diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzrml" "b/data_all_eng_slimpj/shuffled/split2/finalzrml" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzrml" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nWe consider the stochastic Swift-Hohenberg equation on the whole real line.\nThis is one of the prototypes of pattern forming equations and its first instability is supposed \nto be a toy model for the convective instability in Rayleigh-B\\'enard convection.\nIt is given by \n\\begin{equation}\n \\label{e:SHintro}\n \\partial_t u = -(1+\\partial_x^2)^2 u +\\nu\\varepsilon^2 u -u^3 + \\sigma \\varepsilon^{3\/2} \\xi\n\\end{equation}\nwith space-time white noise $\\xi$. \nHere $\\nu\\in\\mathbb{R}$ measures the distance from bifurcation, which scales with $\\varepsilon^2$ \nand $\\sigma\\geqslant0$ measures the noise strength that scales with $\\varepsilon^{3\/2}$, \nfor a small $0\\leqslant \\varepsilon\\ll 1$.\nWe will see later that the scaling is in such a way that close to the bifurcation both terms have an impact on the dynamics. \n\nDue to the presence of the noise we run into several problems. \nFirst, solutions have very poor regularity properties and solutions are at most H\\\"older continuous. \nThus we need to consider weaker concepts of solutions like the mild formulation of the equation. \nMoreover, due to translation invariance of the noise solutions are in general immediately unbounded in space,\nand we need to work in spaces that do allow for growth of solutions for $|x|\\to\\infty$.\nThese weighted spaces are not closed under pointwise multiplication, which is a serious problem in the construction of solutions\ndue to the cubic nonlinearity.\n\nOur main results show that \nclose to the bifurcation, i.e. for small $\\varepsilon>0$, solutions of \\eqref{e:SHintro} are well approximated by a modulated wave \n\\begin{equation*}\n u (t,x) \\approx \\varepsilon A(\\varepsilon^2 t,\\varepsilon x) e^{ix} + c.c. \n\\end{equation*}\nwhere the amplitude $A$ solves a so called modulation or amplitude equation, which is in our case a stochastic complex-valued Ginzburg-Landau equation\n\\[\n\\partial_T A = 4 \\partial_X A +\\nu A -3A|A|^2 + \\eta \n\\]\nfor some complex-valued space-time white noise $\\eta$.\n\n\\subsection{Modulation equations for deterministic PDEs}\nThe Ginzburg-Landau equation as an effective amplitude equation for the \ndescription of pattern forming systems close to the first instability \nhas first been derived in the 1960s by Newell and Whitehead, cf.~\\cite{NW69}.\nThe mathematical justification of this approach beyond pure formal calculations \nhas been done by Mielke and Melbourne together with coauthors \neither with the help of a Lyapunov-Schmidt reduction (see \\cite{Mi92,Mel98,Mel00}),\nor with the construction of special solutions, cf.~\\cite{IMD89}. \nApproximation results showing that there are solutions of the pattern-forming system \nwhich behave as predicted by the Ginzburg-Landau equation has been shown by various authors for instance in \n\\cite{CE90,vH91,KSM92,Schn94a,Schn94c,Takac96}. \nMoreover, there are attractivity results by Eckhaus \\cite{Eck93} and Schneider \\cite{Schn95}, \nshowing that every small solution can be described after a certain time by the Ginzburg-Landau equation. \nVarious results followed in subsequent years: combining the approximation and attractivity results allows to prove the upper semi-continuity of attractors~\\cite{MS:95,Schn99c}, \nshadowing by pseudo-orbits, and global existence results for the pattern-forming systems \\cite{Schn94b,Schn99b}.\nA number of approximation theorems have been proven in slightly modified situations, such as the\ndegenerated case of a vanishing cubic coefficient \\cite{BS07}, the Turing-Hopf case description\nby mean-field coupled Ginzburg-Landau equations \\cite{Schn97}, \nthe Hopf bifurcation at the Fourier wave number k = 0 \\cite{Schn98}, and \nthe time-periodic situation \\cite{SU07}. \nRecently, such results have been established in case of pattern forming \nsystems with conservation laws, too, cf. \\cite{HSZ11,SZ13,DSSZ16}.\nLet us finally point out that this section is just a brief summary of those of the numerous deterministic results existing in the literature which are most closely related to the one presented here.\n\\subsection{SPDEs in weighted spaces on unbounded domains}\n\nThe theory of higher order parabolic stochastic partial differential equations (SPDEs) \non unbounded domains with translation invariant additive noise \nlike space-time white noise is not that well studied, \nwhile for the wave equation with multiplicative noise there are many recent publications. \nSee for example~\\cite{Kho14, DaSS09, Dal09}.\n\nIn older publications often only noise with a spatial cut off or a decay condition at infinity is treated, \nas for example by Eckmann and Hairer~\\cite{EckHai2001}, where the cutoff is in real and in Fourier space, or by Funaki~\\cite{Fun1995}.\nFurthermore, Rougemont \\cite{Ro02} studied the stochastic Ginzburg-Landau using \nexponentially weighted spaces and relatively simple noise that is white in time, but bounded in space.\n\nIn many examples, using trace class noise implies $L^2$-valued Wiener processes \nand thus a decay condition both of solutions and of the noise at infinity. \nThis leads to $L^2$-valued solutions, as for example \nby Brzezniak and Li~\\cite{BreLi2006} or by Kr\\\"uger and Stannat~\\cite{KruSta2014}, where an integral equation is considered.\n In the next paragraph we will comment on the fact that a decay at infinity rules out the effect we want to study here \n using modulation equations.\n\nThe stochastic Ginzburg-Landau Equation in a weighted $L^2$-space was already studied by Bl\\\"omker and Han \\cite{BH:10}. \nThe existence and uniqueness result based on a Galerkin-Approximation is briefly sketched there \nand the asymptotic compactness of the stochastic dynamical system is shown.\n\nRecently, several publications treat SPDEs with space-time-white noise in weighted Besov spaces: \nsee for example R\\\"ockner, Zhu, and Zhu\n\\cite{RoeZhZh:P15} or Mourrat and Weber\\cite{MoWe:P15}. \nThey work with the two-dimensional $\\Phi^4$-model, which is similar to Ginzburg-Landau and \nwhere renormalization is needed to give a meaning to the two-dimensional equation with this choice of noise.\nIn order to construct solutions they consider approximations on the torus and then send the size of the domain to infinity,\nwhich is the method also used in this paper. But the authors work directly in weighted Besov spaces, \nwhile we show our existence and uniqueness result in spaces with less regularity. \nMoreover the result in Besov spaces relies heavily on properties of the heat-semigroup, \nwhich do not seem to hold for fourth order operators like the Swift-Hohenberg operator.\nFor example we will see later, that the operator is not dissipative in weighted $L^p$-spaces, while the Laplacian is.\nThus we cannot derive useful a-priori bounds for Swift-Hohenberg in $L^p$-spaces.\nThis is also the reason that our final approximation result is only valid in a weighted $L^2$-space, \nwhile the residual is bounded in spaces with H\\\"older regularity.\n\nLet us finally remark that spaces without weight like $L^2(\\mathbb{R})$ and the usual Sobolev spaces do not include constant functions and \nmodulated pattern, that do appear close to the bifurcation, and which we want to study here using modulation equations. \nIn order to include these special solutions one needs to consider weighted spaces, see for example~\\cite{BLW:13} or~\\cite{BBY:16}\nfor publications treating random attractors.\n\n \\subsection{Modulation equations for SPDEs}\n \n\nIn Bl\\\"omker, Hairer and Pavliotis \\cite{BHP:05} modulation equations for SPDEs on large domains were treated.\nThe results are quite similar to the ones presented here, but they hold only on large domains of size proportional to $1\/\\varepsilon$ for Swift-Hohenberg \nand thus the Ginzburg-Landau equation is posed on a domain of order $1$. \nThe main advantage is that one can still work with Fourier series, \nand only finitely many modes change stability at the bifurcation. \nMoreover, solutions of the amplitude equation are not unbounded in space and there is no need to consider weighted spaces. \nThe drawback is that various constants depend on the size of the domain and the results do not extend to unbounded domains.\n\nThe first results for modulation equations for Swift-Hohenberg on the whole real line were presented by Klepel, Mohammed, and Bl\\\"omker \n\\cite{MBK:13,KBM:13}. Here the authors used spatially constant noise of a strength of order $\\varepsilon$, which is stronger than the one treated here.\nAlthough the noise does not appear \ndirectly in the amplitude equation, due to nonlinear interaction and averaging \nadditional deterministic terms appear in the Ginzburg-Landau equation. \nDue to the spatial regularity of the noise, the main advantage is that one can work in spaces with much more spatial \nregularity than we have to use here.\nAs a consequence, solutions are still bounded in space and do not grow towards infinity at $|x|\\to\\infty$.