This is the constraint enforced by $E_b$.\n\nAs shown in App.~\\ref{app:sec:stress:free:2D}, $E_s$ enforces constraints on the metric tensor, which is also known as the first fundamental form, while constraints enforced \nby $E_b$ are about curvatures, which are mainly associated with the second fundamental form. Thus, in order to ensure constraints enforced by $E_s$ and $E_b$ are \ncompatible with each other, relations between first and second fundamental forms play a crucial role. It is known that the first and second fundamental forms are related by a set of partial differential equations, known the Gauss-Codazzi equations (in addition to the constraint of Gaussian curvature discussed above). As shown in App.~\\ref{app:sec:stress:free:2D}, the Gauss-Codazzi equations are satisfied if and only if one of the two criteria mentioned above is met.\n\n\n\\section{Minimal surfaces and their floppy mode}\\label{SEC:MiniSurf}\nAs shown in the previous section, when bending energy is taken into account, 2D plates are generically over-constrained and only two special cases can be stress free (a sphere or a minimum surface).  Elastic properties of these two cases are very different. Most importantly, a sphere has a unique ground state, while a minimal-surface 2D plate has infinitely many degenerate ground states, which are connected with each other via a floppy mode. In this section, we focus on the case of minimal surfaces.\n\n\\subsection{Floppy mode}\nMinimal surfaces are 2D surfaces embedded in a 3D space with their local area minimized. The key signature of a minimal surface is the vanishing mean curvature $H=0$. For $E_b$, because the Gaussian curvature $K$ is fixed by $g_0$ as we enforce the constraint $g=g_0$, a minimal surface with $H=0$ offers a natural minimization of $E_b$. This is the key reason why minimal surface can make constraints from $E_b$ and $E_s$ compatible. \n\nAs shown in App.~\\ref{app:sec:minimal:surface}, for any minimal surface, an isometric deformation always exists (i.e. $g$ stays invariant), under which the surface remains minimal (i.e. $H$ remains $0$). One example of such an isometric deformation is shown in Fig.~\\ref{fig:associate:family}.\nFor elasticity, it is easy to notice that this isometric deformation doesn't cost any elastic energy, as $E=E_s+E_b$ stays the same, and thus is a floppy mode. All minimal surfaces that are connected by this deformation is called an associate family, and this floppy mode transfers one minimal surface to other minimal surfaces, which belong to the same associated family.\n\nMinimal surfaces in an associate family can be labeled by a phase angle $\\varphi$ (see App.~\\ref{app:sec:minimal:surface} for details). As a phase angle, only $\\varphi \\mod 2\\pi$ has a real physical meaning. \nIn the  Weierstrass-Enneper parameterization, minimal surfaces are mapped into complex functions, and the phase angle $\\varphi$ is an overall complex phase in the Weierstrass-Enneper parameterization. By varying this phase angle (and keeping the rest of the complex functions invariant), a minimal surface is deformed into other ones in its associate family.\nIn this parameterization, the floppy mode corresponds to adiabatically adjusting the value of $\\varphi$.\n\nIt is worthwhile to emphasize that although for a minimal surface 2D plate $E=E_s+E_b$ has infinite degenerate ground states, in reality, this infinite degeneracy will be lifted by higher order terms in the elastic energy (e.g. if one introduces anisotropy into the elastic medium and one example is shown in App.~\\ref{app:sec:higher:order}). \nAs a result, a real system may prefer one (or two) specific minimal surface(s) as its ground state, while the floppy mode becomes a soft mode, i.e., it costs a small amount of energy to deform the ground state to another minimal surface in its associate family. However, as long as we are focusing on low-energy soft deformation, other configurations beyond the associate family can still be excluded, which cost much higher energies to reach.\n\n\\subsection{Narrow ribbons}\\label{SEC:Ribb}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.9\\columnwidth]{assoicatefamiliy.pdf}\n\t\\caption{Ground state configurations of 2D narrow ribbon with negative Gaussian curvature. Any minimal surface in the helicoid-catenoid associate family is a ground state. These ground states\n\tare labeled by a phase angle $\\varphi$ as marked in the figure.}\n\t\\label{fig:associate:family}\n\\end{figure}\n\nHere, we present one example of stress-free 2D plates, by considering one easy realization of the stress-free criteria. As shown in App.~\\ref{app:sec:ribbons}, for  2D ribbons, as long as the width of a ribbon is narrow enough, the system is guaranteed to satisfy one of the two stress-free criteria regardless of microscopic details, and thus their ground states must be either part of a sphere or a minimal surface. Interesting examples of of minimal surface hyperbolic thin ribbons have been discussed in Ref.~\\onlinecite{Efrati2011}.\n\nIn this section, we focus on 2D narrow ribbons with minimal surface ground states.\nAs discussed above, such a system has infinite number of ground states characterized by a phase angle $\\varphi$. As shown in in Fig.~\\ref{fig:associate:family}, \n$\\varphi=\\pm \\pi\/2$ represent the helicoid ground states with left-hand and right-hand chirality respectively, while $\\varphi=0$ and $\\pi$ represent two catenoid ground states. Other values of $\\varphi$ represent other ground states in this minimal-surface associate family, known as the helicoid-catenoid family.\nIt is worthwhile to mention that as shown in App.~\\ref{app:sec:ribbons}, depending on the linear term of $u$ in $g_{22}$ [Eq.~\\eqref{app:g22_expansion}],\na ribbon's helicoid ground state may or may not contain the rotational axis, i.e. helicoids shown in Fig.~\\ref{fig:associate:family} at $\\varphi=\\pm \\pi\/2$ vs. helicoids in Fig.~\\ref{fig:spin:chain}(e) and (f). These two cases may look different, but they are part of the same helicoid and thus share the same elastic properties.  Specifically, helicoids in Fig.~\\ref{fig:spin:chain}(e) and (f) can be seen as cutting a thin strip from the edge of helicoids in Fig.~\\ref{fig:associate:family}.\n\nAs can be seen from the figure, except for the two catenoid ground states ($\\varphi = 0$ or $\\pi$), all other ground states of the 2D plate simultaneously break the chiral symmetry (of the target space). \nThe two catenoid ground states don't break spatial symmetries of the target space. Instead, they break an internal symmetry of the 2D plate. This can be made more visible if we color the two sides of the plate with two different colors [yellow and blue in Fig.~\\ref{fig:associate:family}]. In our elastic energy, the two sides are fully equivalent and thus the elastic energy is invariant if we swap these two colors. However, a catenoid ground state breaks this symmetry making the yellow and blue inequivalent. Because swapping the two colors is a $Z_2$ symmetry, we should expect two such ground states when this symmetry is broken, which are the two catenoids at $\\varphi= 0$ and $\\pi$. If we swap the two colors, the two catenoids are mapped into each other. This is in analogy to an $Z_2$ internal symmetry breaking, e.g. a ferromagnetic state in the Ising spin model.\n\n\n\n\n\\subsection{Fractional solitons in thin ribbons}\nThe floppy mode in 2D minimal-surface plates has one interesting implication. It means that the system supports fractional topological excitations, in strong analogy to \n fractional topological states such as a $Z_2$ spin liquid.\n\nBecause the system has infinite degenerate ground states $0 \\le \\varphi<2\\pi$, low-energy excitations in such a ribbon shall be characterized as a $\\varphi$ field slowly varying   along the ribbon.\nThe elastic energy for such a low-energy excitation can be described by the following phenomenological theory\n\\begin{align}\nE=\\int d v (\\partial_v \\varphi)^2\n\\end{align}\nwhere $v$ is the coordinate along the ribbon direction.\nBecause any constant $\\varphi$ produces a degenerate ground state, this elastic energy can only depend on derivatives of $\\varphi$, instead of $\\varphi$ itself.  The quadratic term included here is the leading order contribution. Without loss of generality, the elastic modulus for this term is set to unity, which describes the energy cost for an inhomogeneous $\\varphi$.\n\nAs mentioned above, in a real system, the infinite number of minimal-surface states in the associate family cannot be perfectly degenerate, due to higher order corrections in the elastic energy, which can be described by a function of $\\varphi$, $E_{c}(\\varphi)$. Because for a phase angle, $\\varphi$ and $\\varphi+2\\pi$ are fully equivalent as mentioned above, this function must be a periodic function with $E_c(\\varphi)=E_c(\\varphi+2\\pi)$. Thus, in general, the elastic energy shall take the following form\n\\begin{align}\nE=\\int d v \\left[ (\\partial_v \\varphi)^2+ \\sum_n \\gamma_n \\cos  n (\\varphi-\\varphi_{n})\\right]\n\\label{eq:sine_gordon_general}\n\\end{align}\nwith the second term describing higher order terms that lift the infinite degeneracy of the ground state. Here, $n$ are positive integers, while $\\gamma_n$ and $\\varphi_{n}$ are constants for each $n$. This effective theory is a generalized sine-Gordon theory. It has long been known that the sine-Gordon theory supports soliton excitations, i.e., topological excitations \nwith $\\varphi$ changes from one value to another over a finite length scale. A particularly interesting example of a soliton in a mechanical chain described by a sine-Gordon equation is the nonlinear soliton in a 1D topological rotor chain, as discussed in Ref.~\\onlinecite{Chen2014}, where the kink soliton is exactly zero energy due to a boundary term.  \n\nIt must be emphasized that the sine-Gordon theory we use here  is \\emph{compact}, i.e. $\\varphi$ and $\\varphi+2\\pi$ are identical. As will be discussed below, solitons are quantized in a compact sine-Gordon theory, and each soliton has a well-defined quantized charge. \nHere, we first introduce the definition of soliton charge, while its quantization will be addressed below. For a compact field $\\varphi$, the periodicity $2\\pi$ offers a natural reference to measure the varition of $\\varphi$. For a soliton where $\\varphi$ changes by $\\Delta \\varphi$, its charge is defined as $\\Delta \\varphi \/2\\pi$. Solitons with integer charge are called integer solitons, i.e., $\\Delta \\varphi = 2 n \\pi$ with $n$ being an integer, while a fractionally charged  soliton will be called a fractional soliton.\n\nFor a (generalized) sine-Gordon theory shown in Eq.~\\eqref{eq:sine_gordon_general}, generically, solitons  have integer charge (i.e., $\\varphi$ changes by $2 n \\pi$). \nThis charge quantization is because generically these cosine terms will select a unique ground state with a specific value of $\\varphi$ (say $\\varphi=\\varphi_0$). Thus, for a soliton, in order to ensure that the system is at the ground state away from the soliton, $\\varphi$ must change by $2\\pi$ times an integer (e.g. from $\\varphi_0$ to $\\varphi_0+ 2 n \\pi$). This leads to the quantization of the soliton charge. Here, we demonstrate one such example by keeping only the $n=1$ term in the elastic energy, where the standard sine-Gordon theory is recovered $E=\\int d v \\left[ (\\partial_v \\varphi)^2+\\gamma \\cos \\varphi\\right]$. Such a theory is known to support soliton excitations with $\\Delta \\varphi =2\\pi$, i.e. charge-$1$ solitons. \nBy combining multiple charge-1 solitons together, solitons with higher integer charge can be created, but solitons with non-integer (fractional or irrational) charge are prohibited.\n\nFor 2D plates that we considered here, however, the situation differs sharply from the generic case discussed above. \nAs shown in App.~\\ref{app:sec:minimal:surface}, the transformation $\\varphi \\to \\varphi+\\pi$ is a chiral symmetry transformation ($\\mathbf{R} \\to -\\mathbf{R}$). \nBecause the elastic energy of a 2D plate is invariant under a chiral transformation, our elastic energy must be invariant under $\\varphi \\to \\varphi+\\pi$ (although the ground states can spontaneously break this symmetry, as we discuss in Sec.~\\ref{SEC:Ribb}), and thus \n$E_c(\\varphi)=E_c(\\varphi+\\pi)$, in contrast to a generic compact sine-Gordon theory where $E_c(\\varphi)=E_c(\\varphi+2\\pi)$. This means that with the help of the \nchiral symmetry, our system always contains two degenerate ground states (with $\\varphi=\\varphi_0$ and $\\varphi=\\varphi_0+\\pi$), and thus one can introduce\nhalf-charged solitons, which is the domain wall between these two ground states. Because the values of $\\varphi$ for these two ground states differ by $\\pi$, such a soliton \nmust have $\\Delta \\varphi =(2n+1)\\pi$, and thus a fractional charge $n+1\/2$. This fractionalization is in analogy to nematic liquid crystals, where the molecules (and the order parameter)\nare invariant under a $\\pi$ rotation. Such a symmetry also results in fractional topological defects in nematic liquid crystals, i.e., disinclinations or disinclination lines, which can be viewed as\nhalf of a vortex or a vortex line~\\cite{degennes1993, Lubensky2000}.\n\nThe configuration of these fractional solitons can be obtained using the elastic energy. Because the chiral symmetry requires $E_c(\\varphi)=E_c(\\varphi+\\pi)$, \n odd-integer cosine terms in Eq.~\\eqref{eq:sine_gordon_general} are prohibited and thus we have\n\\begin{align}\nE=\\int d v \\left[ (\\partial_v \\varphi)^2+ \\sum_n \\gamma_{2n} \\cos  2 n (\\varphi-\\varphi_{2n})\\right] .\n\\label{eq:sine_gordon_general_chiral}\n\\end{align}\nHere, we demonstrate a fractional soliton by considering the minimal model, which only contains the $\\cos  2 \\varphi$ term\n\\begin{align}\nE=\\int d v \\left[ (\\partial_v \\varphi)^2+\\gamma \\cos 2 \\varphi\\right] .\n\\label{eq:sine_gordon}\n\\end{align}\nThis is similar to  the standard sine-Gordon theory, but with an extra factor of $2$ in the cosine term.  For $\\gamma>0$, the system has two ground states $\\varphi=\\pm \\pi\/2$,  while for $\\gamma<0$, the two ground states are $\\varphi=0$ or $\\pi$. Here, we demonstrate the physics by focusing on the case of $\\gamma>0$. \nFollowing the standard sine-Gordon theory approach,  soliton solutions are expected. Because of the extra factor of $2$ in the cosine term, $\\varphi$ changes by $\\pi$ for such a soliton, instead of $2\\pi$, and thus it contains half charge. Here, the soliton is the domain boundary between the left- and right- handed helicoid ground states. \n\nThis physic picture of fractional excitations can be generalized.  As mention above, for a system whose low-energy \nphysics is described by a compact sine-Gordon theory,  generically, soliton charge is quantized to integer values. However,\nif the Hamiltonian is invariant under the $Z_N$ transformation $\\varphi\\to \\varphi +2\\pi\/N$  ($N$ is an integer),\nthis $Z_N$ symmetry will change the charge quantization from integer to fractional values (integer times $1\/N$). A minimal-surface 2D plate offers such an example with $N=2$, and the $Z_N$ symmetry here is the $Z_2$ \nchiral symmetry $\\varphi\\to \\varphi +\\pi$.\n\n\n\n\\subsection{Numerical verification}\\label{SEC:Nume}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=.2\\columnwidth]{numericdomain1.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.5\\columnwidth]{numericdomain2.pdf}}\n\t\\caption{Soliton configuration from finite-element analysis. By minimizing the elastic energy, here we show the soliton configuration for ribbons with (a)  helicoid or (b) catenoid ground states.}\n\t\\label{fig:numerical}\n\\end{figure}\n\nThese fractional solitons indeed exist in narrow elastic ribbons. \nIn Fig.~\\ref{fig:numerical}, we simulate a narrow ribbon with $E=E_s + E_b$ and a small perturbation is added to lift the infinite degeneracy of the ground states.\nIn this simulation, 10-node triangular elements are utilized. The shape function of such an element preserves the three-fold rotational symmetry, which help minimizing \nanisotropy induced by the shape function. The entire ribbon is composed of $60\\times 4$ nodes. The elastic moduli are set to (in arbitrary units) \n$h B_0 = 6 \\times 10^5$ and $G_0 =B_0\/2$  [Eq.~\\eqref{eq:Es_2D}]. The bending stiffness \n$D_1=\\frac{2 h^3 G (3B+G)}{3(3B+4G)}$ is set to $ 2.4 \\times 10^3$ [Eqs.~\\eqref{eq:Eb} and~\\eqref{eq:Eb:app}].\nWe also added a small perturbation to favor the helicoid (or catenoid) ground states [as shown in Eq.~\\eqref{eq:deltaEb:app}], \nwhose coefficient $\\delta D=0.02 D_1$ in Fig.~\\ref{fig:numerical}.(a) and $-0.01D_1$ in Fig.~\\ref{fig:numerical}(b).\nAs shown in App.~\\ref{app:sec:ribbons}, $g_0$ can always be written in the form of \nEqs.~\\eqref{app:g0_general} and~\\eqref{app:g22_expansion} and here we choose $a_1=0$ and $a_2= 2\\pi\/10$ [Eq.~\\eqref{app:g22_expansion}].\nAll qualitative features that we observed are insensitive to microscopic details and remain stable as we vary the control parameters and the system size.\nThe simulation didn't enforce the excluded-volume condition, and thus the ribbon may intersect with itself. \nEnforcing excluded volume or not doesn't change any qualitative conclusions.\n\nFrom the finite-element analysis, we found that a fractional soliton is indeed a local energy minimum. \nIn particular, for the helicoid ground states, \nby minimizing the elastic energy, we find that such a domain structure\nalways bends the ribbon by nearly $180^\\circ$, i.e. if we move along the direction of the helicoid ribbon, each soliton excitation implies a sharp U-turn. The reason for this sharp turn can be understood by looking at a section of catenoid and trying to connect it with two helicoids with opposite chirality to its two ends. As shown in Fig.~\\ref{fig:associate:family}, if one tries to smoothly connect them (by slowly varying $\\varphi$ along the chain), a $180^\\circ$ degree turn shall naturally arise. \nThis sharp U-turn associated with such a fractional soliton is the key for the formation of kinks in tangled phone-cords.\n\n\\section{Majumdar-Ghosh model and Z$_2$ topological spin liquid}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\columnwidth]{sl.pdf}\n\t\\caption{Fractional solitons in dimerized spin chains. (a-d) and narrow ribbons (e-h). The quantum spin system has two degenerate ground states (a) and (b). A spin-1 excitation\n\tcan be created via local perturbations (c), which splits into two deconfined spin-1\/2 excitations (d). (e) and (f) show two degenerate ground states of a helicoid ribbon. \n\t(g) shows a local excitation, which decays into a pair of fractional solitons in (f).