\n\n\nThe key result towards a full result for amplitude equations \non the whole real line with space-time white noise \nis by Bl\\\"omker and Bianchi~\\cite{BB:15}. Here the full approximation result for linear SPDEs, namely the Swift-Hohenberg and Ginzburg-Landau equations without cubic terms,\nis established. \nThis is very useful in the results presented here, as we use it to approximate \nthe stochastic convolutions in the mild formulation.\n\nLet us finally remark that a decay at infinity of the noise and thus the solution leads to a completely different result.\nUnder the rescaling in space used to obtain the modulation equation,\nwe conjecture to finally obtain a point-forcing at the origin in the Ginzburg-Landau equation, \nwhich is an interesting question in itself.\n\n\n\n\n\n\\subsection{Outline of the paper and main results}\n\n \nIn Section \\ref{sec:set} we introduce basic notation and especially the weighted spaces we are going to work in.\nThe existence and uniqueness of solutions to the Swift-Hohenberg and the Ginzburg-Landau equations is \nbriefly sketched in Section~\\ref{sec:exuni} and again at the beginning of Section~\\ref{sec:reg}.\nThe main results are Theorem~\\ref{thm:exSH} and~\\ref{thm:exGL}, where the existence and uniqueness of solutions is proven, \nfor the Swift-Hohenberg and the Ginzburg-Landau equations respectively.\nFor the proof we use the approximation by finite domains with periodic boundary conditions.\n\nA key technical point is the result of Corollary \\ref{cor:maxregA} in Section \\ref{sec:reg} \nwhere we show that the solution of the amplitude equation \nis H\\\"older up to exponent almost $1\/2$ in space, which is more regularity than we can show for solutions of the Swift-Hohenberg equation.\nThe main idea here is to introduce the standard transformation $B=A-\\mathcal{Z}$ \nusing the Ornstein-Uhlenbeck (OU) process $\\mathcal{Z}$ that solves the stochastic linear Ginzburg-Landau equation and is thus Gaussian. \n\nThe H\\\"older-regularity of $\\mathcal{Z}$ is a well-known result (see Lemma~\\ref{lem:regZ}).\nThe key idea of the transformation is that $B$ solves a random PDE and we can apply energy type estimates in $L^p$- and $W^{1,p}$-spaces for any $p\\geq2$, \nwhich show that $B$ is more regular than $\\mathcal{Z}$ and thus $A$ is as regular as $\\mathcal{Z}$.\n\nLet us remark, that these $L^p$-estimates are not available for the Swift-Hohenberg equation, where we only have $L^2$- or $H^1$-estimates.\nThus we do not know how to establish higher regularity for solutions in that case.\n \n \nThe main approximation result for the amplitude equation is Theorem \\ref{thm:res} in Section \\ref{sec:res}, \nwhere we bound the residual of the approximation uniformly for times up to order $\\varepsilon^{-2}$ in a weighted $C^0$-space. \n\nIn the final Section \\ref{sec:app} we establish in Theorem \\ref{thm:final} the approximation result\n again uniformly for times up to order $\\varepsilon^{-2}$ but now only in a weighted $L^2$-space, as we use $L^2$-energy \n estimates for an equation for the error.\n\n\n\n \\section{Setting}\n \\label{sec:set}\n \n\n \n Consider the stochastic Swift-Hohenberg equation\n \\begin{equation}\n \\label{e:SH}\n \\tag{SH}\n \\partial_t u = {\\mathcal{L}}_\\nu u -u^3 +\\sigma \\partial_t W\\;, \\quad u(0)=u_0\n \\end{equation}\non $\\mathbb{R}$ with a standard cylindrical Wiener process $W$ in $L^2(\\mathbb{R})$,\nwhich means that $\\partial_t W$ is space-time white noise, and the operator ${\\mathcal{L}}_\\nu= -(1+\\partial_x^2)^2 + \\nu\\varepsilon^2 $, where $\\sigma$ and $\\nu$ are constants.\nIn the following, we will also consider ${\\mathcal{L}}_0= -(1+\\partial_x^2)^2 $.\n\nDue to lack of regularity \\eqref{e:SH} is not defined classically. \nIn order to give a rigorous meaning, we use the standard transformation to a random PDE.\nWe define the stochastic convolution \n\\[\nZ(t)= W_{{\\mathcal{L}}_\\nu}(t) = \\int_0^t e^{(t-s){\\mathcal{L}}_\\nu}dW(s)\\;.\n\\]\nWe will see later that $Z$ is for any $\\varrho>0$ in the weighted space $C^0_\\varrho$ of continuous functions defined in the next section in \\eqref{e:C0rho}.\nNote that $\\mathcal{Z}$ is the OU-process corresponding to the Ginzburg-Landau equation.\n\n\nIn order to give a meaning to~\\eqref{e:SH} we define $v=u-Z$ and consider weak solutions of \n\\begin{equation}\n \\label{e:SHt}\n \\partial_t v= {\\mathcal{L}}_\\nu v - (v+Z)^3\\;, \\quad v(0)=u_0\\;. \n\\end{equation}\n\n\n\\begin{definition}[Weak solution]\n\\label{def:weak}\nWe call an $L^3_{\\text{loc}}(\\mathbb{R})$-valued stochastic process $v$ with integrable trajectories a weak solution of \\eqref{e:SHt} \nif for all smooth and compactly supported functions $\\varphi$ one has with probability $1$ that for all $t\\in[0,T]$\n\\[\n\\int_\\mathbb{R} v(t)\\varphi dx = \\int_\\mathbb{R} u_0\\varphi dx + \\int_0^t\\int_\\mathbb{R} v(s) {\\mathcal{L}}_\\nu\\varphi dx - \\int_0^t\\int_\\mathbb{R}(v(s)+Z(s))^3\\varphi dx \\;. \n\\] \n\\end{definition}\n\n\\begin{remark}\nA sufficiently regular weak solution of~\\eqref{e:SHt} is a mild solution of~\\eqref{e:SHt}\ngiven by the integral equation\n\\[\nv(t)=e^{t{\\mathcal{L}}_\\nu}v(0) - \\int_0^t e^{(t-s){\\mathcal{L}}_\\nu}(v(s)+Z(s))^3 ds\n\\]\nwhich is under the substitution $v=u-Z$ a mild solution of~\\eqref{e:SH}. \nThis concept is also known as Duhamel's formula or variation of constants.\nThe equivalence of mild and weak solutions under the assumption of relatively weak regularity can be \nfound for example in~\\cite{paz83,lun95}. \nThe transfer to our situation is straightforward. \n\\end{remark}\n\n\\begin{remark}\n The main problem of mild solutions for existence and uniqueness of solutions is the following. \n In weighted spaces $C^0_\\varrho$ of continuous functions with weights decaying to $0$ at infinity, which are defined in the next section, \n we cannot use the direct fixed point argument for mild solutions, \n as the cubic nonlinearity is an unbounded operator on $C^0_\\varrho$, as it maps\n $C^0_\\varrho$ to~$C^0_{3\\varrho}$. We always have to cube the weight, too.\n Thus one can show that the right hand side of the mild formulation can not be a contraction in $C^0_\\varrho$.\n\n\n Similar problems appear for other weighted spaces, as solutions are allowed to be unbounded in space.\n Thus later in the paper we use the weak formulation to prove existence and uniqueness, \n and then the mild formulation to verify error estimates.\n\\end{remark}\n\n\n\\subsection{Spaces}\n\n\nFor $\\varrho\\in\\mathbb{R}$, denote by $C^0_\\varrho$ the space of continuous functions $v: \\mathbb{R} \\to \\mathbb{R}$ \nsuch that the following norm is finite\n\\begin{equation}\n \\label{e:C0rho}\n \\|v\\|_{C^0_\\varrho} = \\sup_{x\\in\\mathbb{R}} \\Big\\{ \\frac1{(1+x^2)^{\\varrho\/2}} |v(x)| \\Big\\} \\;.\n\\end{equation}\nThis is a monotone increasing sequence of spaces of continuous functions with growth condition at $\\pm\\infty$ for $\\varrho>0$.\nSee also Bianchi, Bl\\\"omker \\cite{BB:15}. \n\n\\begin{definition}[Weights]\n \\label{def:weight}\n We define for $\\varrho>0$ the weight function $w_\\varrho(x)= (1+x^2)^{-\\varrho\/2}$.\n We also define for $c>0$ the scaled weight function $w_{\\varrho,c}(x) = w_\\varrho(cx) = (1+c^2x^2)^{-\\varrho\/2}$\\;.\n\\end{definition}\nWe have the following properties \n\\begin{equation}\\label{eq:propweight}\n|w_\\varrho'(x)| \\leqslant C w_\\varrho(x)\\;,\n\\quad \n|w_\\varrho^{(n)}(x)| \\leqslant C_n w_\\varrho(x)\n\\quad\\text{and}\\quad \n|w_{\\varrho,c}^{(n)}(x)| \\leqslant C_n c^n w_{\\varrho,c}(x)\n\\;.\n\\end{equation}\nMoreover, $w_\\varrho\\in L^1(\\mathbb{R})$ if and only if $\\varrho>1$.\n\nLet us remark that we have the equivalence of norms (see Lemma 2.1 of \\cite{BB:15})\n\\begin{equation}\n \\label{e:equivC0}\n c \\|v\\|_{C^0_\\varrho} \n \\leqslant \n \\sup_{L\\geqslant 1} \\{ L^{-\\varrho} \\|v\\|_{C^0([-L,L])}\\}\n \\leqslant C\n \\|v\\|_{C^0_\\varrho},\n\\end{equation}\nwith strictly positive constants $c$ and $C$.\n\nWe can also define as in Bates, Lu, Wang~\\cite{BLW:13}\nweighted spaces for integrable functions.\n\\[\nL^p_\\varrho = \\{ u \\in L^p_{\\text{loc}}(\\mathbb{R}) \\ :\\ uw_\\varrho^{1\/p} \\in L^p(\\mathbb{R})\\}\n\\]\nwith norm\n\\[\n\\|u\\|_{L^p_\\varrho} = \\Big(\\int_\\mathbb{R} w_\\varrho(x)|u(x)|^p dx\\Big)^{1\/p} \\;.\n\\]\nMoreover, we need weighted Sobolev spaces $W^{k,p}_\\varrho$ and $H^k_\\varrho:=W^{k,2}_\\varrho$ defined by the norm\n\\[\n\\|u\\|_{W^{k,p}_\\varrho} := \\Big( \\sum_{\\ell=0}^k \\|\\partial_x^\\ell u\\|^p_{L^p_\\varrho} \\Big)^{1\/p}\\;. \n\\]\nAs $1 \\leqslant L^{\\varrho} w_\\varrho(x)$ on $[-L,L]$, it is easy to check that,\n\\begin{equation}\n \\label{e:Wkpbound}\n \\sup_{L\\geqslant 1} \\{ L^{-\\varrho\/p} \\|v\\|_{W^{k,p}([-L,L])}\\}\n \\leqslant C \\|v\\|_{W^{k,p}_\\varrho}\\;.\n\\end{equation}\nIn general this is not an equivalence of norms as the opposite inequality is not true.\nNote finally that for $\\varrho>1$ we have an integrable weight $w_\\varrho\\in L^1(\\mathbb{R})$ and thus \nby H\\\"older inequality for all $k\\in\\mathbb{N}$, $p\\geqslant1$ and $\\delta>0$ the embedding\n\\[\nW^{k,p+\\delta}_\\varrho \\subset W^{k,p}_\\varrho\\;. \n\\]\nNote that this is false for $\\varrho<1$ which includes the case of no weight ($\\varrho=0$).\n\nWe also \ndefine weighted H\\\"older spaces $C^{0,\\eta}_\\kappa$ of locally H\\\"older continuous functions such that the following norm is finite:\n\\begin{equation}\n \\label{def:wHoe}\n \\|A\\|_{ C^{0,\\eta}_\\kappa} = \\sup\\{ L^{-\\kappa}\\|A\\|_{C^{0,\\eta}[-L,L]} \\ : \\ L>1\\}\\;.