}\n\t\\label{fig:spin:chain}\n\\end{figure}\n\nIt turns out that the elastic system we discuss here and its fractional soliton excitations show strong analogy to fractional excitations in $Z_2$ spin liquids and other similar quantum dimer systems.  Thus we can utilize insights obtained from the study of fractional excitations in quantum systems to help understanding fractional excitations in 2D plates.\nIn this section, we provide a brief review about some key properties of 2D $Z_2$ spin liquids and 1D quantum dimer systems, which will be compared with 2D plates in the\nnext section.\n\nA $Z_2$ spin liquid in 2D is one of the most important and well-studied fractional topological states, which exhibit exotic properties such as $Z_2$ topological order,  long-range entanglement, fractional excitations and topological degeneracy (see for example a recent review Ref.~\\onlinecite{Wen2017} and references therein).\nThe study of $Z_2$ spin liquids originates from Anderson's resonating-valence-bond (RVB) senario~\\cite{Anderson1973, Fazekas1974} in frustrated quantum spin systems and quantum dimer models~\\cite{Kivelson1987, Rokhsar1988, Moessner2001}. \nThis exotic quantum phase of matter is characterized by a topological Ising gauge theory and gives rise to deconfined fractional excitations, e.g. spinons which carry spin-$1\/2$ \nbut no charge~\\cite{Read1991,Wen1991,Mudry1994, Senthil2000, Moessner2001b}. Later, an exactly sovable model with the same topological order was introduced, known as the toric code model of Kitaev~\\cite{Kitaev2003}.\n\n\\begin{figure}[t]\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{spinliquid1.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{catenoid1.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{spinliquid2.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{catenoid2.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{spinliquid3.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{catenoid3.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{spinliquid4.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.4\\columnwidth]{catenoid4.pdf}}\n\t\\caption{Fractional excitations and topological degeneracy. The left column demonstrates a quantum $Z_2$ spin liquid (e.g. a RVB spin liquid) defined on an annulus, which has two degenerate ground states (a) and (g) due to topological degeneracy. Via local perturbation, a spin-1 excitation can be introduced to the first ground state, which can split into two spin-1\/2 fractional excitations as show in Fig.~(c). If these two fractional excitations are moved around the annuls (e) and then annihilated with each other (g), the system turns into the other ground state, different from the original one where we start from. A catenoid with the same geometry setup shows the same property as show in the second column. Here, we also have two degenerate ground states, which correspond to swap the two sides of the 2D manifold. One can create two charge-1\/2 solitons (d) and move them around the catenoid (f) before annihilate them. This procedure also flips a ground state into the other one.}\n\t\\label{fig:spin:liquid}\n\\end{figure}\n\n\nHere, instead of providing a comprehensive review about spin liquids in 2D, we examine a highly-simplified 1D version of dimer states, which exhibits a lot of the interesting ingredients of 2D spin liquids. Consider the one-dimensional Majumdar-Ghosh model~\\cite{Majumdar1969},\nwhich studies a 1D chain of spin-$1\/2$ Heisenberg spins with frustrated nearest and next-nearest-neighbor anti-ferromagnetic couplings. This model has two degenerate ground states as shown in Fig.~\\ref{fig:spin:chain}(a) and (b), where neighboring spins form singlet pairs and each spin can only participate in the formation of one spin singlet with one of its two neighbors. Such a singlet pair is called a ``dimer'' and the ground states of the 1D Majumdar-Ghosh model are called dimer states. This 1D chain is in strong analogy to\n$Z_2$ spin liquids in 2D, whose ground states are also composed of dimerized  singlet pairs.\nIn the 1D Majumdar-Ghosh model, the two-fold ground-state degeneracy is due to spontaneous breaking of the lattice translational symmetry. \nFor a $Z_2$ spin liquid defined on a 2D annulus  [Fig.~\\ref{fig:spin:liquid} (a) and (g)], although the ground state breaks no symmetry, a similar \ntwo-fold degeneracy is expected due to topological reasons, which is known as topological degeneracy (see e.g. Ref.~\\onlinecite{Wen2017} and references therein). In contrast to degeneracy from spontaneous symmetry breaking,\ntopological degeneracy implies that the number of degenerate ground states varies  according to the topology of \nthe underlying manifold of the system in real space. A $Z_2$ spin liquid has $2^{n_L}$ ground states, where $n_L$ counts the number of \nindependent non-contractible loops of the underlying manifold, e.g. a sphere or a disk has $n_L=0$. An annulus has $n_L=1$, while a torus or a double torus has $n_L=2$ or $n_L=3$ respectively. In this manuscript, to compare with narrow ribbons,\nwe focus on spin liquid defined on an annulus, with $n_L=1$ and thus $2$ degenerate ground states.\n\nNow, we consider excitations in the Majumdar-Ghosh model. The obvious excitation in a dimerized ground state is to break a dimer, i.e. transfer a singlet into a triplet. Such a triplet excitation carries spin-1, which is a local excitation and can be introduced via a local perturbation. In the Majumdar-Ghosh model, such a local excitations can fractionalize into two spin-1\/2 fractional excitations, as shown in Fig~\\ref{fig:spin:chain} (c) and (d). Similar fractional excitations arise in 2D spin liquids. In both 1D and 2D, such a fractional excitation cannot be directly created in the bulk. Instead, they need to be created in pairs. More importantly, although each spin-1\/2 excitation may look like an individual particle, fractional excitations in each pair are connected by a ``string'', which distinguish them form ordinary local excitations.\n\nIn a conventional material, the energy cost increases rapidly as one tries to split an integer-charged excitation into two parts and separate them away from each other.  \nIn a $Z_2$ spin liquid or a 1D Majumdar-Ghosh chain, however, the energy cost for separating two fractional excitations saturate quickly as their separation gets large and thus the attraction between the two fractional excitations decreases to zero as an exponential function of the separation. Therefore, once far apart, such a fractional excitation behaves just like a point particle, and is called a deconfined fractional excitation.\nIn addition to $Z_2$ spin liquids, this deconfinement mechanism is applicable to a wide variety of fractional excitations as well, such as fractional and non-abelian particles in fractional quantum Hall systems~\\cite{Nayak2008}.\n\nIn topological states such as a Z$_2$ spin liquid, there exists one important connection between topological degeneracy and fractional excitations, known as braiding. \nThis terminology usually refers to move a fractional particle around another one in 2D~\\cite{Nayak2008}. However, similar phenomena often arise as long as \nthe trajectory of a fractional particle forms a non-contractible loop, i.e. a closed loop which cannot smoothly shrink into a point, not necessarily due to the existence of another particle. Thus, in this paper, we will use this terminology loosely to refer to any non-contractible loops.\nImagine that we create a set of fractional excitations via certain local perturbation in a fractional topological state (e.g. a $Z_2$ spin \nliquid or a fractional quantum Hall system), and then adiabatically move their locations in a non-contractible way, i.e., the path of certain fractional excitation form a non-contractible closed loop. Afterwards, these fractional excitations are annihilated with one another and thus the system goes back to the ground state. Although this procedure starts from a ground states and ends also as a ground state, the initial and final states may be two distinct quantum states orthogonal to each other. \nOne such example in a $Z_2$ spin liquid is demonstrated in Fig.~\\ref{fig:spin:liquid}(a) (c) (e) and (g). \nAs mentioned above, on a 2D annulus, this system has two degenerate ground states due to topological degeneracy. \nBy creating a pair of fractional excitations and moving them around the annulus, the system is transformed from one ground state\nto the other. In other words, braiding offers a pathway for topologically degenerate ground states to evolve into each other. This phenomenon plays a crucial role in topological quantum computing. If we consider two degenerate ground states as a quantum two-level system, these two states \nforms a q-bit and braiding serves as logical gates that flips and control this q-bit (For more details, see a review article Ref.