\n\\end{equation}\nThis is the natural space for solutions of the SPDE, as the stochastic convolution $Z$ \nwill be in such spaces. See for example Lemma~\\ref{lem:regZ} later.\n\n\n\n\n\n\\section{Existence and Uniqueness of solutions}\n\\label{sec:exuni}\n\n\nHere in the presentation we mainly focus on the Swift-Hohenberg equation and state later \nthe analogous result for the Ginzburg-Landau equation without proof, as they are very similar.\nMoreover, there is already the result of Mourrat and Weber \\cite{MoWe:P15} for the two-dimensional real-valued Ginzburg-Landau \n(or Allen-Cahn) equation that is a similar from the technical point of view, although it is proven in Besov spaces.\nThe main result of this section is:\n\n\\begin{theorem}\n \\label{thm:exSH} \nFor all $u_0\\in L^2_\\varrho$ and $T>0$ with $\\varrho>3$ there is a stochastic process such that $\\mathbb{P}$-almost surely \n\\[\nv\\in L^\\infty(0,T,L^2_\\varrho) \\cap L^2(0,T,H^2_\\varrho) \\cap L^4(0,T,L^4_\\varrho) \n\\]\nand $v$ is a weak solution of \\eqref{e:SHt} in the sense of Definition \\ref{def:weak}. \nMoreover, for any other such weak solution $\\tilde{v}$ we have\n\\[\n\\mathbb{P}\\Big( \\sup_{t\\in[0,T]}\\|v(t)- \\tilde{v}(t)\\|_{L^2_\\varrho}=0 \\Big) =1\\;.\n\\]\n\\end{theorem}\n\n\\begin{remark} \\label{rem:rho3}\nAs we are looking at periodic solutions, the weight $w_\\varrho$ with $\\varrho>3$ has to decay sufficiently fast, so that it guarantees \nthat all boundary terms at $\\pm\\infty$ arising in integration by parts formula \nin the following proof do all vanish.\n\\end{remark}\n\n\nFor the relatively straightforward proof of Theorem \\ref{thm:exSH} we could follow some ideas of \\cite{BH:10} for the Ginzburg-Landau equation,\nwhere a Galerkin method based on an orthonormal basis of $L^2_\\varrho$ was used.\nBut here we consider the approximation using finite domains and periodic boundary conditions. \nThis is a fairly standard approach also presented in~\\cite{MoWe:P15} for the $\\Phi^4$-model, which is similar to the Ginzburg-Landau equation.\nNevertheless, the approach of \\cite{MoWe:P15} in Besov spaces does not seem to work for the Swift-Hohenberg equation, \nas we are for example not able to establish a-priori bounds in Besov spaces.\n\n\nLet $v^{(n)}$ be a $2n$-periodic solution of~\\eqref{e:SHt} on $[-n,n]$ with initial condition \n$v^{(n)}(0)=u_0|_{[-n,n]} $ and forcing $Z^{(n)}=Z|_{[-n,n]}$ both $2n$-periodically extended.\n\nBy standard parabolic theory there is for all $n\\in\\mathbb{N}$ a $2n$-periodic solution \n\\[\nv^{(n)}\n \\in L^\\infty(0,T,L^2([-n,n])) \\cap L^2(0,T,H^2([-n,n])) \\cap L^4(0,T,L^4([-n,n]))\\;,\n\\]\nwhich extends by periodicity to the whole real line $\\mathbb{R}$.\n\nUsing a weight $\\varrho>3$, so that all integrals and integrations by parts are well defined,\nwe obtain:\n\\begin{align}\n\\label{e:aprio}\n \\frac12\\partial_t \\|v^{(n)}\\|^2_{L^2_\\varrho} \n & = \\int_\\mathbb{R} w_\\varrho v^{(n)} \\partial_t v^{(n)} dx \\nonumber\\\\\n &\\leqslant \\int_\\mathbb{R} w_\\varrho v^{(n)} {\\mathcal{L}}_\\nu v^{(n)} dx - \\int_\\mathbb{R} w_\\varrho v^{(n)}(v^{(n)}+Z^{(n)})^3 dx \\\\\n &\\leqslant \\int_\\mathbb{R} w_\\varrho v^{(n)} {\\mathcal{L}}_\\nu v^{(n)} dx - \\frac12 \\int_\\mathbb{R} w_\\varrho |v^{(n)}|^4 dx + C \\int_\\mathbb{R} w_\\varrho |Z^{(n)}|^4 dx \\nonumber \\;.\n\\end{align}\nNow we first use that $Z$ is uniformly bounded in $C^0_\\gamma$ for any small $\\gamma>0$. See Lemma \\ref{lem:regZ} below.\nThus the $L^4_\\varrho$-norm of $Z$ is finite for $\\varrho>1$, and we now want to uniformly bound $\\|Z^{(n)}\\|^4_{L^4_\\varrho}$ in the formula \\eqref{e:aprio} above by $\\|Z\\|^4_{L^4_\\varrho}$.\nFor this first a simple calculation verifies for $k,n\\in\\mathbb{N}_0$ \n\\[\n\\sup_{|x|\\leqslant n} \\frac{w_\\varrho(x+2nk)}{w_\\varrho(x)} \n= \\left[ \\sup_{|z|\\leqslant 1}\\frac{1+z^2n^2}{1+(z+2k)^2n^2} \\right]^{\\varrho\/2} \\leqslant \\left[ \\frac{2}{(2|k|-1)^2} \\right]^{\\varrho\/2} \\;.\n\\]\nThus we obtain by periodicity for $\\varrho>1$\n\\begin{align*}\n \\|Z^{(n)}\\|^4_{L^4_\\varrho}\n = \\int_\\mathbb{R} w_\\varrho |Z^{(n)}|^4 dx \n & = \\sum_{k\\in\\mathbb{Z}} \\int_{(2k-1)n}^{(2k+1)n} w_\\varrho(x) |Z(x)|^4 dx \\\\\n & = \\sum_{k\\in\\mathbb{Z}} \\int_{-n}^n w_\\varrho(x+2nk) |Z(x)|^4 dx \\\\\n& \\leqslant \\sum_{k\\in\\mathbb{Z}} 2^{\\varrho\/2} |2|k|-1|^{-\\varrho} \\int_{-n}^n w_\\varrho(x) |Z(x)|^4 dx \\\\\n& \\leqslant 2^{1+\\varrho\/2} \\sum_{k\\in\\mathbb{N}_0} |2k-1|^{-\\varrho} \\|Z\\|^4_{L^4_\\varrho}\\;.\n\\end{align*}\n\nTo proceed with \\eqref{e:aprio}, we need a bound on the quadratic form of the operator~${\\mathcal{L}}_\\nu$. For this we use the following Lemma (compare to Lemma 3.8 of Mielke, Schneider~\\cite{MS:95}).\n\n\\begin{lemma}\n\\label{lem:spec}\n For any weight $w_\\varrho$ with $\\varrho>0$ given by Definition~\\ref{def:weight} we have \n \\[\n \\int_\\mathbb{R} w_\\varrho v {\\mathcal{L}}_0 v\\, dx \\leqslant - \\frac{C_2}{1+2C_2 } \\| v''\\|_{L^2_\\varrho} \n + C_2 (3 +\\frac52 C_2) \\| v\\|_{L^2_\\varrho}^2 \\;.\n \\]\n\\end{lemma}\n\n\n\\begin{remark}\nThis is not sufficient for the approximation result later, as we need $C_2 = O(\\varepsilon^2)$, which is achieved \nif we consider $w_{\\varrho,\\varepsilon}$ instead of $w_\\varrho$. \n\\end{remark}\n\n\\begin{proof}\nWe have to prove the Lemma first for smooth compactly supported or periodic $v$ and then also extend by \ncontinuity to any $v\\in H^2_\\varrho$.\nResults like this are standard.\nIn order to not overload the subsequent presentation with indices, we do not recall this approximation step.\nThe same proof as presented below would hold for the approximation, \nand one just needs to check in the final estimate that we can pass to the limit. \n\nIntegration by parts and H\\\"older's inequality yield\n\\begin{multline*}\n \\int_\\mathbb{R} w_\\varrho v {\\mathcal{L}}_0 v\\, dx =\\\\\n \\begin{aligned}\n &= - \\int_\\mathbb{R} w_\\varrho v^2\\, dx - 2 \\int_\\mathbb{R} w_\\varrho v v''\\, dx + \\int_\\mathbb{R} w_\\varrho' v v''' \\, dx + \\int_\\mathbb{R} w_\\varrho v' v''' \\, dx\\nonumber\\\\\n &= - \\int_\\mathbb{R} w_\\varrho v^2\\, dx - 2 \\int_\\mathbb{R} w_\\varrho v v''\\, dx - \\int_\\mathbb{R} w_\\varrho'' v v'' \\, dx -2 \\int_\\mathbb{R} w_\\varrho' v' v'' \\, dx - \\int_\\mathbb{R} w_\\varrho v'' v'' \\, dx \\nonumber \\\\\n &= - \\int_\\mathbb{R} w_\\varrho v^2\\, dx - 2 \\int_\\mathbb{R} w_\\varrho v v''\\, dx - \\int_\\mathbb{R} w_\\varrho'' v v'' \\, dx + \\int_\\mathbb{R} w_\\varrho'' (v')^2 \\, dx - \\int_\\mathbb{R} w_\\varrho v'' v'' \\, dx \\nonumber \\\\\n &\\leqslant - \\| v\\|^2_{L^2_\\varrho} - \\|v''\\|^2_{L^2_\\varrho}\n - \\int_\\mathbb{R} (2 w_\\varrho+ w_\\varrho'') v v'' \\, dx + C_2\\int_\\mathbb{R} w_\\varrho |v'|^2 \\, dx \\nonumber \\;.\n \\end{aligned}\n \\end{multline*}\nNow we use the following interpolation inequality\n\\begin{align*} \n \\int_\\mathbb{R} w_\\varrho (v')^2 \\, dx \n & = - \\int_\\mathbb{R} w_\\varrho' vv' \\, dx - \\int_\\mathbb{R} w_\\varrho vv'' \\, dx \\\\\n & = \\frac12 \\int_\\mathbb{R} w_\\varrho'' v^2 \\, dx - \\int_\\mathbb{R} w_\\varrho vv'' \\, dx \\\\\n & \\leqslant \\frac{C_2}{2} \\| v\\|_{L^2_\\varrho}^2 + \\| v\\|_{L^2_\\varrho}\\| v''\\|_{L^2_\\varrho} \n\\end{align*}\nto obtain:\n \\begin{multline*} \n \\int_\\mathbb{R} w_\\varrho v {\\mathcal{L}}_0 v\\, dx\\leqslant\\\\\n \\begin{aligned}\n &\\leqslant - (1- \\frac{C_2^2}{2}) \\| v\\|^2_{L^2_\\varrho} - \\|v''\\|^2_{L^2_\\varrho}\n - \\int_\\mathbb{R} (2 w_\\varrho+ w_\\varrho''+ C_2 w_\\varrho) v v'' \\, dx \\nonumber\\\\\n &\\leqslant - (1- \\frac{C_2^2}{2}) \\| v\\|^2_{L^2_\\varrho} - \\|v''\\|^2_{L^2_\\varrho}\n + 2 (1 + C_2) \\| v\\|_{L^2_\\varrho}\\|v''\\|_{L^2_\\varrho} \\nonumber\\\\\n &\\leqslant - (1- \\frac{C_2^2}{2}-(1+C_2)\\delta) \\| v\\|^2_{L^2_\\varrho} - (1-(1+C_2)\\delta^{-1})\\|v''\\|^2_{L^2_\\varrho} \\nonumber\\\\\n & = - \\frac{C_2}{1+2C_2 } \\| v''\\|_{L^2_\\varrho} + C_2 (3 +\\frac52 C_2) \\| v\\|_{L^2_\\varrho}^2 \\nonumber \\;,\n \\end{aligned}\n\\end{multline*}\nwhere we used Young's inequality with $\\delta=1+2C_2$. This finishes the proof of the Lemma.\n\\end{proof}\n\n \nGoing back to \\eqref{e:aprio} and using Lemma \\ref{lem:spec} \nwe obtain the following result for the $2n$-periodic approximation $v^{(n)}$.\n\n \\begin{lemma}\n \\label{lem:apprio}\n Let $u_0$ be in $L^2_\\varrho$ for some $\\varrho>3$, then \n there is a small constant $c>0$ and a large constant $C>0$ such that for all $t>0$\n \\[\n \\partial_t \\|v^{(n)}\\|^2_{L^2_\\varrho} \n \\leqslant - c \\| v^{(n)}\\|^2_{H^2_\\varrho} - \\| v^{(n)}\\|_{L^4_\\varrho}^4 + C \\| v^{(n)}\\|_{L^2_\\varrho}^2 + C\\| Z\\|_{L^4_\\varrho}^4 \\;.\n \\]\n \\end{lemma}\n \n As already mentioned this result at least on bounded domains is well known. \n Usually one would estimate the $v^2-v^4$ by $C-v^2$ in order to obtain bounds on the $L^2$-norm that are uniform in time.\n But as we are only after the existence of solutions in this section, \n we keep the $L^4$-norm in order to exploit that regularity.\n \n \n The following corollary is standard for a-priori estimates as in Lemma \\ref{lem:apprio}.\n First by neglecting \n the negative terms on the right hand side and by applying Gronwall inequality \n we obtain an $L^\\infty(0,T,L^2_\\varrho)$-bound.\n The final two estimates follow by integrating in time the inequality in Lemma \\ref{lem:apprio}.\n\n \n \\begin{corollary}\nUnder the assumptions of the previous Lemma~\\ref{lem:apprio} the sequence $\\{v^{(n)}\\}_{n\\in\\mathbb{N}}$ is uniformly bounded \n in $L^\\infty(0,T,L^2_\\varrho) \\cap L^2(0,T, H^2_\\varrho)\\cap L^4(0,T,L^4_\\varrho)$\nfor all $T>0$. Moreover, $\\{v^{(n)}\\}_{n\\in\\mathbb{N}}$ is also uniformly bounded \n in \n \\[L^\\infty(0,T,L^2([-L,L])) \\cap L^2(0,T, H^2([-L,L]))\\cap L^4(0,T,L^4([-L,L]))\n \\]\n for all $T>0$ and all $L>0$.\n \\end{corollary}\n\n Now we can finalize the proof of\nTheorem \\ref{thm:exSH}. \n By taking consecutively subsequences \n we obtain a $v \\in L^\\infty(0,T,L^2_\\varrho) \\cap L^2(0,T, H^2_\\varrho)\\cap L^4(0,T,L^4_\\varrho)$\n such that \n \\[\nv^{(n_k)} \\rightharpoonup v\n\\quad\\text{in } L^2(0,T, H^2_\\varrho)\\cap L^4(0,T,L^4_\\varrho)\n \\]\n and \n \\[\nv^{(n_k)}\\stackrel{*}{\\rightharpoonup} v\n\\quad\\text{in } L^\\infty(0,T,L^2_\\varrho)\\;.\n \\]\nFurthermore using a diagonal argument, \n \\[\nv^{(n_k)} \\rightharpoonup v\n\\quad\\text{in } L^2(0,T, H^2_\\text{loc})\\cap L^4(0,T,L^4_\\text{loc})\n \\]\n and \n \\[\nv^{(n_k)}\\stackrel{*}{\\rightharpoonup} v\n\\quad\\text{in } L^\\infty(0,T,L^2_\\text{loc})\\;.\n \\]\nThis yields by compactness on bounded intervals that \n\\[\nv^{(n_k)} \\to v \\quad\\text{in all } L^2([0,T],H^1[-L,L])\n\\]\nand thus \n\\[\nv^{(n_k)}(t,x) \\to v(t,x) \\quad\\text{for almost all } (t,x)\\;.\n\\]\nThus, by passing to the limit in the weak formulation for $v^{(n)}$\nwe obtain for any $\\varphi\\in C^\\infty_c(\\mathbb{R})$ (smooth and compactly supported)\nthat\n\\[\n\\langle v(t) ,\\varphi\\rangle = \\langle u_0 ,\\varphi\\rangle \n- \\int_0^t \\langle (1+\\partial_x^2) v(s), (1+\\partial_x^2)\\varphi \\rangle ds \n- \\int_0^t \\langle ( v(s)+Z(s))^3 , \\varphi \\rangle ds\\;.\n\\]\nThis implies that $v$ is a weak solution of\n\\[\n\\partial_t v= {\\mathcal{L}}_\\nu v - (v+Z)^3\\;, \\quad v(0)=u_0\\;.\n\\]\nFurthermore by regularity of $v$ we can take the scalar product with $v$ here. \nThis will be used in the proof of uniqueness.\n\n\\begin{remark}\nOne needs to be careful here, as the resulting \nlimit (i.e., the solution) is in general not a measurable random variable.\nThis is well known and due to the fact that we take subsequences that might depend on the given realization of $Z$, and thus the limit is in general not measurable. \nUniqueness, which is proved in the next step, \nenforces that the whole sequence $v^{(n)}$ converges to the solution $v$ and thus the limit is a measurable random variable.\n\\end{remark}\n\n\nFor uniqueness consider two weak solutions $v_1$ and $v_2$ \nin $L^\\infty(0,T,L^2_\\varrho) \\cap L^2(0,T, H^2_\\varrho)\\cap L^4(0,T,L^4_\\varrho)$ with $\\varrho>3$.\nDefine \n\\[d=v_1-v_2\n\\]\nwhich solves \n\\[\n\\partial_t d= {\\mathcal{L}}_\\nu d - (v_1+Z)^3 + (v_2+Z)^3\\;, \\quad d(0)=0\\;.\n\\]\nBy the regularity of $d$ we have $\\partial_t d \\in L^2(0,T,H^{-2}_\\varrho)$, thus we can take the $L^2_\\varrho$-scalar product with $d$ \nto obtain \n\\begin{align*}\n \\frac12 \\partial_t \\|d\\|^2_{L^2_\\varrho}\n &= \\int_\\mathbb{R} w_\\varrho d[ {\\mathcal{L}}_\\nu d - (v_1+Z)^3 + (v_2+Z)^3] dx\\\\\n &= \\int_\\mathbb{R} w_\\varrho d[ {\\mathcal{L}}_\\nu d - (d+v_2+Z)^3 + (v_2+Z)^3] dx\\\\\n &= \\int_\\mathbb{R} w_\\varrho d[ {\\mathcal{L}}_\\nu d - d^3 - 3 d^2(v_2+Z) - 3d(v_2+Z)^2] dx\\\\\n &\\leqslant \\int_\\mathbb{R} w_\\varrho d {\\mathcal{L}}_\\nu d dx - \\int_\\mathbb{R} w_\\varrho [d^4 + 3d^2(v_2+Z)^2] - 3 \\int_\\mathbb{R} w_\\varrho d^3(v_2+Z) dx\\\\ \n &\\leqslant - c \\|d\\|^2_{H^2_\\varrho} + C \\|d\\|^2_{L^2_\\varrho} \n\\end{align*}\nusing that $3d^3b \\leqslant d^4 + \\frac94 d^2b^2$ and Lemma \\ref{lem:spec}. \nNeglecting now all negative terms and using Gronwall's inequality yields (as $d(0)=0$) that\n\\[d(t)=0\\quad \\text{for all }t\\geqslant0 \\;,\n\\]\nand thus uniqueness of solutions.\n \n\n\n\\section{Additional regularity for the Ginzburg-Landau equation}\n\\label{sec:reg}\n\n\nAt the moment we need a very strong weight for the existence and uniqueness of solutions, \nand also related results like the one of \\cite{MoWe:P15} always use Besov spaces with integrable weights.\nRecall the amplitude equation for the complex-valued amplitude $A$\n\\begin{equation}\n\\label{e:GL}\n \\partial_T A =4\\partial_X^2 A + \\nu A -3A|A|^2 + \\partial_T \\mathcal{W}\\;, \\qquad A(0)=A_0\\;,\n \\tag{GL}\n\\end{equation}\nwith complex-valued space time white noise $ \\partial_T \\mathcal{W}$.\nNow we use again the standard substitution \n\\[\nB=A- \\mathcal{Z}\n\\]\nwith stochastic convolution for $\\Delta_\\nu = 4\\partial_X^2+\\nu$ defined by\n\\begin{equation*}\n \\mathcal{Z}(T) =\\mathcal{W}_{\\Delta_\\nu}(T)\n = \\int_0^T {\\mathrm{e}}^{(T-s)\\Delta_\\nu} d \\mathcal{W}(s)\\;.\n\\end{equation*}\nNow $B$ solves \n\\begin{equation}\n \\label{e:GLt}\n \\partial_T B =4\\partial_X^2 B + \\nu B -3(B+\\mathcal{Z})|B+\\mathcal{Z}|^2\\;, \\qquad B(0)=A_0\\;.\n\\end{equation}\nIn the regularity results of this section, we will try to weaken the weight as much as possible. \nMoreover, we show spatial H\\\"older regularity, which is the most we can hope for, \nas we are limited by the regularity of the stochastic convolution~$\\mathcal{Z}$. See Lemma \\ref{lem:regZ} below.\n\nThe key idea is to use energy estimates together with a classical bootstrap argument:\n\\begin{itemize}\n\\item Using the $L^2_\\varrho$-energy estimate we obtain \n $A-\\mathcal{Z}\\in L^\\infty(0,T,L^2_\\varrho) \\cap L^2(0,T,H^1_\\varrho) \\cap L^4(0,T,L^4_\\varrho)$\n in the proof of existence in Theorem \\ref{thm:exGL}.\n\\item Using the $L^p_\\varrho$-norm we derive $A \\in L^\\infty (0,T,L^q_\\varrho)$ in Lemma \\ref{lem:5.1}.\n\\item The $H^1_\\varrho$-norm yields $ A-\\mathcal{Z} \\in L^\\infty (0,T,H^1_\\varrho) \\cap L^2 (0,T,H^2_\\varrho)$ \nin Lemma \\ref{lem:apH1}\n\\item Sobolev embedding yields H\\\"older regularity $A\\in L^\\infty(0,T, C^0_\\kappa)$ for arbitrarily small weight $\\kappa>0$. \nSee Theorem \\ref{thm:apHoeld}.\n\\item Using the $W^{1,p}_\\varrho$-norm we derive $A - \\mathcal{Z} \\in L^\\infty(0,T, W^{1,2p}_\\varrho)$\nin Lemma \\ref{lem:apW1p}.\n\\item The final result again by Sobolev embedding is $ A \\in L^\\infty(0,T,C^{0,\\eta}_\\kappa )$\nfor all H\\\"older exponents $\\eta\\in(0,1\/2)$ and for arbitrarily small weight $\\kappa>0$. \nSee Corollary \\ref{cor:maxregA}.\n\\end{itemize}\n\nThis procedure can only be done for the amplitude equation, but not for the Swift-Hohenberg equation.\nFor example for the $L^p$-estimate we need $\\langle u^{p-1}, \\Delta u \\rangle_{L^2_\\varrho} \\leq c \\|u\\|_{L^p_\\varrho}$, which holds for \nthe Laplacian for any $p\\geq2$, but\nfor the Swift-Hohenberg operator only for $p=2$.\n\nLet us now start with the bootstrap argument.\nThe proof of existence and uniqueness with a strong weight is the same as for the Swift-Hohenberg equation before.\nWe only need the slightly weaker assumption $\\varrho>2$ in this case. \nThe precise theorem is:\n\\begin{theorem}\n \\label{thm:exGL} \nFor $T>0$ and all $A_0\\in L^2_\\varrho$ with $\\varrho>2$ there is a complex-valued stochastic process $B$ such that $\\mathbb{P}$-almost surely \n\\[\nB\\in L^\\infty(0,T,L^2_\\varrho) \\cap L^2(0,T,H^1_\\varrho) \\cap L^4(0,T,L^4_\\varrho) \n\\]\nand $B$ is a weak solution of \\eqref{e:GLt}. \nMoreover, for any other such weak solution $\\tilde{B}$ we have\n\\[\n\\mathbb{P}( \\sup_{t\\in[0,T]}\\|B(t)- \\tilde{B}(t)\\|_{L^2_\\varrho}=0 ) =1\\;.\n\\]\n\\end{theorem}\nNote that in the previous theorem we only assumed $\\varrho>2$ for the weight. \nThis is due to the fact that for the Laplacian we need one integration by parts less. See Remark \\ref{rem:rho3}.\n\n\nWhile for $B$, in the following we can go all the way up to H\\\"older exponent $1$ and even show $W^{1,p}_\\varrho$-regularity.\nFor the amplitude $A$ we are limited by the following Lemma on the regularity of the stochastic convolution $\\mathcal{Z}$.\n\\begin{lemma}\\label{lem:regZ}\nFor all $\\eta<1\/2$, $T>0$ and small weight $\\kappa>0$ one has $\\mathbb{P}$-almost surely \n\\[\n\\mathcal{Z} \\in L^\\infty(0,T,C^{0,\\eta}_\\kappa )\\;.\n\\]\nActually, for all $p>0$ there exists a constant $C_p$ such that\n\\[\n\t\\mathbb{E}[\\sup_{[0,T]}\\|\\mathcal{Z}\\|^p_{C^{0,\\eta}_\\kappa}]\\leqslant C_p.\n\\]\n\\end{lemma}\n\n\\begin{proof}[Sketch of Proof]\nWe refrain from giving all the lengthy details of this proof here.\nMore details on the estimates used can for example be found \nin Lemma~2.4 and Lemma~3.1 in~\\cite{BB:15}, where all tools necessary to prove this \nlemma are presented.\n\nThe proof for regularity of the stochastic convolution is fairly standard and based on the proof \nof the Kolmogorov test for H\\\"older continuity of stochastic processes.\nFirst note that by H\\\"older's inequality it is enough to verify the claim for large $p$.\nFor spatial regularity we consider the embedding of $C^{0,\\gamma}([-L,L])$ into \n$W^{\\alpha,p}([-L,L])$ for $\\gamma<\\alpha<1\/2$ and $p\\to\\infty$. \nThen we can use explicit representation of these norms in terms of integrated H\\\"older quotients.