~\\onlinecite{Nayak2008} and references therein). \nFor a Z$_2$ spin liquid, this braiding relation makes it possible to use this topological states as a topological quantum memory, where information stored in such a memory is robust against any local perturbations or quantum decoherence. The same principle can also be used for topological quantum computing. However, for such an objective, topological states with more complicated fractional particles and braiding algebra are needed, such as Majorana or Fibonacci anyons, where the later one can even achieve \nuniversal quantum computation~\\cite{Freedman2002}.\n\nThis relation between degeneracy and fractional excitations also arises in 1D dimer states, and in Fig.~\\ref{fig:spin:chain} (c) and (d), we can already see that moving\nfractional particles flip the ground state (from blue to yellow).\n\nFinally, it needs to be highlighted that their exists a deep connection between fractional excitations in a quantum spin chain and those in a minimal-surface narrow ribbon, \nif one realizes that low-energy physics in both systems are described the compact sine-Gordon theory. \nIn the study of spin chains, one well-known and very powerful mathematical tool is the Luttinger liquid approach~\\cite{Emery1979, \nAffleck1989, Francesco2012,Fradkin2013}. As shown by Haldane~\\cite{Haldane1982}, in this approach, \nthe low-energy effective theory of a dimerized spin chain is a compact sine-Gordon field theory, same as these ribbon systems. \nRemarkably, in the same work, Haldane also pointed out that spin-1\/2 and spin-1 excitations there indeed correspond \nto charge-1\/2 and charge-1 solitons respectively.\nThis analogy is the fundamental reason why these elastic ribbons share common properties with exotic quantum states formed by frustrated quantum spins.\n\n\n\\section{Fractional excitations in narrow ribbons}\\label{SEC:FracRibb}\nIn this section, we discuss physical properties of fractional excitations in minimal-surface narrow ribbons.\n\n\\subsection{Integer and fractional solitons}\nWe first consider a minimal-surface narrow ribbon and assume that higher order terms in the elastic energy  favor the helicoid ground state.\nAs discussed above, key properties of such a ribbon can be characterized by the phenomenological theory of Eq.~\\eqref{eq:sine_gordon} with $\\gamma>0$. Most of our\nconclusions can be easily generalized to other cases, e.g. for $\\gamma<0$, where the ground state is a catenoid.\n\nAs shown in Fig.~\\ref{fig:spin:chain} (e) and (f), this system has two degenerate ground states, i.e. left- or right- handed helicoids.\nNow, we introduce soliton excitations to one of the ground states (e.g. right-handed) as shown in Fig.~\\ref{fig:spin:chain} (g) and (h). \nWith local deformations, only integer charged solitons can be introduced [Fig.~\\ref{fig:spin:chain} (g)], which can fractionalize into two 1\/2-charge fractional solitons\n[Fig.~\\ref{fig:spin:chain} (h)]. It is worthwhile to point out that for better visualization, we keep these helicoids straight even after solitons are introduced, whose \ntrue lowest-energy configuration should involve a kink as shown in Fig.~\\ref{fig:numerical}.  \nSame as the dimer chain [Fig.~\\ref{fig:spin:chain} (a-d)], moving a pair of fractional solitons here flips one ground state into the other one.\nOnce created and becomes deconfined, a fractional soliton can no longer be removed by any local adjustment. Instead, it can only be neutralized by globally adjust \nthe entire chain or by annihilating with an anti-soliton.\n\nIn Fig.~\\ref{fig:catenoid:solitons}, we consider a ribbon with catenoid ground states ($\\gamma<0$) and assume that the catenoid form a closed loop as shown in the figure. \nInterestingly, creating a half-integer-charge fractional soliton must sacrifice the orientability, turning the orientable cylinder-like structure into an non-orientable surface.\nwhich offers another hint about the fractional nature of these solitons.\n\n\n\\begin{figure}[t]\n\t\\subfigure[]{\\includegraphics[width=.45\\columnwidth]{catenoid-no-sol.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.45\\columnwidth]{catenoid-half-sol.pdf}}\n\t\\subfigure[]{\\includegraphics[width=.45\\columnwidth]{catenoid-one-sol.pdf}}\n\t\t\\subfigure[]{\\includegraphics[width=.45\\columnwidth]{catenoid-threehalf-sol.pdf}}\n\t\\caption{Integer and fractional excitations in a catenoid. (a) shows a catenoid ground state without soliton excitations. (b-d) contain solitons. In contrast to an integer charged soliton (c), which preserves the orientability, a half-integer charged soliton makes the manifold non-orientable as shown in (b) and (d).}\n\t\\label{fig:catenoid:solitons}\n\\end{figure}\n\n\n\\subsection{Two unique features}\n\\label{sec:sub:two:features}\nFractional solitons in a narrow helical ribbon have two unique features, which are not generally expected for most other fractional excitations.\nFirs of all, these solitons are \\emph{holographic}, which means that if there is only one charge-1\/2 soliton in a helicoid, we can pin-point and control its location via \ncontrolling the two ends of the helicoid.\nThis is because the soliton here is the domain boundary between left- and right- handed sections. For a helicoid with length $L$, we define the left-handed section length to be $x$, and thus the right-handed section shall have length $L-x$. For simplicity, here we ignore the width of the soliton, which can be included easily without alternating any main conclusions. For a helicoid, we can define the total helicity of the entire ribbon as $(L-2x)\/\\lambda$ where $\\lambda$ is the pitch of the helicoid. This quantity describes how many times the ribbon twists,\nwith right-handed twists defined as positive. It is easy to notice that this quantity directly connects helicity with the location of the fraction soliton $x$. For an elastic ribbon, the helicity can be adjust by twisting the two ends of the ribbon in the opposite direction and each $2\\pi$ twist increases\/decreases the helicity by $1$, which moves the fractional soliton by one \npitch $\\lambda$. This holographic control is not a general property of fractional solitons, but a special feature for solitons in helicoids. \nIn addition to control fractional solitons, the holographic property also provide a natural way to generate these fractional excitations. If we twist the two ends of a helicoid such that \nthe helicity decreases from the ground state value ($L\/\\lambda$), this process will eventually create a fractional soliton (i.e. a non-zero $x$) to reduce energy. As will be discussed \nin the next section, this is the mechanism how solitons in telephone cords is generated.\n\n\nThe second feature of these fractional solitons is that they kink the helical ribbon by turning its direction by almost $180^\\circ$ as discussed in Sec.~\\ref{SEC:Nume}. Again, this is a special feature, not generally expected for fractional excitations.\nFor a telephone cord, this is the reason why solitons  tangle up the cord.\n\n\\subsection{Braiding}\nIn this section we demonstrate how braiding changes the ground state in narrow ribbons by considering a catenoid ground state $g<0$. As mentioned early on, here\nwe are using this terminology loosely, which includes any non-contractable loop trajectory.\n\nHere, we consider one orientable cylinder-like catenoid [Fig.~\\ref{fig:spin:liquid}(b)] and introduce a pair of fractional solitons by locally flipping a section of \nthe the catenoid inside out [Fig.~\\ref{fig:spin:liquid}(d)]. Then, we gradually move the pair of fractional solitons around the catenoid [Fig.~\\ref{fig:spin:liquid}(f)], where they meet\nagain and annihilate with each other. It is easy to realize that such a procedure changes the ground state, in analogy to the same procedure in a $Z_2$ spin liquid. \n\nIt is worthwhile to mention that here, as we annihilate two solitons, large dissipation is assumed, such that a system can quickly release its elastic energy and dissipate from an excited state (with a pair of solitons) to  the ground state (without solitons). In the absence of dissipation, the system will remain in excited states and thus solitons cannot be annihilated, even if a pair of solitons  collide. In such a scenario, the solitons will pass through each other, which is in fact part of the definition of solitons. \n\n\n\n\n\\section{discussion}\nNow we come back to telephone cords. \nA helical telephone cord is not a thin 2D helicoid ribbon, but we can consider it as a stack of 2D helicoid ribbons and thus they are expected to share similar qualitative features\nas our minimal-surface ribbons.  The insight we gain from the discussions above on fractional solitions provide us with answers to the three questions we raised in the Introduction.\n\nFirst, when we use a telephone headset, we often unintentionally twist\/rotate the headset before we put it back. As shown in Sec.~\\ref{sec:sub:two:features}, \ndue to the holographic property, this type of twisting introduces fractional solitons.  This is the origin, from which these solitons emerge in  phone cords.\nSecond, in Sec.~\\ref{sec:sub:two:features}, we showed that such a soliton bend the ribbon, which is why they result in kinks in phone cords and turn them into a tangled mess. \nThird, because these fractional excitations are topological defects, they cannot be removed by any local deformations, and this is the reason why these annoying kinks are \nso hard to get rid off.