\nFor the bound in time, we can use the celebrated factorization method of Da Prato, Kwapie\\'n and Zabzcyck~\\cite{DKZ87}. \n\\end{proof}\n\n \nLet us first start with a standard energy estimate for the $L^{2p}_\\varrho$-norm. \nHere and in all other energy estimates, we need to perform these estimates \nfor the approximating sequence from the proof of existence, \nand then pass to the limit later. \nBut for simplicity of presentation, we do not state the index $n$ in the estimate.\nThe proof for approximating sequence is the same as the one presented below. One just needs to check in the final estimate, \nwhether one can pass to the limit or not.\n\\begin{lemma}\\label{lem:5.1}\nLet $A$ be such that $B=A-\\mathcal{Z}$ is a weak solution of \\eqref{e:GLt} given by Theorem \\ref{thm:exGL} and fix $T>0$.\nIf $\\varrho>1$ and $q$ such that $A(0)\\in L^q_\\varrho$,\nthen $\\mathbb{P}$-almost surely\n\\[A, B\\in L^\\infty (0,T,L^q_\\varrho)\\;.\n\\]\nMoreover, for all $q\\geqslant 1$ \nthere exists a constant $C_q$ such that\n\\[\n\t\\mathbb{E}[\\int_0^T \\|A\\|^q_{L^q_\\varrho}ds]\\leqslant C_q, \\qquad \\mathbb{E}[\\int_0^T \\|B\\|^q_{L^q_\\varrho}ds]\\leqslant C_q.\n\\]\n\\end{lemma}\n\n\\begin{proof} In view of Lemma \\ref{lem:regZ} it is sufficient to consider only $B$. We use $q=2p$,\nthe notation $B'=\\partial_X B$, and the estimate $\\text{Re}(z) \\leqslant |z|$ to obtain:\n %\n\\begin{align*}\n\\frac1p \\partial_T \\|B\\|^{2p}_{L^{2p}_\\varrho} \n = {}& \\frac1p \\partial_T \\int_\\mathbb{R} w_\\varrho B^p \\overline{B}^p dx \\\\\n = {}& 2\\text{Re} \\int_\\mathbb{R} w_\\varrho B^{p-1}\\partial_T B \\cdot \\overline{B}^p dx\\\\\n = {}& 8\\text{Re} \\int_\\mathbb{R} w_\\varrho B^{p-1} B'' \\cdot \\overline{B}^p dx + 2\\nu \\int_\\mathbb{R} w_\\varrho |B|^{2p} dx\\\\\n& - 6\\text{Re} \\int_\\mathbb{R} w_\\varrho B^{p-1}(B+\\mathcal{Z}) |B+\\mathcal{Z}|^2 dx \\\\\n = {}& - 8\\text{Re} \\int_\\mathbb{R} w_\\varrho' B^{p-1} B' \\overline{B}^p dx \n+ 2\\nu \\|B\\|^{2p}_{L^{2p}_\\varrho} \n- 3 \\|B\\|^{2p+2}_{L^{2p+2}_\\varrho} \n+ C \\|\\mathcal{Z}\\|^{2p+2}_{L^{2p+2}_\\varrho} \\\\\n& - 8(p-1)\\text{Re} \\int_\\mathbb{R} w_\\varrho B^{p-2} (B')^2 \\overline{B}^p dx \n- 8p \\int_\\mathbb{R} w_\\varrho |B|^{2p-2} |B'|^2dx \\\\\n\\leqslant {}& 8C_1 \\int_\\mathbb{R} w_\\varrho |B|^{2p-1} |B'| dx + 2\\nu \\|B\\|^{2p}_{L^{2p}_\\varrho} \n- 3 \\|B\\|^{2p+2}_{L^{2p+2}_\\varrho} \n+ C \\|\\mathcal{Z}\\|^{2p+2}_{L^{2p+2}_\\varrho} \\\\\n& - 8 \\int_\\mathbb{R} w_\\varrho |B|^{2p-2} |B'|^2dx \\\\\n\\leqslant {}& (2C_1^2+\\nu) \\|B\\|^{2p}_{L^{2p}_\\varrho} - 3 \\|B\\|^{2p+2}_{L^{2p+2}_\\varrho} + C \\|Z\\|^{2p+2}_{L^{2p+2}_\\varrho} \\\\\n\\leqslant {}& C - 2 \\|B\\|^{2p+2}_{L^{2p+2}_\\varrho} + C \\|\\mathcal{Z}\\|^{2p+2}_{L^{2p+2}_\\varrho}\\;,\n\\end{align*}\nwhere we need that the weight is integrable while applying H\\\"older's inequality in the last step.\nIntegrating and taking expectations yields the claim.\n\\end{proof}\nWe also need $L^\\infty(0,T,H^1_\\varrho)$-regularity.\n\\begin{lemma}\n\\label{lem:apH1}\nLet $A$ be such that $B=A-\\mathcal{Z}$ is a weak solution of~\\eqref{e:GLt} given by Theorem \\ref{thm:exGL} and fix $T>0$.\nIf $\\varrho>2$ and $A(0)\\in H^1_\\varrho \\cap L^6_\\varrho$,\nthen we have $\\mathbb{P}$-almost surely\n\\[\nB= A-\\mathcal{Z} \\in L^\\infty (0,T,H^1_\\varrho) \\cap L^2 (0,T,H^2_\\varrho)\\;.\n\\]\nMoreover, there exists a constant $C$ such that\n\\[\n\\mathbb{E}[\\int_0^T \\|B\\|^2_{H^2_\\varrho}ds]\\leqslant C, \\qquad \\mathbb{E}[\\sup_{[0,T]}\\|B\\|^2_{H^1_\\varrho}]\\leqslant C\\;.\n\\]\n\\end{lemma}\n\n\\begin{proof} Consider \n\\begin{align*}\n\\frac12 \\partial_T \\|\\partial_XB\\|^{2}_{L^{2}_\\varrho} \n={}& \\frac12 \\partial_T \\int_\\mathbb{R} w_\\varrho B' \\cdot \\overline{B'}\\; dx \\\\\n={}& \\text{Re} \\int_\\mathbb{R} w_\\varrho \\partial_T B' \\cdot \\overline{B'}\\; dx\\\\\n={}& 4\\text{Re} \\int_\\mathbb{R} w_\\varrho B''' \\cdot \\overline{B'} dx + \\nu \\int_\\mathbb{R} w_\\varrho |B'|^{2} dx - 3 \\text{Re} \\int_\\mathbb{R} w_\\varrho \\left(A |A|^{2}\\right)' \\cdot \\overline{B'} \\;dx \\\\\n={}& - 4\\text{Re} \\int_\\mathbb{R} w_\\varrho' B'' \\overline{B'} \\; dx - 4 \\int_\\mathbb{R} w_\\varrho |B''|^2 \\; dx \n+ \\nu \\|B'\\|^{2}_{L^{2}_\\varrho} \n\\\\& + 3 \\text{Re} \\int_\\mathbb{R} w_\\varrho A |A|^{2} \\cdot \\overline{B''} \n+ 3 \\text{Re} \\int_\\mathbb{R} w_\\varrho' A |A|^{2} \\cdot \\overline{B'} \\\\\n\\leqslant {}& - 4\\int_\\mathbb{R} w_\\varrho |B''|^2 \\; dx + C_1 \\int_\\mathbb{R} w_\\varrho |B''| |B'| \\; dx + \\nu \\|B'\\|^{2}_{L^{2}_\\varrho}\n\\\\&\n+ 3 \\int_\\mathbb{R} w_\\varrho |A|^{3} |B''| \\; dx + C_1 \\int_\\mathbb{R} w_\\varrho |A|^{3}|B'| dx \\\\\n\\leqslant {}& (\\nu + C_1^2 +\\frac14) \\|B'\\|_{L^2_\\varrho}^2 + (\\frac12-4) \\|B''\\|_{L^2_\\varrho}^2 \n + (C_1^2+9) \\|A\\|_{L^6_\\varrho}^6 \\\\\n\\leqslant {}& -\\frac72 \\|B''\\|_{L^2_\\varrho}^2 + C \\|B'\\|_{L^2_\\varrho}^2 + C^\\ast \\|B\\|_{L^6_\\varrho}^6 + C^\\ast \\|\\mathcal{Z}\\|_{L^6_\\varrho}^6\\;,\n\\end{align*}\nwhere we used Young's inequality $ab\\leqslant \\frac14a^2+b^2$, partial integration and the given form of the weight $ w_\\varrho(x)$ and its properties \\eqref{eq:propweight}.\nIn order to finish the proof, we first drop the term with $\\|B''\\|_{L^2_\\varrho}^2$ thanks to the negative constant in front of it and apply Gronwall's \ninequality to obtain the $L^\\infty(0,T,H^1_\\varrho)$-regularity. Secondly, we just integrate and take expectations to obtain the $L^2 (0,T,H^2_\\varrho)$-regularity.\n\\end{proof}\nNow we are aiming at space-time regularity for $B$ and thus $A$ in $L^\\infty(0,T,C^0_\\kappa)$.\nThis is achieved by a pointwise interpolation of $C^0([-L,L])$ between $L^p([-L,L])$ and $H^1([-L,L])$ \nfor each fixed $t\\in[0,T_0]$.\nBut we need to be careful in the arguments as the interpolation-constants depend on the spatial domain size $L$. \nWe will prove:\n\\begin{theorem}\n\\label{thm:apHoeld}\nLet $A$ be such that $B=A-\\mathcal{Z}$ is a weak solution of \\eqref{e:GLt} given for $T>0$ by Theorem \\ref{thm:exGL}.\nIf $\\varrho>2$, $A(0)\\in H^1_\\varrho$ and\n$A(0)\\in L^p_\\varrho$ for all large~$p$, \nthen $\\mathbb{P}$-almost surely\n\\[ A, B \\in L^\\infty(0,T, C^0_\\kappa)\\quad \\text{for all small }\\kappa>0\\;.\n\\]\nMoreover, for all $p>0$ there exists a constant $C_p$ such that\n\\[\n\\mathbb{E}[\\sup_{[0,T]}\\|A\\|^p_{C^{0}_\\kappa}]\\leqslant C_p\\;,\n\\qquad \\mathbb{E}[\\sup_{[0,T]}\\|B\\|^p_{C^{0}_\\kappa}]\\leqslant C_p\\;.\n\\]\n\\end{theorem}\n\n\\begin{proof}\nFor a bounded interval $I=[-1,1]$ we use for $1\/2 > \\alpha >1\/p$ first the Sobolev embedding of $W^{\\alpha,p}(I)$ into $C^0(I)$,\nthen interpolation between $W^{\\alpha,p}$ spaces and finally \nthe Sobolev embedding of $H^1(I)$ into $W^{1\/2 ,p}(I)$ for all $p\\in(1,\\infty)$ \nto obtain \n\\[\n\\|A\\|_{C^0(I)} \\leqslant C \\|A\\|_{W^{\\alpha,p}(I)} \\leqslant C \\|A\\|_{L^p(I)}^{1-2\\alpha}\\|A\\|_{W^{1\/2 ,p}(I)}^{2\\alpha}\n\\leqslant C \\|A\\|_{L^p(I)}^{1-2\\alpha}\\|A\\|_{H^1(I)}^{2\\alpha}.\n\\]\nNow we use the scaling for $L\\geqslant 1$ in order to derive the precise scaling in the domain size $L$\nof the constant in the Sobolev embedding.\n\\[\n\t\\begin{split}\n\t\t \\|B(L \\cdot )\\|_{L^p(I)} &= \\Big(\\int_I |B(L x )|^p\\; dx\\Big)^{1\/p} = \\Big(\\frac1L \\int_{-L}^L |B(x )|^p\\; dx\\Big)^{1\/p} \\\\\n\t\t & = L^{-1\/p} \\|B \\|_{L^p([-L,L])} \\;.\n\t\\end{split}\n \\]\nThus we obtain \n \\begin{align*}\n \\|B(L \\cdot )\\|_{H^1(I)}\n & = \n L \\|B'( L \\cdot )\\|_{L^2(I)} + \\|B( L \\cdot )\\|_{L^2(I)}\\\\ \n &=L^{1\/2} \\|B'\\|_{L^2([-L,L])} + L^{-1\/2} \\|B\\|_{L^2(I)} \\leqslant L^{1\/2} \\|B\\|_{H^1([-L,L])}\\;.\n \\end{align*}\n %\n Moreover, we obtain using \\eqref{e:Wkpbound}\n \\begin{equation*}\n\\begin{split}\n \\|B\\|_{C^0([-L,L])} & = \\|B(L\\cdot)\\|_{C^0([-1,1])} \\\\\n &\\leqslant C \\|B(L\\cdot)\\|_{L^p([-1,1])}^{1-2\\alpha}\\|B(L\\cdot)\\|_{H^1([-1,1])}^{2\\alpha} \\\\\n &\\leqslant C L^{-\\frac1p(1-2\\alpha)} \\|B\\|_{L^p([-L,L])}^{1-2\\alpha} L^{\\frac12(2\\alpha)} \\|B\\|_{H^1([-L,L])}^{2\\alpha} \\\\\n &\\leqslant C L^{-\\frac1p(1-2\\alpha)} L^{\\alpha} L^{\\varrho(1-2\\alpha)\/p} \\|B\\|_{L^p_\\varrho}^{1-2\\alpha} L^{2\\varrho\\alpha\/2} \\|B\\|_{H^1_\\varrho}^{2\\alpha} \\\\\n &\\leqslant C L^{\\frac1p(1-2\\alpha)(\\varrho-1)} L^{\\alpha(1+\\varrho)} \\|B\\|_{L^p_\\varrho}^{1-2\\alpha} \\|B\\|_{H^1_\\varrho}^{2\\alpha} \\;.\n\\end{split}\n \\end{equation*}\nNow we can first choose $\\alpha >0$ small and then $p>1\/\\alpha$ sufficiently large, so that for any given $\\kappa>0$ we have a $C>0$\nso that \n\\begin{equation*}\n\\|B\\|_{C^0([-L,L])} \n\\leqslant \n C L^\\kappa \\|B\\|_{L^p_\\varrho}^{1-2\\alpha} \\|B\\|_{H^1_\\varrho}^{2\\alpha} \\;.\n\\end{equation*}\nThus the claim for $B$ follows from the equivalent definition of the $C^0_\\kappa$-norm (see~\\eqref{e:equivC0}).\nFor $A=B-\\mathcal{Z}$ we just use the fact that the stochastic convolution $\\mathcal{Z}$ is more regular.\n\nWe can get the bounds for all sufficiently large moments from \n\\[\\|B\\|_{C^0_\\kappa} \n\\leqslant C \\|B\\|_{L^p_\\varrho}^{1-2\\alpha} \\|B\\|_{H^1_\\varrho}^{2\\alpha}\n\\]\nby carefully choosing $\\alpha>0$ sufficiently small and taking the supremum in $[0,T]$,\nas we have any moments in $L^p_\\varrho$ and second moments in $H^1_\\varrho$.