\nWith the origin of the kinks understood, we can use this knowledge as guidance to remove\/avoid these kinks. By avoiding rotating the headset, these kinks would not be \ncreated. For telephone cords that already have these kinks, we just need to twist the headset, which moves the solitons, making them annihilate with each other or driving them out\nof the cord.\n\nSimilar types of kinks\/solitons between domains of helical structures with opposite handedness have been observed previously in various situations such as perversion of tendrils on climbing plants~\\cite{Goriely1998}, intrinsically curved rods~\\cite{domokos2005multiple}, self-assembled structures of Janus colloidal particles~\\cite{Chen2011supra},\nelastic bi-strips~\\cite{liu2014structural}, helical strings~\\cite{Nisoli2015}, and minimal surface liquid films~\\cite{Machon2016}.  In this paper we show how this type of solitons arises  in narrow elastic ribbons and are fractional excitations flipping the ribbon between ground states that belong to the same minimal surface associate family.  We further demonstrate that they share the same topological description as fractional excitations in $Z_2$ spin liquids.\n\nThe fractional soliton excitations we discussed in the helicoid-catenoid family are protected by the chiral symmetry of the elastic theory.  Interestingly, when this symmetry is weakly broken, e.g., the left- and right- handed helicoids have slightly different elastic energy but both remain local energy minima, the soliton propagation between these two domains will have a preferred direction, such that it flips the higher energy domain into the lower energy domain.  Interestingly, because this soliton is holographic, its propagation requires the rotation of the domains, the sizes of which change as the soliton propagates.  \nThis results in a locked speed for the soliton propagation, as the released elastic energy from the domain-energy difference \n  turns into kinetic energy and fuels the rotation of the domain (the size of which increases).  This mechanism of locked soliton speed was first discussed in the context of solitons in helical strings, in Ref.~\\onlinecite{Nisoli2015}.  In contrast, if the chiral symmetry is preserved and the two domains have exactly the same elastic energy, the speed of the soliton changes at it propagates, because the domain size changes but the total energy is conserved.  This phenomenon of locked soliton speed may have broad applications as a constant actuation\/propulsion mechanism without the need of fine-tuning.\n  \n\nTopological solitons have also been found in discrete 1D rotor chains, which are  Maxwell lattices with exactly one floppy mode under open boundary conditions~\\cite{Chen2014}.  Interestingly, solitons in these 1D chains exhibit a kink-antikink asymmetry~\\cite{Zhou2017}.  \nIf we follow the same soliton charge defined above, this asymmetry implies the breaking of the charge conjugation symmetry for such solitons, i.e., positively and negatively charged solitons are no longer equivalent,  in sharp contrast to conventional solitons. \nWhether similar type of excitations can arise in continuum would be an interesting question for future studies.\n\nIn addition to narrow ribbons, similar physics of associate family arises in any minimal-surface 2D plates and their low-energy properties share the same sine-Gordon\ndescription, from which fractional excitations can also arise. Generalizing this knowledge about fractional excitations to other 2D plates will be an interesting subject for future studies,\nfrom which a universal understanding about fractional excitations in systems at the verge of mechanical stability may eventually emerge. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the most important tasks in high energy physics in the next decades\nwill be to determine if Higgs boson(s) exist, and to measure the\nproperties of any that are found. Only then will a full understanding\nof the Higgs sector and its role in the complete underlying\ntheory be possible.\nHere we outline the very substantial\nand, in some areas, unmatched capabilities of a muon collider \nfor these tasks.~\\cite{bbgh} We restrict our discussions\nto the Standard Model (SM) with a single neutral \nHiggs boson, $h_{SM}$, and to the minimal\nsupersymmetric extension of the SM (MSSM) with a light SM-like\nHiggs boson, $h^0$,\nand two heavier neutral Higgs scalars, $H^0$ and $A^0$.\n\nAlthough muon colliders have only received substantial attention in\nthe last few years, so far it seems~\\cite{muonreports} that there are\n``no barriers'' to either of the following two designs:\n\\begin{itemize}\n\\item $\\sqrt s\\alt 500~{\\rm GeV}$, \n${\\cal L}\\sim {\\rm few} \\times 10^{33}$cm$^{-2}$s$^{-1}$,\n$L_{\\rm tot}=50~{\\rm fb}^{-1}$\/yr;\n\\item $\\sqrt s\\alt 4~{\\rm TeV}$, ${\\cal L}\\sim {\\rm few} \\times 10^{35}$\ncm$^{-2}$s$^{-1}$, $L_{\\rm tot}=200~{\\rm fb}^{-1}$\/yr.\n\\end{itemize}\nA $\\mu^+\\mu^-$ collider has some natural advantages \nas compared to an $e^+e^-$ collider, including some\nthat are crucial for Higgs boson physics:\n\\begin{itemize}\n\\item there is essentially no beamstrahlung; \n\\item there is substantially reduced bremsstrahlung;~\\footnote{Still,\nbremsstrahlung depletes the central Gaussian peak in $\\sqrt s$\nby $\\sim 40\\%$ --- this effect is included in the computations.~\\cite{bbgh}}\n\\item there is no final focus problem (storage rings are used to build up\nthe effective instantaneous luminosity);\n\\item as required for detailed Higgs studies in the $s$-channel,\nexcellent beam energy resolution of $R=0.01\\%$ and the ability \nto set $E_{\\rm beam}$ to 1 part in $10^6$ using spin\nprecession techniques are both possible if the necessary technology is\nbuilt into the machine;\n\\item $\\sqrt s_{\\rm max}> 500~{\\rm GeV}$ can probably be reached more easily; and\n\\item there is hope that the cost might be lower.\n\\end{itemize}\nThe negatives regarding a muon collider include:\n\\begin{itemize}\n\\item the design is immature, and five years of research and development\nprojects are needed before a full-fledged proposal would be possible ---\nin particular, cooling tests are required to see if multistage \ncooling will be sufficiently efficient;\n\\item the exact nature of the detector backgrounds, and how to manage them,\nis still under investigation --- certainly the detector will be more\nexpensive due to higher shielding and segmentation requirements;\n\\item significant polarization probably implies significant loss in ${\\cal L}$\n --- for $s$-channel Higgs production, \n$S\/\\sqrt B\\propto \\sqrt{L_{\\rm tot}}\\sqrt{(1+P^2)^2\/(1-P^2)}$\nimplies that\n$[P=0,L_{\\rm tot}]$ is equivalent to $[P=0.8,L_{\\rm tot}\/10]$;\n\\item it is not possible to have a $\\gamma\\gam$ collider facility.\n\\end{itemize}\n\n\n\\begin{figure}[h]\n\\leavevmode\n\\begin{center}\n\\centerline{\\psfig{file=schannelhiggsdiagram.ps,width=5cm}}\n\\end{center}\n\\fcaption{Feynman diagram for $s$-channel production of a Higgs boson.}\n\\label{schanfig}\n\\end{figure}\n\nA muon collider can explore Higgs physics in two modes. \nAway from the $s$-channel Higgs pole, {\\it e.g.}\\ at\nthe maximum available energy $\\sqrt s_{\\rm max}$, a $\\mu^+\\mu^-$ collider\ncan discovery and study Higgs bosons in exactly \nthe same ways as an $e^+e^-$ collider\nwith the same $\\sqrt s$ and ${\\cal L}$ (barring\nunmanageable detector backgrounds at the muon collider).\nOn the $s$-channel pole at $\\sqrt s\\sim m_{\\h}$, \n$\\mu^+\\mu^-\\to h$ collisions (see Fig.~\\ref{schanfig}) imply\nunique capabilities for:\n\\begin{itemize}\n\\item precision studies of $(b\\overline bh)^2:(WWh)^2:(ZZh)^2$ coupling-squared\n ratios and of $\\Gamma_{\\h}^{\\rm tot}$ for a SM-like Higgs (either $h_{SM}$ or $h^0$)\nwith $m_{\\h}\\alt 2m_W$; and\n\\item discovery and study\nof the heavier $H^0,A^0$ of the MSSM up to $\\sqrt s_{\\rm max}$.\n\\end{itemize}\nFor $s$-channel Higgs physics,\nthe size of the $s$-channel cross section, $\\overline \\sigma_{\\h}$, is crucial.\nWe obtain $\\overline \\sigma_{\\h}$ \nby convoluting the standard Breit-Wigner shape for the Higgs\nwith a Gaussian distribution of width \n$\\sigma_{\\tiny\\sqrt s}^{}$ centered at $\\sqrt s=m_{\\h}$; $\\overline \\sigma_{\\h}$ is given by \n$\\overline \\sigma_{\\h}\\sim 4\\pim_{\\h}^{-2}BF(h\\to\\mu^+\\mu^-)$ if $\\sigma_{\\tiny\\sqrt s}^{}\\ll\\Gamma_{\\h}^{\\rm tot}$\nand by\n$\\overline \\sigma_{\\h}\\sim2\\pi^2m_{\\h}^{-2}\\Gamma(h\\to\\mu^+\\mu^-)\/(\\sqrt{2\\pi}\\sigma_{\\tiny\\sqrt s}^{})$\nif $\\sigma_{\\tiny\\sqrt s}^{}\\gg\\Gamma_{\\h}^{\\rm tot}$. To get near maximal $\\overline \\sigma_{\\h}$ and\nto have sensitivity to $\\Gamma_{\\h}^{\\rm tot}$ via scanning in $\\sqrt s$ it\nis important that $\\sigma_{\\tiny\\sqrt s}^{}$ be no larger than $2-3\\times\\Gamma_{\\h}^{\\rm tot}$.\nVery small $\\Gamma_{\\h}^{\\rm tot}$ is not uncommon.\nFig.~\\ref{hwidths} shows that $\\Gamma_{\\h}^{\\rm tot}< 1-10~{\\rm MeV}$ is typical of the $h_{SM}$\nfor $m_{\\hsm}\\alt 140~{\\rm GeV}$ and of {\\it all} \nthe MSSM Higgs bosons if $\\tan\\beta\\alt 2$ and $m_{\\ha}\\alt 2m_W$.\nUsing the parameterization \n$\\sigma_{\\tiny\\sqrt s}^{}\\simeq 7~{\\rm MeV}\\left({R\\over 0.01\\%}\\right)\\left({\\sqrt s\\over \n100~{\\rm GeV}}\\right)$ for $\\sigma_{\\tiny\\sqrt s}^{}$ in terms of the beam energy resolution, $R$,\nwe see that very excellent resolution $R\\sim0.01\\%$ is required\nfor $\\sigma_{\\tiny\\sqrt s}^{}< 2-3\\times\\Gamma_{\\h}^{\\rm tot}$ in the above cases.\n\n\\begin{figure}[h]\n\\leavevmode\n\\begin{center}\n\\centerline{\\psfig{file=hwidths.ps,width=3.25in}}\n\\end{center}\n\\fcaption{\nTotal width versus mass of the SM and MSSM Higgs bosons\nfor $m_t=175~{\\rm GeV}$.\nIn the case of the MSSM, we have plotted results for\n$\\tan\\beta =2$ and 20, taking $m_{\\stop}=1~{\\rm TeV}$ and\nincluding two-loop\/RGE-improved Higgs mass corrections and\nneglecting squark mixing; SUSY decay channels are assumed to be absent.}\n\\label{hwidths}\n\\end{figure}\n\nA final important note is that high ${\\cal L}$ is\nneeded at all $\\sqrt s$ values where a Higgs boson \nis discovered or might exist.  This will possibly imply that several\nfinal storage rings (which fortunately are relatively cheap),\ndesigned to maintain near-optimal ${\\cal L}$ over a span of $\\sqrt s$ values, \nwill need to be constructed\n\n\n\\section{{\\boldmath $s$}-channel studies of a Standard-Model-like Higgs boson}\n\nMost probably, one would first\ndiscover the $h$ at the LHC (e.g. in the $h\\to\\gamma\\gam$ or $h\\to 4\\ell$\ndiscovery modes), \nor at the NLC  (e.g. in the $Z^\\star\\to Zh$ production mode)\nand then set up the muon collider for running at $\\sqrt s\\simm_{\\h}$. \nAt the LHC or NLC, $m_{\\h}$ will be quite well determined. \nFor $m_{\\h}\\alt 2m_W$ one finds:~\\cite{snowmass96}\n$\\Deltam_{\\h}\\sim 100~{\\rm MeV}\\left({600~{\\rm fb}^{-1}\\over L}\\right)^{1\/2}$ at the LHC, \nusing $h\\to\\gamma\\gam,4\\ell$ decays and reconstructing the\nresonance peak in $m_{\\gamma\\gam},m_{4\\ell}$;\n$\\Deltam_{\\h}\\sim 100~{\\rm MeV}\\left({50~{\\rm fb}^{-1}\\over L}\\right)^{1\/2}$ for NLC $\\sqrt s=500~{\\rm GeV}$\nrunning and reconstructing the $m_{b\\overline b}$ resonance peak\nin the $Zh$ ($h\\to b\\overline b$) mode; and $\\Deltam_{\\h}< 100~{\\rm MeV}$ at the\nNLC, assuming $L=50~{\\rm fb}^{-1}$ devoted to a $Zh$ threshold study\nat $\\sqrt s=m_Z+m_{\\h}+0.5~{\\rm GeV}$. If there is no LHC or NLC, one would accumulate\n$L\\sim 1~{\\rm fb}^{-1}$ at $\\sqrt s=500~{\\rm GeV}$ at the muon collider\nin order to observe $\\mu^+\\mu^-\\to Zh$ and\ndetermine $m_{\\h}$ to within $\\Deltam_{\\h}\\sim 1~{\\rm GeV}$, and then turn to $\\sqrt s\\simm_{\\h}$\nrunning.\n\nFor a SM-like Higgs with $m_{\\h}\\agt 2m_W$, \n$\\Gamma_{\\h}^{\\rm tot}$ is large (as a result\nof $h\\to WW,ZZ$ decays), see Fig.~\\ref{hwidths}, \n$\\overline \\sigma_{\\h}\\proptoBF(h\\to\\mu^+\\mu^-)$ is tiny,\nand $\\mu^+\\mu^-\\toh$ will not be useful.~\\cite{bbgh}\nBut, if $m_{\\h}\\alt 2m_W$ then $\\Gamma_{\\h}^{\\rm tot}$ is very small,\n$\\sigma_{\\tiny\\sqrt s}^{}\\sim 2-3\\times\\Gamma_{\\h}^{\\rm tot}$ is typical\nfor $R=0.01\\%$, $\\overline \\sigma_{\\h}\\propto\\Gamma(h\\to\\mu^+\\mu^-)\/\\sigma_{\\tiny\\sqrt s}^{}$ \nwill  be much larger,  and the $\\mu^+\\mu^-$ collider\nbecomes a Higgs factory. There are several reasons to suppose\nthat the SM-like Higgs will have mass $\\alt 2m_W$:\na) precision electroweak measurements currently\nfavor (but only weakly) a relatively light SM-like Higgs boson;\nb) light Higgs masses are preferred in the SM \nif the SM couplings are required to remain perturbative up to the GUT scale;\nand c) in the minimal supersymmetric model (MSSM), the light SM-like Higgs has \n$m_{\\hl}\\alt 130~{\\rm GeV}$.~\\cite{dpfreport}\n\nIf a SM-like $h$ with $m_{\\h}\\leq 2m_W$ has been discovered, \nthe FMC final ring would\nbe optimized for $\\sqrt s\\sim m_{\\h}$ and a scan of the region\nof size $2\\Deltam_{\\h}$ in the vicinity of the known $m_{\\h}$ \nwould be performed in order to pin down $m_{\\h}$ more\nprecisely (roughly within $\\pm \\sigma_{\\tiny\\sqrt s}^{}\/2$).~\\footnote{Details of\nthe strategy\nfor this scan~\\cite{bbgh} will not be discussed here;\nthe scan is most efficient for small $R$ (as desirable in any case)\nin which case $\\sqrt s$ must be reset rapidly with high precision.}~\nIn the  ``typical case'' of $m_{\\h}\\sim 110~{\\rm GeV}$, $\\sigma_{\\tiny\\sqrt s}^{}\\sim 8~{\\rm MeV}$\n(for $R\\sim 0.01\\%$) and\n$2\\Deltam_{\\h}\\sim 200~{\\rm MeV}$ (see above), we would require $\\agt 25$ scan points\nwith $\\sim 0.01~{\\rm fb}^{-1}$ per scan point (the luminosity for a $5\\sigma$\nsignal if $\\sqrt s=m_{\\h}$), corresponding to a total luminosity\nof $L_{\\rm tot}\\sim 0.25~{\\rm fb}^{-1}$.\n(One must reset $\\sqrt s$ with $\\Delta\\sqrt s<\\sigma_{\\tiny\\sqrt s}^{}$ hourly.)\nCentering on $\\sqrt s\\simeqm_{\\h}$ does not require much luminosity      \nfor such Higgs masses. The worst mass is $m_{\\h}\\simeqm_Z$.\nFor the $R=0.01\\%$ value of $\\sigma_{\\tiny\\sqrt s}^{}\\sim 6.5~{\\rm MeV}$,\n$L_{\\rm tot}\\sim 25~{\\rm fb}^{-1}$ would be needed to center on $\\sqrt s\\simeq m_{\\h}$\nby scanning. A smaller $\\Deltam_{\\h}$, such as could be obtained\nusing ``super'' tracking or other improvements\nat the NLC,~\\cite{snowmass96} would be very helpful for avoiding wasting\nluminosity on the  centering process in this case.\n\nOnce we have determined $m_{\\h}$ to within $\\sim \\pm\\sigma_{\\tiny\\sqrt s}^{}\/2$, we would \nperform a three point scan of the Higgs resonance peak employing\nluminosity distributed as follows:\n$L_1$ at $\\sqrt s\\simeqm_{\\h}$; $2.5L_1$ at\n$\\sqrt s\\simeq m_{\\h}+2\\sigma_{\\tiny\\sqrt s}^{}$; $2.5L_1$ at\n$\\sqrt s\\simeq m_{\\h}-2\\sigma_{\\tiny\\sqrt s}^{}$.\nThe total luminosity for the scan is then $L=6L_1$.  \nThis scan would simultaneously yield measurements of $\\Gamma_{\\h}^{\\rm tot}$\nand of the $\\mu^+\\mu^-\\toh\\to b\\overline b$, $WW^\\star$ and\n$ZZ^\\star$ rates.  Assuming that $L=200~{\\rm fb}^{-1}$ is devoted over a period\nof several years to this three point scan (presumably beginning with\n$\\sqrt s\\simeqm_{\\h}$), one obtains accuracies for\nthe event rates as given in Table~\\ref{fmcsigbrerrors} and\nerrors for $\\Gamma_{\\h}^{\\rm tot}$ and coupling-squared ratios as given in\nTable~\\ref{fmcerrors}.\n\n\\begin{table}[h]\n\\tcaption{Errors for\n$\\sigma(\\mu^+\\mu^-\\toh_{SM})BF(h_{SM}\\to b\\overline b, WW^\\star, ZZ^\\star)$,\nfor $R=0.01\\%$ and $L_{\\rm scan}=200~{\\rm fb}^{-1}$ \n(equivalent to $L_{\\protect\\sqrt s=m_{\\h}}=50~{\\rm fb}^{-1}$).}\n\\footnotesize\n\\begin{center}\n\\small\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n Quantity & \\multicolumn{5}{c|}{Errors} \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 80} & {\\bf 90} & {\\bf 100} & {\\bf 110} & {\\bf 120} \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( b\\overline b) $ & \n$\\pm 0.2\\%$ & $\\pm 1.6\\%$ & $\\pm 0.4\\%$ & $\\pm 0.3\\%$ & $\\pm 0.3\\%$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( WW^\\star) $ &\n$-$ & $-$ & $\\pm 3.5\\%$ & $\\pm 1.5\\%$ & $\\pm 0.9\\%$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( ZZ^\\star) $ &\n$-$ & $-$ & $-$ & $\\pm 34\\%$ & $\\pm 6.2\\%$ \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 130} & {\\bf 140} & {\\bf 150} & {\\bf 160} & {\\bf 170} \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( b\\overline b) $ & \n$\\pm 0.3\\%$ & $\\pm 0.5\\%$ & $\\pm 1.1\\%$ & $\\pm 59\\%$ & $-$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( WW^\\star) $ &\n$\\pm 0.7\\%$ & $\\pm 0.5\\%$ & $\\pm 0.5\\%$ & $\\pm 1.1\\%$ & $\\pm 9.4\\%$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( ZZ^\\star) $ &\n$\\pm 2.8\\%$ & $\\pm 2.0\\%$ & $\\pm 2.1\\%$ & $\\pm 22\\%$ & $\\pm 34\\%$ \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 180} & {\\bf 190} & {\\bf 200} & {\\bf 210} & {\\bf 220} \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( WW^\\star) $ &\n $\\pm 18\\%$ & $\\pm 38\\%$ & $\\pm 58\\%$ & $\\pm 79\\%$ & $-$ \\\\\n\\hline\n$\\sigma(\\mu\\mu\\toh_{SM})BF( ZZ^\\star) $ &\n$\\pm 25\\%$ & $\\pm 27\\%$ & $\\pm 35\\%$ & $\\pm 45\\%$ & $\\pm 56\\%$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{fmcsigbrerrors}\n\\end{table}\n\n\\begin{table}[h]\n\\tcaption{Errors for \ncoupling-squared ratios and $\\Gamma_{\\hsm}^{\\rm tot}$ for\n$s$-channel Higgs production at the FMC, assuming $L=6L_1=200~{\\rm fb}^{-1}$\ntotal scan luminosity and $R=0.01\\%$.}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n Quantity & \\multicolumn{4}{c|}{Errors} \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 80} & {\\bf $m_Z$} & {\\bf 100} & {\\bf 110} \\\\\n\\hline\n$(W\\wstarh_{SM})^2\/(b\\overline bh_{SM})^2$ & $-$ & $-$ & $\\pm 3.5\\%$ & $\\pm 1.6\\%$ \\\\\n\\hline\n$(Z\\zstarh_{SM})^2\/(b\\overline bh_{SM})^2$ & $-$ & $-$ & $-$ & $\\pm 34\\%$ \\\\\n\\hline\n$(Z\\zstarh_{SM})^2\/(W\\wstarh_{SM})^2$ & $-$ & $-$ & $-$ & $\\pm 34\\%$ \\\\\n\\hline\n $\\Gamma_{\\hsm}^{\\rm tot}$ & $\\pm 2.6\\%$ & $\\pm 32\\%$ & $\\pm 8.3\\%$ & \n  $\\pm 4.2\\%$ \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 120} & {\\bf 130} & {\\bf 140} & {\\bf 150} \\\\\n\\hline\n$(W\\wstarh_{SM})^2\/(b\\overline bh_{SM})^2$ & \n $\\pm 1\\%$ & $\\pm 0.7\\%$ & $\\pm 0.7\\%$ & $\\pm 1\\%$ \\\\\n\\hline\n$(Z\\zstarh_{SM})^2\/(b\\overline bh_{SM})^2$ & \n $\\pm 6\\%$ & $\\pm 3\\%$ & $\\pm 2\\%$ & $\\pm 2\\%$ \\\\\n\\hline\n$(Z\\zstarh_{SM})^2\/(W\\wstarh_{SM})^2$ & \n $\\pm 6\\%$ & $\\pm 3\\%$ & $\\pm 2\\%$ & $\\pm 2\\%$ \\\\\n\\hline\n $\\Gamma_{\\hsm}^{\\rm tot}$ & $\\pm 3.6\\%$ & $\\pm 3.6\\%$ & $\\pm 4.1\\%$ &\n  $\\pm 6.5\\%$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{fmcerrors}\n\\end{table}\nVery good errors in the $c\\overline c$ and $\\tau^+\\tau^-$ channels\nmight also possible, depending on the detector.