\n\\end{proof}\nFor H\\\"older continuity of $B$ and thus $A$, we need to proceed with the bootstrap-argument\nand show $W^{1,2p}_\\varrho$-regularity for $B$ first. \nHere our proof is based again on energy estimates, but now for $\\|B'\\|^{2p}_{L^{2p}}$.\n\\begin{lemma}\n\\label{lem:apW1p}\nLet $A$ be such that $B=A-\\mathcal{Z}$ is a weak solution of \\eqref{e:GLt} given for $T>0$ by Theorem \\ref{thm:exGL}.\nIf $\\varrho>2$, $p>2$, $A(0)\\in W^{1,2p}_\\varrho$ and\n$A(0)\\in L^{6p}_\\varrho$,\nthen \n\\[ \nB= A - \\mathcal{Z} \\in L^\\infty(0,T, W^{1,2p}_\\varrho)\\;.\n\\]\nMoreover for all $p>2$ there exists a constant $C_p$ such that\n\\[\n\\mathbb{E}[\\sup_{[0,T]}\\|B\\|^{2p}_{W^{1,2p}_\\varrho}]\\leqslant C_p.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nRecall\n\\[\n\\partial_T B= B'' +\\nu B - 3|A|^2A\\;.\n\\]\nThen, using the same ideas as in Lemma~\\ref{lem:5.1}, we obtain the following bound:\n\\begin{equation*}\n\t\\begin{split}\n\t\\partial_T \\| B' \\|^{2 p}_{L_{\\varrho}^{2 p}} & = \\partial_T \\int_\\mathbb{R} w_\\varrho\n\t(B')^p (\\overline{B}')^p\\\\\n\n\t& = 2 p \\,\\text{Re} \\int_\\mathbb{R} w_\\varrho (B')^{p -\n\t\t1} (\\overline{B}')^p \\partial_T B' d x\\\\\n\t& = 8p \\text{Re} \\int_\\mathbb{R} w_\\varrho (B')^{p -\n\t\t1} (\\overline{B}')^p B''' d x \n\t + 2p\\nu \\text{Re}\\int_\\mathbb{R} w_\\varrho (B')^{p} (\\overline{B}')^p d x\\\\\n\t& \\qquad -6p \\text{Re}\\int_\\mathbb{R} w_\\varrho (B')^{p -\n\t\t1} (\\overline{B}')^p (|A|^2A)' d x\\;.\n\t\\end{split}\n\\end{equation*}\n\nNow the second term is $2p\\nu \\|B'\\|^{2p}_{L^{2p}_\\varrho}$, \nand it remains to control the first and third terms. \nLet us start with the first one:\n\\begin{align*}\n\t8p \\text{Re} \\int_\\mathbb{R} w_\\varrho (B')^{p -\n\t\t1} (\\overline{B}')^p B''' d x ={}& -8p \\text{Re} \\int_\\mathbb{R} w_\\varrho' (B')^{p -\n\t\t1} (\\overline{B}')^p B'' d x \\\\\n\t& -8p(p-1) \\text{Re} \\int_\\mathbb{R} w_\\varrho (B')^{p -2}(B'')^2 (\\overline{B}')^p d x \\\\\n\t& -8p^2 \\text{Re} \\int_\\mathbb{R} w_\\varrho (B')^{p -1} (\\overline{B}')^{p-1} \\overline{B}'' B'' d x \\\\\n\t\\leqslant {}& 8pC_1^2\\int_\\mathbb{R} w_\\varrho|B'|^{2p}d x - (8p-\\frac{8p}{4})\\int_\\mathbb{R} w_\\varrho |B'|^{2p-2}|B''|^2 d x\\\\\n\t ={}& 8pC_1^2\\|B'\\|^{2p}_{L^{2p}_\\varrho}- 6p\\int_\\mathbb{R} w_\\varrho |B'|^{2p-2}|B''|^2 d x\\;.\n\\end{align*}\nConcerning the third term we have:\n\\begin{align*}\n\t-6p \\text{Re}\\int_\\mathbb{R} w_\\varrho (B')^{p -1} (\\overline{B}')^p (|A|^2A)' d x \n\t= {}& 6p \\text{Re}\\int_{\\mathbb{R}} w_\\varrho'(B')^{p-1}(\\overline{B}')^{p}(|A|^2A) d x\\\\\n\t &+6p(p-1)\\text{Re}\\int_{\\mathbb{R}} w_\\varrho(B')^{p-2}B''(\\overline{B}')^{p}(|A|^2A) d x\\\\\n\t& +6p^2 \\text{Re}\\int_{\\mathbb{R}} w_\\varrho(B')^{p-1}(\\overline{B}')^{p-1}\\overline{B}''(|A|^2A) d x\\\\\n\n\n\n\t\\leqslant{}& 6C_1 p \\int_\\mathbb{R} w_\\varrho |B'|^{2p-1}|A|^3 d x \\\\\n\t& + (12p^2-6p) \\int_{\\mathbb{R}} w_\\varrho|B'|^{2p-2}|B''| |A|^3 d x\\\\\n\n\t \\leqslant {}& C\\int_\\mathbb{R} w_\\varrho|B'|^{2p}d x + C\\int_\\mathbb{R} w_\\varrho|A|^{6p}d x\\\\\n\t&+ p \\int_\\mathbb{R} w_\\varrho |B'|^{2p-2}|B''|^2 d x + C \\int_\\mathbb{R} B'|^{2p-2} |A|^6 \\\\\n\t \\leqslant {}& C \\|B'\\|^{2p}_{L^{2p}_\\varrho} + C \\|A\\|^{6p}_{L^{6p}_\\varrho}+p \\int_\\mathbb{R} w_\\varrho |B'|^{2p-2}|B''|^2 d x\\;,\n\\end{align*}\nwhere the constant $C$ depends only on $p$ and $C_1$.\nSo we can conclude, putting the estimates for all three terms together,\n\\begin{equation*}\n\\begin{split}\n\t\\partial_T \\| B' \\|^{2 p}_{L_{\\varrho}^{2 p}} & \\leqslant C \\|B'\\|^{2p}_{L^{2p}_\\varrho} + C \\|A\\|^{6p}_{L^{6p}_\\varrho}\n-5p\\int_{\\mathbb{R}} w_\\varrho |B'|^{2p-2}|B''|^2 d x\\\\\n\t&\\leqslant C\\|B'\\|^{2p}_{L^{2p}_\\varrho}+C \\|A\\|^{6p}_{L^{6p}_\\varrho}\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we dropped a negative term.\n \nNow, we can finish the proof using Gronwall's inequality, where we need for the initial condition \n$A (0) \\in W_{\\varrho}^{1, 2p}$ for the $W^{1,2p}_\\varrho$-bound on $B$, and \nfurthermore $A (0) \\in L^{6p}_{\\varrho}$ for the ${L^{6p}_\\varrho}$-bound on $A$.\nThe bound for the moments is again straightforward.\n\\end{proof}\n\nNow we turn to the regularity in weighted H\\\"older spaces defined in \n\\eqref{def:wHoe}.\n\n\n\\begin{theorem}\\label{thm:whreg}\nLet $B=A-\\mathcal{Z}$ with $A$ a weak solution of \\eqref{e:GLt} given for $T>0$ by Theorem \\ref{thm:exGL}. \nIf for some $\\varrho>2$ and all sufficiently large $p\\geqslant 2$ we have $A(0)\\in W^{1,p}_\\varrho $ \nthen for all $\\eta\\in(0,1)$\n\\[ \n B \\in L^\\infty(0,T,C^{0,\\eta}_\\kappa ) \\quad \\text{for all sufficiently small }\\kappa>0\\;.\n\\]\nMoreover, for all $p>0$ there exists a constant $C_p$ such that\n\\[\n\\mathbb{E}[\\sup_{[0,T]}\\|B\\|^{p}_{C^{0,\\eta}_\\kappa}]\\leqslant C_p.\n\\]\n\\end{theorem}\nUsing Lemma~\\ref{lem:regZ} for the regularity of $\\mathcal{Z}$ we obtain:\n\\begin{corollary}\n\\label{cor:maxregA}\n Under the assumptions of the previous theorem for all $\\eta\\in(0,1\/2)$\n\\[ \n A \\in L^\\infty(0,T,C^{0,\\eta}_\\kappa ) \\quad \\text{for all sufficiently small }\\kappa>0\\;.\n\\]\nMoreover, for all $p>0$ there exists a constant $C_p$ such that\n\\[\n\\mathbb{E}[\\sup_{[0,T]}\\|A\\|^{p}_{C^{0,\\eta}_\\kappa}]\\leqslant C_p.\n\\]\n\\end{corollary}\n\n\\begin{proof}(Proof of Theorem~\\ref{thm:whreg})\nWe proceed\nby using the Sobolev embedding of $W^{\\alpha,p}([-L,L])$ into $C^{0,\\alpha-1\/p}([-L,L])$ for $p+1>\\alpha p>1$ and then an interpolation inequality.\nAs before, we need to take care of the scaling of the constants with respect to $L$. Recall that $I=[-1,1]$. First,\n\\begin{equation*}\n\\|B\\|_{C^{0,\\alpha-1\/p}(I)} \n\\leqslant C\\|B\\|_{W^{\\alpha,p}(I)}\n\\leqslant C\\|B\\|^{1-\\alpha}_{L^p(I)}\\|B\\|^{\\alpha}_{W^{1,p}(I)}\\;.\n\\end{equation*}\nRescaling for $L\\geqslant1$ yields\n\\begin{equation*}\n\\begin{split}\n\\|B\\|_{C^{0,\\eta}[-L,L]}&=\\sup_{\\xi,\\zeta\\in [-L,L]}\\frac{|B(\\xi)-B(\\zeta)|}{|\\xi-\\zeta|^\\eta}+\\|B\\|_{C^0[-L,L]}\\\\\n&=\\sup_{x,y\\in I} \\frac{|B(xL)-B(yL)|}{|x-y|^\\eta L^\\eta}+\\|B(\\cdot L)\\|_{C^0(I)}\\\\\n &\\leqslant L^{-\\eta}\\|B(\\cdot L)\\|_{C^{0,\\eta}(I)} + \\|B(\\cdot L)\\|_{C^0(I)} \\;.\n\\end{split}\n\\end{equation*}\nNow we take $\\varsigma=\\alpha-1\/p$ (recall $\\alpha p>1$) and interpolate:\n\\begin{equation*}\n\\|B\\|_{C^{0,\\varsigma}[-L,L]} \\leqslant L^{-\\varsigma} \\|B(\\cdot L)\\|^\\alpha_{W^{1,p}(I)}\\cdot\\|B(\\cdot L)\\|^{1-\\alpha}_{L^{p}(I)} + \\|B(\\cdot L)\\|_{C^0(I)} \\;.\n\\end{equation*}\nNow we rescale back all the norms to the original length scale. For the first one we obtain\n\\begin{equation*}\n\\begin{split}\n \\|B(\\cdot L)\\|^p_{W^{1,p}(I)}&= L^p\\int_I |B'(Lx)|^p dx + \\int_I|B(Lx)|^p dx\\\\\n &= L^{p-1}\\int_{-L}^{L}|B'|^p dz + \\frac{1}{L} \\int_{-L}^{L}|B|^p dx\n \\leqslant L^{p-1}\\|B\\|^p_{W^{1,p}[-L,L]}\n\\end{split}\n\\end{equation*}\nand thus\n\\[\n \\|B(\\cdot L)\\|^\\alpha_{W^{1,p}(I)}\\leqslant L^{\\frac{p-1}{p}\\alpha}\\|B\\|^\\alpha_{W^{1,p}[-L,L]} \\;.\n\\]\nFor the second norm in the interpolated part we have, by a substitution,\n\\[\n\\|B(\\cdot L)\\|^p_{L^p(I)}\\leqslant L^{-1}\\|B\\|^p_{L^p[-L,L]}\n\\quad \\text{and}\\quad\n \\|B(\\cdot L)\\|^{1-\\alpha}_{L^{p}(I)} \\leqslant L^{-\\frac{1-\\alpha}{p}}\\|B\\|^{1-\\alpha}_{L^{p}[-L,L]} \\;.\n\\]\nWe can now put these estimates together and derive\n\\begin{equation*}\n \\begin{split}\n\t\\| B \\|_{C^{0, \\varsigma} [- L, L]} \n\t& \\leqslant L^{- \\varsigma} \\| B (\\cdot L)\\|^{\\alpha}_{W^{1, p}(I)} \\| B (\\cdot L) \\|^{1 - \\alpha}_{L^p(I)}+ \\|B(\\cdot L)\\|_{C^0[-1,1]}\\\\\n\t& \\leqslant L^{- \\varsigma} L^{\\frac{p - 1}{p} \\alpha} \\| B\\|^{\\alpha}_{W^{1, p} [- L, L]} \\cdot L^{- \\frac{1 - \\alpha}{p}} \\| B \\|^{1- \\alpha}_{L^p [- L, L]} + \\|B\\|_{C^0[-L,L]}\\\\\n\n\t& = \\| B \\|^{\\alpha}_{W^{1, p} [- L, L]} \\cdot \\| B \\|^{1 - \\alpha}_{L^p[- L, L]} + \\|B\\|_{C^0[-L,L]}\n\t\\end{split}\n\\end{equation*}\nwith $\\varsigma = \\alpha -\\frac{1}{p}$, \nwhere $\\varsigma \\in (0, 1)$, $\\alpha \\in (\\varsigma, 1)$ and $p > 1\/\\alpha$ sufficiently large. It is somewhat remarkable here that the constant above is $1$ and thus independent of $L$.\n\nNow by \\eqref{e:Wkpbound} we can change to the weighted space to\nobtain\n\\begin{equation*}\n\\begin{split}\n\t\\| B \\|_{C^{0, \\varsigma} [- L, L]} \n\t& \\leqslant \\| B \\|^{\\alpha}_{W^{1, p} [-L, L]} \\| B \\|^{1 - \\alpha}_{L^p [- L, L]} + \\|B\\|_{C^0[-L,L]}\\\\\n\t& \\leqslant \\| B \\|^{\\alpha}_{W^{1, p}_{\\varrho}} \\| B \\|^{1 -\\alpha}_{L^p_{\\varrho}} L^{\\varrho \/ p} + \\|B\\|_{C^0[-L,L]} \\;.\n\\end{split}\n\\end{equation*}\nNow for $\\varsigma \\in (0, 1)$, $\\varsigma =\n\\alpha - \\frac{1}{p}$, $\\alpha \\in (\\varsigma, 1)$ and $p > 1\/\\alpha$,\nusing the definition of the weighted H\\\"older norms from \\eqref{def:wHoe} \nand the equivalent representation of the $C^0_\\kappa$-norm (see \\eqref{e:equivC0}) we derive\n\\begin{equation*}\n\\begin{split}\n\\|B\\|_{C^{0,\\varsigma}_\\kappa} &= \\sup_{L\\geqslant1}\\{L^{-\\kappa}\\|B\\|_{C^{0,\\varsigma}[-L,L]}\\}\\\\\n&\\leqslant \\sup_{L\\geqslant1}\\{L^{\\varrho\/p-\\kappa}\\}\\| B\n\\|^{\\alpha}_{W^{1, p}_{\\varrho}} \\| B \\|^{1 - \\alpha}_{L^p_{\\varrho}} + \\|B\\|_{C^0_\\kappa},\n\\end{split}\n\\end{equation*}\nand as soon as we choose $p$ large enough that $\\varrho\/p-\\kappa\\leqslant 0$ we have finished the proof.