\nFor example, $c\\overline c$ isolation requires\ntopological tagging in which one must\ndistinguish primary, secondary, and \ntertiary vertices for a $b$-jet vs. primary and\nsecondary only for a $c$-jet. The ability to do so\ndepends on how close to the beam the first layer \nof the vertex detector can be placed.\n\nWhat do the errors of Table~\\ref{fmcerrors} \nmean in terms of our ability to discriminate between the SM $h_{SM}$\nand the MSSM $h^0$? Plots~\\cite{dpfreport,snowmass96}\nof the $WW^\\star\/b\\overline b$ coupling-squared\nratio as computed in the MSSM divided by that computed in the SM\nshow: a) there is almost no dependence of the ratio \nupon the squark mixing scenario; \nb) the ratio is essentially independent of $\\tan\\beta$ in the\nallowed portion of the standard $(m_{\\ha},\\tan\\beta)$ parameter space;\nand c) roughly $-50\\%$ ($-20\\%$) deviations from the SM result\nare predicted for $m_{\\ha}\\sim 250~{\\rm GeV}$ ($m_{\\ha}\\sim 400~{\\rm GeV}$).\nBy the time the muon collider is in operation, theoretical systematic\nerrors (primarily from uncertainty in the running mass, $m_b(m_{\\h})$)\nin the ratio should be in the $\\pm 5\\%$ to $\\pm 10\\%$ range.\nTable~\\ref{fmcerrors} shows that statistical errors will be much smaller\nover most of the relevant $m_{\\h}$ range.  Thus, the $WW^\\star\/b\\overline b$\nevent rate ratio will distinguish between the SM and the MSSM\nat the $\\geq 2\\sigma$ level for $m_{\\ha}\\alt 400~{\\rm GeV}$. For $m_{\\ha}< 400~{\\rm GeV}$,\na rough determination of $m_{\\ha}$ will be possible. (This will be important\nfor the $H^0,A^0$ discussion in the next section.) \nSimilar results apply for the $ZZ^\\star\/b\\overline b$ ratio.  \n\nIn contrast to the event rate ratios, deviations in $\\Gamma_{\\h}^{\\rm tot}$ \nin the MSSM vs. the SM as a function of $(m_{\\ha},\\tan\\beta)$ are very dependent\nupon the squark mixing scenario, the amount of SUSY decays present,\nand so forth; large deviations in $\\Gamma_{\\h}^{\\rm tot}$ \nfrom SM expectations are the rule, but do not pin down either $m_{\\ha}$\nor $\\tan\\beta$. However, if one~\\cite{snowmass96}\nuses the very accurate direct determination\nof $\\Gamma_{\\h}^{\\rm tot}$ from the FMC scan in combination with other measurements\nperformed with $L=200~{\\rm fb}^{-1}$ and $\\sqrt s=500~{\\rm GeV}$ at the NLC~\\footnote{The same\nmeasurements are also possible at an FMC, but we use NLC notation\nin what follows.}~\na large variety of very important coupling-squared magnitudes can be extracted.\nThe ratio of the MSSM prediction to the SM prediction for a squared coupling\nis always very squark-mixing independent and gives new opportunities\nfor determining $m_{\\ha}$.\nAs an example, there are four ways to determine\n$\\Gamma(h\\to\\mu^+\\mu^-)$ by combining NLC and FMC data:~\\footnote{Note\nthat since $\\sigma_{\\tiny\\sqrt s}^{}\\sim 2-3\\times\\Gamma_{\\h}^{\\rm tot}$, \nthe measured $\\mu^+\\mu^-\\toh\\to X$ rate is more or less proportional\nto $\\Gamma(h\\to\\mu^+\\mu^-)\/\\sigma_{\\tiny\\sqrt s}^{}$ so that $\\Gamma(h\\to\\mu^+\\mu^-)$\ncan be computed given the known $\\sigma_{\\tiny\\sqrt s}^{}$; small corrections\nfrom the influence of $\\Gamma_{\\h}^{\\rm tot}$ can be made using the measured $\\Gamma_{\\h}^{\\rm tot}$.}~\n$\\Gamma(h\\to\\mu^+\\mu^-)=$\n\n\\begin{tabular}{ll}\n 1)~ ${[\\Gamma(h\\to\\mu^+\\mu^-)BF(h\\to\nb\\overline b)]_{\\rm FMC}\\over BF(h\\to b\\overline b)_{\\rm NLC}}$;&\n 2)~ ${[\\Gamma(h\\to\\mu^+\\mu^-)BF(h\\to\nWW^\\star)]_{\\rm FMC}\\overBF(h\\to WW^\\star)_{\\rm NLC}}$;\\\\\n 3)~ ${[\\Gamma(h\\to\\mu^+\\mu^-)BF(h\\to\nZZ^\\star)\\Gamma_{\\hsm}^{\\rm tot}]_{\\rm FMC}\\over\\Gamma(h\\to ZZ^\\star)_{\\rm NLC}}$;&\n 4)~ ${[\\Gamma(h\\to\\mu^+\\mu^-)BF(h\\to\nWW^\\star)\\Gamma_{\\hsm}^{\\rm tot}]_{\\rm FMC}\\over\\Gamma(h\\to WW^\\star)_{\\rm NLC}}$.\\\\\n\\end{tabular}\n\n\\noindent\nIn the above, $\\Gamma_{\\h}^{\\rm tot}|_{\\rm FMC}$ refers to the scan measurement at the FMC.\nResulting errors for $h=h_{SM}$ are tabulated in Table~\\ref{nlcfmcerrors},\nlabelled $(\\mupmumh_{SM})^2|_{\\rm NLC+FMC}$.  The predicted\n$(\\mupmumh_{SM})^2|_{\\rm MSSM}\/(\\mupmumh_{SM})^2|_{\\rm SM}$ coupling-squared ratio\nis independent of $\\tan\\beta$: one finds\nvalues of 1.5, 1.2, $1.15$ for $m_{\\ha}=300$, 475, $600~{\\rm GeV}$, respectively (for\ntheoretically allowed $\\tan\\beta$ values at the given $m_{\\ha}$).\nSince there is\nno systematic error due to uncertainty in the muon mass\nand since experimental systematics should be well below 10\\%, \nthe above numbers imply that the expected $\\alt \\pm 5\\%$ statistical\nerror probes out to very high $m_{\\ha}$.\n\n\n\\begin{table}[h]\n\\tcaption{Errors for combining NLC ($L=200~{\\rm fb}^{-1}$) data, \nthe NLC + LHC ($L=600~{\\rm fb}^{-1}$,\nATLAS+CMS) $BF(h_{SM}\\to\\gamma\\gam)$ determination, \n$\\gamma\\gam$ collider ($L=50~{\\rm fb}^{-1}$) data and FMC $s$-channel ($L=200~{\\rm fb}^{-1}$) data.}\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n Quantity & \\multicolumn{4}{c|}{Errors} \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 80} & {\\bf 100} & {\\bf 110} & {\\bf 120} \\\\\n\\hline\n $(b\\overline bh_{SM})^2|_{\\rm NLC+FMC} $ & $\\pm6\\%$ & $\\pm 9\\%$ & $\\pm 7\\%$ &\n  $\\pm6\\%$ \\\\\n\\hline\n $(c\\overline ch_{SM})^2|_{\\rm NLC+FMC} $ & $\\pm9\\%$ & $\\pm 11\\%$ & $\\pm 10\\%$ &\n  $\\pm9\\%$ \\\\\n\\hline\n $(\\mupmumh_{SM})^2|_{\\rm NLC+FMC}$ & \n$\\pm 5\\%$ & $\\pm 5\\%$ & $\\pm 4\\%$ & $\\pm 4\\%$ \\\\\n\\hline\n $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2|_{\\rm FMC}$ & $\\pm 15\\%$ & $\\pm 16\\%$ & $\\pm 14\\%$ &\n $\\pm 13\\%$ \\\\\n\\hline\n $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2|_{\\rm NLC+FMC}$ & $\\pm 9\\%$ & $\\pm 10\\%$ & $\\pm 9\\%$ &\n $\\pm 9\\%$ \\\\\n\\hline\n\\hline\n{$\\bfm_{\\hsm}$}{\\bf (GeV)} & {\\bf 130} & {\\bf 140} & {\\bf 150} & {\\bf 170} \\\\\n\\hline\n $(b\\overline bh_{SM})^2|_{\\rm NLC+FMC}$ & $\\pm7\\%$ & $\\pm7\\%$ & $\\pm10\\%$ &\n $\\pm23\\%$ \\\\\n\\hline\n $(c\\overline ch_{SM})^2|_{\\rm NLC+FMC} $ & $\\pm10\\%$ & \\multicolumn{3}{c|}{$?$} \\\\\n\\hline\n $(\\mupmumh_{SM})^2|_{\\rm NLC+FMC}$ & \n$\\pm 3\\%$ & $\\pm 3\\%$ & $\\pm 4\\%$ &  $\\pm 10\\%$ \\\\\n\\hline\n $(W\\wstarh_{SM})^2|_{\\rm FMC}$ & $\\pm 16\\%$ & $\\pm 9\\%$ & $\\pm 9\\%$ &\n $-$ \\\\\n\\hline\n $(W\\wstarh_{SM})^2|_{\\rm NLC+FMC}$ & $\\pm 5\\%$ & $\\pm 4\\%$ & $\\pm 6\\%$ &\n $\\pm 10\\%$ \\\\\n\\hline\n $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2|_{\\rm FMC}$ & $\\pm 14\\%$ & $\\pm 18\\%$ & $\\pm 36\\%$ & $-$ \\\\\n\\hline\n $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2|_{\\rm NLC+FMC}$ & \n    $\\pm 10\\%$ & $\\pm 13\\%$ & $\\pm 23\\%$ & $-$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{nlcfmcerrors}\n\\end{table}\n\nTable~\\ref{nlcfmcerrors} gives a number of other quantities that are\ndetermined with remarkable precision by combining $\\sqrt s=500~{\\rm GeV}$ NLC data\nand $s$-channel FMC data\n[and, in the case of $(\\gamma\\Gamma_{\\h}^{\\rm tot}_{SM})^2$, including the (NLC+LHC) determination\nof $BF(h_{SM}\\to\\gamma\\gam)$].~\\cite{snowmass96}  The advantages\nof having high $L$ data from all three machines are enormous. \nIn particular, \nif there is a SM-like Higgs bosons with $m_{\\h}\\alt 2m_W$, \nit would be much more beneficial to have \nboth an $e^+e^-$ collider operating at full energy, $\\sqrt s\\sim 500~{\\rm GeV}$,\n{\\it and} a $\\mu^+\\mu^-$  collider operating at $\\sqrt s\\sim m_{\\h}$, as opposed\nto two NLC's.\n\n\\section{The Heavy MSSM Higgs Bosons}\n\nColliders other than the FMC offer various mechanisms\nto directly search for the $A^0,H^0$, but have significant limitations:\n\\begin{itemize}\n\\item There are regions in $(m_{\\ha},\\tan\\beta)$ parameter space at moderate\n$\\tan \\beta$, $m_{\\ha}\\agt 200~{\\rm GeV}$ in which the $H^0,A^0$ cannot be detected\nat the LHC.\n\\item At the NLC one can use the mode $e^+e^-\\to Z^\\star\\to H^0A^0$,\nbut it is limited to $m_{\\hh}\\sim m_{\\ha}\\alt \\sqrt{s}\/2$.\n\\item A $\\gamma \\gamma$ collider could probe heavy Higgs up to masses of\n$m_{\\hh}\\sim m_{\\ha}\\sim 0.8\\sqrt s$, but this would quite likely require\n$L> 100~{\\rm fb}^{-1}$, especially if the Higgs bosons are at the upper\nend of the  $\\gamma \\gamma$ collider energy spectrum.~\\cite{ghgamgam}\n\\end{itemize}\nIn contrast, there is an excellent chance of being able to detect\nthe $H^0,A^0$ at a $\\mu^+\\mu^-$ collider provided only that $m_{\\ha}$ is smaller\nthan the maximal $\\sqrt s$ available. This could prove to be very important\ngiven that GUT MSSM models usually predict $m_{\\ha}\\agt 200~{\\rm GeV}$.\n\nA detailed study of $s$-channel production\nof the $H^0,A^0$ has been made.~\\cite{bbgh} \nThe signals become viable when $\\tan\\beta>1$\n(as favored by GUT models) since the $\\mupmumH^0$ and $\\mupmumA^0$\ncouplings are proportional to $\\tan\\beta$. In particular, \neven though $\\Gamma_{\\hh}^{\\rm tot},\\Gamma_{\\ha}^{\\rm tot}$ are big (see Fig.~\\ref{hwidths}) at high $\\tan\\beta$, \ndue to large $b\\overline b$ decay widths, $BF(H^0,A^0\\to\\mu^+\\mu^-)$\napproaches a constant value that is large enough to imply\nsubstantial $\\overline \\sigma_{\\hh},\\overline \\sigma_{\\ha}$. The optimal strategy \nfor $H^0,A^0$ detection and study depends upon the circumstances.