\nOnce again the bounds on all the moments follow easily, as we have all moments of the terms on the right hand side.\n\\end{proof}\n\n\n\n\\section{Residual}\n\\label{sec:res}\n\nDefine the approximation \n\\begin{equation}\n\\label{def:uA}\nu_A(t,x) = \\varepsilon A(\\varepsilon^2 t, \\varepsilon x){\\mathrm{e}}^{ix} + c.c. \\;,\n\\end{equation}\nwhere $A$ is both a weak and a mild solution of the amplitude equation given by \n\\begin{equation}\n \\label{e:GLmild}\n A(T) = {\\mathrm{e}}^{(4\\partial_X^2+\\nu) T } A(0) - 3\\int_0^T {\\mathrm{e}}^{(4\\partial_X^2+\\nu)(T-s) } A|A|^2(s) ds + \\mathcal{Z}(T) \\;.\n\\end{equation}\nDefine the residual\n\\begin{equation}\n \\label{e:residual}\n \\text{Res}(\\varphi)(t) = \\varphi(t) - {\\mathrm{e}}^{t{\\mathcal{L}}_\\nu}\\varphi(0) + \\int_0^t {\\mathrm{e}}^{(t-s){\\mathcal{L}}_\\nu} \\varphi(s)^3 ds + \\varepsilon^{3\/2} W_{{\\mathcal{L}}_\\nu} (t) \\;,\n\\end{equation}\nwhich measures how close $u_A$ is to a solution.\nIn this section we bound $\\text{Res}(u_A)$. \nThis is a key result to prove the error estimate later. \nFirst we plug in the definition of $u_A$ to obtain\n\\begin{align*}\n\\text{Res}(u_A)(t,\\cdot) \n = {}& \\varepsilon \\Big[A( \\varepsilon^2 t, \\varepsilon \\cdot){\\mathrm{e}}_1 - {\\mathrm{e}}^{t{\\mathcal{L}}_\\nu}[A(0,\\varepsilon \\cdot){\\mathrm{e}}_1] \n\\\\&\n+ 3\\varepsilon^2 \\int_0^t {\\mathrm{e}}^{(t-s){\\mathcal{L}}_\\nu} A|A|^2(\\varepsilon^2 s ,\\varepsilon \\cdot) {\\mathrm{e}}_1 ds \\Big]\\\\\n& + \\varepsilon^3 \\int_0^t {\\mathrm{e}}^{(t-s){\\mathcal{L}}_\\nu}A^3(\\varepsilon^2 s , \\varepsilon \\cdot ){\\mathrm{e}}_3 ds\n+ c.c. - \\varepsilon^{3\/2} W_{{\\mathcal{L}}_\\nu} (t,\\cdot)\\;,\n\\end{align*}\nwhere we used the notation \n\\[\n{\\mathrm{e}}_n(x)={\\mathrm{e}}^{inx}\\;.\n\\]\n\nRescaling to the slow time-scale, we find\n\\begin{equation}\n\\label{e:Res1}\n\\begin{split}\n \\text{Res}(u_A)(T\\varepsilon^{-2}, \\cdot) \n = {}& \\varepsilon A( T,\\varepsilon \\cdot ){\\mathrm{e}}_1 - \\varepsilon {\\mathrm{e}}^{T\\varepsilon^{-2}{\\mathcal{L}}_\\nu}[A(0,\\varepsilon \\cdot){\\mathrm{e}}_1] \n \\\\ &\n + \\varepsilon \\int_0^{T} {\\mathrm{e}}^{(T-s)\\varepsilon^{-2}{\\mathcal{L}}_\\nu} \\left[A^3(s, \\varepsilon \\cdot){\\mathrm{e}}_3 + 3A|A|^2(s, \\varepsilon \\cdot ) {\\mathrm{e}}_1\\right] ds \n \\\\ &\n + c.c. - \\varepsilon^{3\/2} W_{{\\mathcal{L}}_\\nu} (T\\varepsilon^{-2},\\cdot)\\;.\n \\end{split}\n\\end{equation}\nNow we need to transform this to obtain the mild formulation \\eqref{e:GLmild}.\nThis will remove all the $\\mathcal{O}(\\varepsilon)$-terms.\n\n\n\\subsection{Stochastic Convolution}\n\n\nThe stochastic convolution in \\eqref{e:Res1} can be replaced by (see \\cite{BB:15})\n\\[ \n\\varepsilon^{1\/2} W_{{\\mathcal{L}}_\\nu} (t,x) \\approx \\mathcal{Z}(\\varepsilon^2 t,\\varepsilon x)e_1 +c.c. \\quad\\text{uniformly on $[0,T\\varepsilon^{-2}]$ in } C^0_\\gamma \\;.\n\\]\nThe precise statement from \\cite{BB:15} is \n\\begin{theorem}\nFor all $T>0$, for all $\\kappa >0$, for all $p>1$ and all sufficiently small $\\gamma>0$ \nthere is a constant $C>0$ such that\n\\begin{equation*}\n\\mathbb{P}\\Big(\\sup_{[0,T\\varepsilon^{-2}]}\n\\|\\varepsilon^{ 1 \/ 2} W^{(\\varepsilon)}_{\\mathcal{L}_{\\varepsilon}} (t, x) - [ \n \\mathcal{Z} (t \\varepsilon^2, x \\varepsilon) e_1 +\n \\textrm{c.c}.] \\|_{C^0_{\\gamma}}\\geqslant\\varepsilon^{1-\\kappa}\\Big)\\leqslant C \\varepsilon^p\n\\end{equation*}\nfor all $\\varepsilon\\in (0,\\varepsilon_0)$.\n\\end{theorem}\nWe use the following short-hand notation in order to reformulate this lemma.\n\\begin{definition} \n\t\\label{def:0}\n\tWe say that a real valued stochastic process $X$ is \n $\\mathcal{O}(f_\\varepsilon)$ with high probability, if there is a constant $C>0$ such that for all $p>1$ there is a constant $C_p>0$ such that for $\\varepsilon\\in(0,1)$\n\\[ \\mathbb{P}\\Big(\\sup_{[0,T]} |X| \\geqslant C f_\\varepsilon\\Big) \\leqslant C_p\\varepsilon^p\\;.\n\\]\n\\end{definition}\n\\begin{lemma} \n\\label{lem:stoch}\nWe can write\n\t\\[\n\t\t\\varepsilon^{3\/2} W_{{\\mathcal{L}}_\\nu} (t,\\cdot) = \\varepsilon \\mathcal{Z}(\\varepsilon^2 t, \\varepsilon\\cdot){\\mathrm{e}}_1 +c.c. + E_s(t)\n\t\\]\n\twhere $\\|E_s(\\varepsilon^{-2}\\cdot)\\|_{C^0_{\\gamma}}=\\mathcal{O}(\\varepsilon^{1-\\kappa})$ for any $\\kappa>0$ in the sense of the previous definition.\n\\end{lemma}\n\n\n\\subsection{Exchange Lemmas and estimates for the residuals}\n\n\nLet us now come back to \\eqref{e:Res1}. In the following we present lemmas to exchange the Swift-Hohenberg \nsemigroup generated by ${\\mathcal{L}}_\\nu={\\mathcal{L}}_0 + \\varepsilon^2 \\nu$ with \nthe Ginzburg-Landau semigroup generated by $\\Delta_\\nu=4\\partial_x^2 + \\nu$.\nThe first one is for the initial condition, which is the most difficult one, as we cannot allow for a pole in time.\nThe second one is for the term in \\eqref{e:Res1} that contains $A|A|^2$, while the third one shows that the term in \\eqref{e:Res1} with $A^3$ is negligible.\nAfter applying all the exchange Lemmas to \\eqref{e:Res1} we will see that in \\eqref{e:ResExch} below\nall of the remaining terms of order $\\mathcal{O}(\\varepsilon)$ \nwill cancel due to the mild formulation of the Amplitude Equation in \\eqref{e:GLmild}.\n\n\nWe will state all lemmas here and first prove the bound on the residual, before verifying the Exchange Lemmas.\n\n\\begin{lemma}[Exchange Lemma - IC]\n\\label{lem:lemma1} Let $A_0\\in C_\\kappa^{0,\\alpha}$ for some $\\alpha\\in(\\frac12,1)$ and sufficiently small $ \\kappa>0$. \n\tThen the following holds \n\t\\[\n\t\te^{t{\\mathcal{L}}_\\nu}[A_0(\\varepsilon \\cdot){\\mathrm{e}}_1]=(e^{\\Delta_\\nu T}A_0)(\\varepsilon \\cdot){\\mathrm{e}}_1 + E_1(A_0)\n\t\\]\n\twhere $T=\\varepsilon^2 t$, and the error $E_1$ is bounded uniformly in time for all small $\\kappa>0$ by\n\t\\[\n\t\t\\|E_1\\|_{C^0_\\kappa} \\leqslant C \\varepsilon^{\\alpha-\\frac12} \\|A_0\\|_{ C_\\kappa^{0,\\alpha}}\\;.\n\t\\]\n\\end{lemma}\n\n\\begin{remark}\nHere we allow some dependence on higher norms of the initial conditions, i.e. we assume more regularity for the initial conditions in order to avoid the pole in time that appears \nin the exchange lemma below.\n\\end{remark}\n\\begin{remark}\nFor the solution $A$ of the amplitude equation we showed in Section~\\ref{sec:reg} that it splits into a more regular part $B$ and the Gaussian $\\mathcal{Z}$.\nThe process $B$ has $W^{1,p}_\\varrho$-regularity as assumed for the initial condition $A(0)=A_0$ in the previous exchange Lemma.\nThe term $\\mathcal{Z}$ is less regular, but thanks to the fact that it is Gaussian, we can still prove the exchange Lemma for initial conditions $A(0)$ which split into the more regular and the Gaussian part (see the application of Lemma 3.5 in the proof of Theorem 4.2 of \\cite{BB:15}.\n\nThus for our result we could take initial conditions that are as regular, as the solution of the amplitude equation, but\nin order to simplify the statement of the result we refer from adding the less regular Gaussian part here.\n\\end{remark}\nThe following lemma is applied in \\eqref{e:Res1} to the term in the residual associated to the nonlinearity $|A|^2A$,\nin order to exchange the semigroups there.\n\n\\begin{lemma}[Exchange Lemma I]\n\\label{lem:lemma2}\n\tFor any function $D \\in C^{0,\\alpha}_\\kappa$ with small $\\kappa>0$ and $\\alpha\\in(0,1)$, we have for all $t \\in[0,T_0\\varepsilon^{-2}]$\n\t\\[\n\t\te^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] = (e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1 + E_2(T,D)\n\t\\]\n\twith $T=\\varepsilon^2t$ where the error term $E_2$ is bounded by\n\t\\[\n\t\\|E_2(T,D) \\|_{C^{0}_\\kappa} \\leqslant C \\left[(\\varepsilon^{2\\gamma-1}+\\varepsilon^{\\alpha-\\kappa} )T^{-1\/2}+\\varepsilon T^{-3\/4}\\right] \\|D\\|_{C^{0,\\alpha}_\\kappa}\\;.\n\t\\]\n\\end{lemma}\nThe third lemma is needed in \\eqref{e:Res1} for the term in the residual associated to $A^3$, which should be small.\n\\begin{lemma}[Exchange Lemma II]\n\\label{lem:lemma3}\n\tFor any function $D \\in C^{0,\\alpha}_\\kappa$ for all $\\kappa>0$ and $\\alpha\\in(0,1)$, we have for all $t \\in[0,T_0\\varepsilon^{-2}]$\n\t\\[\n\t\te^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_3] = E_3(T,D) \n\t\\]\t\n\twhere for any $\\gamma\\in[1\/8,1\/2)$ \n\tthe error term $E_3$ on the slow time-scale $T=\\varepsilon^2t$\n\tis bounded by \n\t\\[\\|E_3(T, D)\\|_{C^{0}_\\kappa} \\leqslant C ( \\varepsilon^{2\\gamma-1\/2}+ \\varepsilon^{\\alpha-\\kappa})T^{-1\/2}\n\t\\|D\\|_{C^{0,\\alpha}_\\kappa}\\;.\n\t\\]\n\\end{lemma}\n\nNow we apply all Exchange Lemmas \\ref{lem:lemma1}, \\ref{lem:lemma2}, and \\ref{lem:lemma3} \ntogether with the result for the stochastic convolution from Lemma \\ref{lem:stoch} \nto the definition of $\\text{Res}(u_A)$ from~\\eqref{e:Res1} to obtain\n\\begin{equation}\n\\label{e:ResExch}\n\\begin{split}\n\t\\text{Res}(u_A)(t) ={}& \\varepsilon \\Big[A(\\varepsilon \\cdot, \\varepsilon^2 t ) - {\\mathrm{e}}^{\\Delta_\\nu T}[A(\\varepsilon \\cdot, 0)]\n\t\\\\ &\n\t+ 3\\varepsilon^2 \\int_0^t {\\mathrm{e}}^{(T-\\varepsilon^2s)\\Delta_\\nu} A|A|^2(\\varepsilon^2 s,\\varepsilon \\cdot ) ds - \\mathcal{W}_{\\Delta_\\nu}(\\varepsilon^2 t, \\varepsilon\\cdot)\\Big]{\\mathrm{e}}_1 + c.c.\\\\\n\t& + \\varepsilon [ E_1(A_0)+ E_s] \n\t+ \\varepsilon \\int_0^T \\left[E_2\\left(T-S, A|A|^2\\right)+ E_3\\left(T-S, A^3\\right)\\right] \\; dS.