\nFirst, it could be that the $H^0$ and\/or $A^0$ will already have been\ndiscovered at the LHC.  \nWith $L=300~{\\rm fb}^{-1}$ (ATLAS+CMS) of integrated           \nluminosity, this would be the case if $\\tan\\beta\\alt 3$ or $\\tan\\beta$ \nis above an $m_{\\ha}$-dependent lower bound ({\\it e.g.}\\ $\\tan\\beta\\agt 10$ for $m_{\\ha}\\sim\n400~{\\rm GeV}$).~\\footnote{For \n$\\tan\\beta\\alt 3$, one makes use of modes such as $H^0\\toh^0\\hl\\to\nb\\overline b \\gamma\\gam$ and $H^0 \\to ZZ^{(*)}\\to 4\\ell$, when $m_{\\hh}\\alt 2m_t$,\nor $H^0,A^0\\to t\\overline t$, when $m_{\\hh},m_{\\ha}\\agt 2m_t$.  At high $\\tan\\beta$,\nthe enhanced production rates for $b\\overline b H^0,b\\overline bA^0$ with\n$H^0,A^0\\to\\tau^+\\tau^-$ are employed.}~\nEven if the $H^0,A^0$ have not been detected,\nstrong constraints on $m_{\\ha}$ are possible if precision measurements\nof the properties of the $h^0$ (such as the $b\\overline b\/WW^\\star$\nand $c\\overline c\/b\\overline b$ event rate ratios and the $(\\mupmumh^0)^2$\ncoupling-squared, as discussed earlier) are made via $s$-channel production\nat the FMC or in $\\sqrt s=500~{\\rm GeV}$ running at the NLC~\\cite{snowmass96} or\nby combining these two types of data.\nBy limiting the $\\sqrt s$ scan for the $H^0$ and $A^0$\nin the $s$-channel to the $m_{\\ha}\\simm_{\\hh}$ \nmass region preferred by $h^0$ measurements,\nwe would greatly reduce the luminosity needed to\nfind the $A^0$ and $H^0$ via an $s$-channel scan as compared to that required\nif $m_{\\ha}$ is not constrained.\n\nWith such pre-knowledge\nof $m_{\\ha}$, it will be possible to detect and perform detailed\nstudies of the $H^0,A^0$ for all $\\tan\\beta\\geq 1$\nprovided only that $m_{\\ha}\\alt \\sqrt s_{\\rm max}$.~\\footnote{We\nassume that a final ring optimized for maximal luminosity at $\\sqrt s\\sim m_{\\ha}$\nwould be constructed.}~ If $\\tan\\beta\\alt 3$ and $m_{\\hh},m_{\\ha}\\alt 2m_t$, then\nexcellent resolution, $R\\sim 0.01\\%$, will be necessary for detection\nsince the $A^0$ and $H^0$ are relatively narrow (see Fig.~\\ref{hwidths}).\nFor higher $\\tan\\beta$ values, $R\\sim 0.1\\%$ is adequate\nfor $H^0,A^0$ detection, but $R\\sim 0.01\\%$ would be required in order\nto separate the rather overlapping $H^0$ and $A^0$ peaks (as a function\nof $\\sqrt s$) from one another.\n\n\nEven without pre-knowledge of $m_{\\ha}$, \nthere would be an excellent chance for discovery of the $A^0,H^0$\nHiggs bosons in the $s$-channel at a $\\mu^+\\mu^-$ collider if they\nhave not already been observed at the LHC.\nThis is because non-observation at the LHC implies \nthat $\\tan\\beta\\agt 3$ while it is precisely for\n$\\tan\\beta\\agt 2.5-3$ that detection of the $A^0,H^0$ is possible~\\cite{bbgh}\nin the mass range from 200 to 500 GeV via an $s$-channel scan in \n$\\mu^+\\mu^-$ collisions. (The lower $\\tan\\beta$ reach given\nassumes that $L_{\\rm tot}=200~{\\rm fb}^{-1}$ is devoted to the scan.\nA detailed strategy as to how\nmuch luminosity to devote to different $\\sqrt s$ scan settings\nin the $200-500~{\\rm GeV}$ range must be employed.~\\cite{bbgh})\nThat the LHC and the FMC are complementary in this\nrespect is a very crucial point. Together, the LHC and FMC\nguarantee discovery of the $A^0,H^0$ after 3 to 4 years of\nhigh luminosity operation each, provided $m_{\\ha}\\alt 500~{\\rm GeV}$.\nOnce $m_{\\ha},m_{\\hh}$ are known, very precise measurements of some of\nthe crucial properties of the $H^0,A^0$ \n(including a scan determination of their\ntotal widths) become possible.~\\cite{bbgh}\n\n\nIn the event that the NLC has not been constructed, it could be\nthat the first mode of operation of the FMC would be to optimize\nfor and accumulate luminosity at, say, $\\sqrt s=500~{\\rm GeV}$.\nIn this case, there is still a high probability\nfor detecting the $H^0,A^0$ if they have not been\nobserved at the LHC. First, if $m_{\\ha}\\simm_{\\hh}\\alt \\sqrt s\/2\\sim 250~{\\rm GeV}$ then\n$\\mu^+\\mu^-\\to H^0A^0$ (and $H^+H^-$) pair production will be observed. Second,\nalthough reduced in magnitude compared to an electron\ncollider, there is a long low-energy bremsstrahlung tail \nat a muon collider that provides a\nself-scan over the full range of $\\sqrt s$ values below \nthe nominal operating energy.\nObservation of $A^0,H^0$ $s$-channel peaks in the $b\\overline b$ mass \n($m_{b\\overline b}$) distribution\ncreated by this bremsstrahlung tail may be possible.\nThe region of the $(m_{\\ha},\\tan\\beta)$ parameter space plane for which\na peak is observable depends strongly on the $m_{b\\overline b}$\nresolution. For excellent $m_{b\\overline b}$\nresolution of order $\\pm 5~{\\rm GeV}$ and integrated luminosity\nof $L=200~{\\rm fb}^{-1}$ at $\\sqrt s=500~{\\rm GeV}$, \nthe $A^0,H^0$ peak(s) are observable for $\\tan\\beta\\agt 4-5$\nif $500~{\\rm GeV}\\geqm_{\\ha}\\geq 250~{\\rm GeV}$.~\\footnote{Required $\\tan\\beta$ values increase\ndramatically as one moves into the $m_{\\ha}\\sim m_Z$ zone,\nbut this region is covered by $H^0A^0$ pair production.}\n\n\n\nFinally, if neither the LHC nor a FMC scan of the $\\leq 500~{\\rm GeV}$ region\nhas discovered the $H^0,A^0$, but supersymmetric particles and the $h^0$\nhave been observed, we would believe that the $H^0,A^0$ must exist\nbut have $m_{\\ha}\\simm_{\\hh}\\geq\n500~{\\rm GeV}$ . Analyses of the SUSY spectrum in the GUT context\nand precision $h^0$ studies\nmight have yielded some prejudice for the probable $m_{\\ha}$, and\nan extension of the FMC energy up to the appropriate $\\sqrt s\\sim m_{\\ha}$ \nfor $s$-channel discovery of the $H^0,A^0$\ncould be considered. But,\na machine with much higher $\\sqrt s$, such as the earlier-mentioned $\\sqrt s=4~{\\rm TeV}$,\nmight be most worthwhile.\nIt has been shown~\\cite{gk} \nthat such an energy with appropriately matched luminosity\nwould allow discovery of $\\mu^+\\mu^-\\to\\haH^0$ \nand $H^+H^-$ pair production,\nvia the $b\\overline b$ or $t\\overline t$ decay channels of the $H^0,A^0$\nand $t\\overline b,\\overline t b$ decay channels of the $H^+,H^-$,\nup to masses very close to $m_{\\ha}\\sim m_{\\hh}\\sim m_{\\hpm} \\sim\n2~{\\rm TeV}$, even if SUSY decays of the $H^0,A^0,H^{\\pm}$ are substantial. \n(This mass range certainly includes that expected in\nany supersymmetric model that provides a solution to the\nnaturalness and hierarchy problems.) \nDetailed studies of the $H^0,A^0,H^{\\pm}$ would be possible \nonce they were discovered.~\\cite{gk,fengmoroi}\n\n\n\\section{Conclusions and Discussion}\n\nIn this report, we have briefly reviewed \nthe capabilities of a muon collider to explore Higgs physics\nin the Standard Model and its minimal supersymmetric extension\nvia direct $s$-channel Higgs production. If there is a light ($m_{\\h}\\alt 2m_W$)\nSM-like Higgs boson, an $s$-channel scan of the Higgs resonance\npeak provides direct measurements of a number of its important \nproperties (in particular $\\Gamma_{\\h}^{\\rm tot}$)\nthat cannot be duplicated using $e^+e^-$ (or $\\mu^+\\mu^-$) \ncollisions at high $\\sqrt s$. Thus, if a light SM-like Higgs boson\nhas already been observed at the LHC or in early operation at the first NLC,\nit would be much more useful to follow the first NLC with an FMC,\nrather than a second NLC.\nIn the context of the minimal supersymmetric extension of the SM,\nthe FMC takes on added value.  At the FMC, \nan appropriately designed $s$-channel scan will allow $H^0,A^0$ discovery \nin the $200\\alt m_{\\ha}\\alt 500~{\\rm GeV}$ range for all $\\tan\\beta\\agt 3$. Thus, if\n$H^0A^0$ pair production is not seen in $\\sqrt s=500~{\\rm GeV}$\nrunning at the NLC (or FMC), implying $m_{\\ha}\\agt 200-240~{\\rm GeV}$,\nnor at the LHC, implying $\\tan\\beta\\agt 3$ (and below an \n$m_{\\ha}$-dependent upper limit) if $m_{\\ha}\\agt 200~{\\rm GeV}$,\nthe $H^0,A^0$ {\\it will} be observed at the FMC (provided $m_{\\ha}\\alt 500~{\\rm GeV}$).\nOnce discovered (at any collider), the FMC $s$-channel \nproduction mode allows (for any $\\tan\\beta>1$)\na detailed study of some of their key properties\nand a scan determination of their total widths. As reviewed\nelsewhere,~\\cite{bbgh,snowmass96,dpfreport,perspectives97} a muon collider will\nbe at least as valuable if the Higgs sector is still more\nexotic than the constrained two-doublet MSSM sector. Particularly noteworthy\nis the ability of $\\mu^-\\mu^-$ collisions to probe the\n$\\mu^-\\mu^-$ coupling of a doubly-charged\nHiggs boson~\\footnote{Such a Higgs is\nvery likely to be seen at the LHC if it has mass below 1 TeV.~\\cite{glp}}~\n(as present in many Higgs triplet models)\ndown to an extraordinarily small coupling magnitude. Overall,\nif there are elementary Higgs bosons, a muon collider will\nalmost certainly be mandated\npurely on the basis of its ability to explore the Higgs sector\nvia $s$-channel factory-like production of Higgs bosons.\n\n\n\\section{Acknowledgements}\nI would like to acknowledge the many contributions\nof my collaborators, V. Barger, M. Berger, and T. Han,\nto the muon collider Higgs physics results. This work was supported\nin part by the Department of Energy and by the Davis Institute for\nHigh Energy Physics.\n\n\\section{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}