\n\\end{split}\n\\end{equation}\n\nWith this representation we are done. By substituting $S=s\\varepsilon^2$ in the integral, \nwe obtain that the whole bracket $[\\cdots]{\\mathrm{e}}_1$ is the mild formulation of the Ginzburg-Landau equation (see~\\eqref{e:GLmild}) and thus cancels.\nUsing the bounds on the error terms and the regularity of $A$, we obtain the main result.\n\nNote that all poles from the error terms are integrable and that we choose $\\alpha, \\gamma<1\/2$, arbitrarily close to $1\/2$.\n\n\\begin{theorem}[Residual]\n\\label{thm:res}\nLet $A$ be a solution of the amplitude equation \\eqref{e:GL} and assume that there is a $\\varrho>2$ such that for all $p>1$ one has $A(0)\\in W^{1,p}_{\\varrho}$.\nThen for the approximation $u_A$ defined in \\eqref{def:uA} and the residual defined in \\eqref{e:residual} we have\nfor all small $\\kappa>0$ that\n\\begin{equation}\n\\label{e:thmmain}\n\\| \\text{\\rm Res}(u_A)( \\varepsilon^{-2}\\;\\cdot)\\|_{C^0_\\kappa} =\\mathcal{O}(\\varepsilon^{\\frac32 - 2\\kappa }) \\;.\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\n\tNote that under the assumptions of the previous theorem \n\tby the regularity results in Section \\ref{sec:reg} we have for all small $\\kappa>0$, $\\gamma\\in(0,1)$ and $\\alpha\\in(0,\\frac12)$ that $A(0)\\in C^{0,\\gamma}_\\kappa$ and $A\\in L^\\infty([0,T], C^{0,\\alpha}_\\kappa)$\n\\end{remark}\n\n\n We remark without proof that one could replace $- 2\\kappa $ on the right hand side of \n \\eqref{e:thmmain} by an arbitrarily small $\\delta>0$. But as $\\kappa$ is small also,\n we state this simpler but weaker statement. \n\n\n\n\\subsection{Fourier Estimates}\n\n\n\n\nNow we present three results that have the same focus, as\nthey all bound convolution operators with a kernel such that the support of the Fourier transform is bounded away from $0$. \nThe bounds are established in weighted H\\\"older norms and are the backbone of the proofs of the exchange lemmas.\n \nIn the first one, Lemma \\ref{lem:guidos}, we consider some smooth projection on a region in Fourier space that is far away from the origin.\nUsing H\\\"older regularity, we show that this is an operator with small norm, when considered from $C^{0,\\alpha}_\\kappa$ to $C^0_\\kappa$.\nLater in Lemma \\ref{lem:guidoext} we modify the proof to give bounds on convolution operators using the $H^2$-norm of the Fourier transform of the kernel.\nWhile in Corollary \\ref{cor:new} we finally modify the result even more, by showing that we do not need the $L^2$-norm of the Fourier transform of the kernel. \nWhile Lemma \\ref{lem:guidoext} will be sufficient for most of the estimates used in the proofs of the exchange lemmas, at one occasion we need Corollary \\ref{cor:new}.\n\n\\begin{lemma}\\label{lem:guidos}\n\tLet $\\widehat{P}:\\mathbb{R}\\to [0,1]$ be a smooth function with bounded support, such that $0\\notin\\textup{supp}(\\widehat{P})$. Let also $D\\in \\mathcal{C}^{0,\\alpha}_\\kappa$, with $\\alpha\\in (0,1)$, $\\kappa>0$. \n\tThen \n\t\\[\n\t\t\\|P\\ast D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\\leqslant C\\varepsilon^\\alpha\\|D\\|_{C^{0,\\alpha}_\\kappa}.\n\t\\]\n\\end{lemma}\n\n\\begin{proof}\n\tLet us define $\\widehat{G}=1-\\widehat{P}$. Then, by taking the inverse Fourier transform we have for $x\\in \\mathbb{R}$\n\t\\[\n\t\tP(x)+G(x)=\\sqrt{2\\pi}\\delta_0(x).\n\t\\]\n\tNote also that $\\widehat{G}(0)=1$.\n\t\n\tNow \n\t\\begin{equation*}\n\t\\begin{split}\n\t\tP\\ast D(\\varepsilon\\cdot) & = \\sqrt{2\\pi}D(\\varepsilon\\cdot) - G\\ast D(\\varepsilon\\cdot)\\\\\n\t\t& = D(\\varepsilon\\cdot)\\int_{\\mathbb{R}}G(z)dz - \\int_{\\mathbb{R}}G(z)D(\\varepsilon(z-\\cdot))dz\\\\\n\t\t& = \\varepsilon^\\alpha\\int_{\\mathbb{R}}G(z)|z|^\\alpha\\frac{D(\\varepsilon\\cdot)-D(\\varepsilon(z-\\cdot))}{\\varepsilon^\\alpha|z|^\\alpha}dz.\n\t\\end{split}\n\t\\end{equation*}\n\t%\n\tLet us consider\n\t\\[\n\t\t\\begin{split}\n\t\t\t\\|P\\ast D(\\varepsilon\\cdot)\\|_{C^0[-L,L]} & \\leqslant \\varepsilon^\\alpha\\int_{\\mathbb{R}}|G(z)||z|^\\alpha\\|D\\|_{C^{0,\\alpha}[-L_z,L_z]}dz\\\\\n\t\t\t& \\leqslant \\varepsilon^\\alpha\\int_{\\mathbb{R}}|G(z)||z|^\\alpha L_z^\\kappa\\|D\\|_{C^{0,\\alpha}_\\kappa}dz,\n\t\t\\end{split}\n\t\\]\n\twhere $L_z=\\max\\{\\varepsilon L+\\varepsilon |z|,2\\}$. We can now divide both sides by $L^\\kappa$ to obtain\n\t\\begin{equation}\\label{e:so}\n\t\t\\|P\\ast D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\\leqslant \\varepsilon^\\alpha\\int_{\\mathbb{R}}|G(z)||z|^\\alpha \\Big(\\frac{L_z}{L}\\Big)^\\kappa dz \\|D\\|_{C^{0,\\alpha}_\\kappa}.\n\t\\end{equation}\n\tNow recall $\\varepsilon\\in (0,1)$ and $L>1$, so we derive\n\t\\[\n\t\t\\frac{L_z}{L}=\\max\\Big\\{\\varepsilon + \\varepsilon\\frac{|z|}{L},\\frac{2}{L}\\Big\\}\\leqslant 2+|z| \\;.\n\t\\]\n\tGoing back to \\eqref{e:so}\n\t\\[\n\t\t\\|P\\ast D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\\leqslant \\varepsilon^\\alpha\\int_{\\mathbb{R}}|G(z)||z|^\\alpha (2+|z|)^\\kappa dz \\|D\\|_{C^{0,\\alpha}_\\kappa}.\n\t\\]\n\tThe integral is actually finite and bounded by a constant $C_{\\alpha,\\kappa}$, as $\\widehat{G}$ is sufficiently smooth. \n\tSince any derivative has bounded support so $G$ decays sufficiently fast for the existence of the integral.\n\n\\end{proof}\n\nA simple modification of the previous proof yields the following Lemma:\n\n\\begin{lemma}\n \\label{lem:guidoext} \n\tLet $\\widehat{P}:\\mathbb{R}\\to [0,1]$ be a smooth function and $P$ its inverse Fourier transform. \n\tLet also $D\\in \\mathcal{C}^{0,\\alpha}_\\kappa$, with $\\alpha\\in (0,1)$, $\\kappa\\in(0,1\/2)$. \n\tThen \n\t\\[\n\t\t\\|P\\ast D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\\leqslant C \\varepsilon^\\alpha \\big[ \\|\\widehat{P}\\|_{L^2}^{1-\\alpha} \\|\\widehat{P}''\\|_{L^2}^\\alpha + \\|\\widehat{P}''\\|_{L^2} \\big]\\|D\\|_{C^{0,\\alpha}_\\kappa} + |\\widehat{P}(0)|\\|D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\\;.\n\t\\]\n\\end{lemma}\n\n\\begin{proof}\n First note that \n \\[\n[ P\\ast D(\\varepsilon\\cdot)](x) = \\int_{\\mathbb{R}} P(y) [D(\\varepsilon (x-y)) - D(\\varepsilon x) ] dy + \\widehat{P}(0) D(\\varepsilon x)\\;.\n \\]\n Then we can proceed as in the proof before to obtain \n \\begin{equation}\n \\label{e:cent}\n \\|P\\ast D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\n\\leqslant C\\varepsilon^\\alpha \\int_{\\mathbb{R}}|P(z)||z|^\\alpha (2+|z|)^\\kappa dz \\|D\\|_{C^{0,\\alpha}_\\kappa} + |\\widehat{P}(0)|\\|D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\\;.\n \\end{equation}\nNow we bound the integral. For $|z|\\geqslant1$ as $\\alpha+\\kappa \\leqslant 3\/2$\n\\[\n\\begin{split}\n \\int_{\\{|z|\\geqslant1\\}} |P(z)||z|^{\\alpha+\\kappa} dz \n & \\leqslant \\Big(\\int_{\\mathbb{R}}|P(z)|^2 |z|^4 dz \\Big)^{\\frac12} \n \\Big(\\int_{\\mathbb{R}} |z|^{2(\\alpha+\\kappa-2)} dz\\Big)^{\\frac12}\\\\\n & \\leqslant C \\|\\widehat{P}''\\|_{L^2}\\;,\n\\end{split}\n\\]\nwhere we used Plancherel theorem in the last step. For $|z|\\leqslant 1$ we have \n\\[\n\\begin{split}\n \\int_{\\{|z|\\leqslant 1\\}} |P(z)||z|^{\\alpha} dz \n & \\leqslant C \\Big(\\int_{\\{|z|\\leqslant 1\\}} |P(z)|^2|z|^{2\\alpha} dz\\Big)^{1\/2} \\\\\n&\\leqslant C \\Big(\\int_{\\mathbb{R}}|P(z)|^2 dz \\Big)^{(1-\\alpha)\/2} \n \\Big(\\int_{\\mathbb{R}} |P(z)|^2 |z|^{2} dz\\Big)^{\\alpha\/2}\\\\\n & \\leqslant C \\|\\widehat{P}\\|_{L^2}^{1-\\alpha} \\|\\widehat{P}'\\|_{L^2}^\\alpha\\;,\n\\end{split}\n\\]\nwhere we used first the Cauchy-Schwarz inequality, then H\\\"older's one with $p=1\/(1-\\alpha)$ and $q=1\/\\alpha$, and finally Plancherel theorem.\n\nPutting together all estimates finishes the proof.\n\\end{proof}\n\nUnfortunately, the previous two lemmas are not sufficient in Lemma~\\ref{lem:lemma1}.\nWhen the support of $\\widehat{P}$ is unbounded we have problems with the $L^2$-norm of $\\widehat{P}$, \nwhile higher derivatives are easier to bound. \n\n\\begin{corollary}\n \\label{cor:new}\n Let $\\widehat{P}:\\mathbb{R}\\to [0,1]$ be a smooth function, $P$ its inverse Fourier transform, \n and suppose that there is some $\\delta>0$ such that $\\text{supp}(\\widehat{P}) \\cap (-\\delta,\\delta)=\\emptyset$. \n Fix $\\alpha\\in (0,1)$, $\\kappa\\in(0,1\/2)$ and suppose that there is a $\\gamma>1$ such that $\\alpha+\\kappa+\\gamma\/2 \\in(1,2)$. \n \nThen for all $D\\in \\mathcal{C}^{0,\\alpha}_\\kappa$ we have\n\t\\[\n\t\t\\|P\\ast D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\n\t\t\\leqslant C \\varepsilon^\\alpha \\|\\widehat{P}'\\|_{L^2}^{2-\\alpha-\\kappa-\\frac{\\gamma}2} \\|\\widehat{P}''\\|_{L^2}^{\\alpha+\\kappa+\\frac{\\gamma}2-1 } \\|D\\|_{C^{0,\\alpha}_\\kappa}\\;.\n\t\\]\n\\end{corollary}\n\n\n\\begin{proof}\n\tFrom~\\eqref{e:cent} we obtain using H\\\"older's inequality\n\\[\n\\begin{split} \n\\|P\\ast D(\\varepsilon\\cdot)\\|_{C^0_\\kappa}\n& \\leqslant C\\varepsilon^\\alpha \\int_{\\mathbb{R}}|P(z)||z|^\\alpha (2+|z|)^\\kappa dz \\|D\\|_{C^{0,\\alpha}_\\kappa} \\\\\n&\\leqslant C\\varepsilon^\\alpha\\int_{\\{|z|\\geqslant \\delta\\}} |P(z)||z|^{\\alpha+\\kappa} dz\\|D\\|_{C^{0,\\alpha}_\\kappa} \\\\\n& \\leqslant C\\varepsilon^\\alpha \\Big(\\int_{\\{|z|\\geqslant \\delta\\}} |P(z)|^2|z|^{2\\alpha+2\\kappa+\\gamma} dz\\Big)^{1\/2} \\|D\\|_{C^{0,\\alpha}_\\kappa}\\\\\n& \\leqslant C\\varepsilon^\\alpha \\Big(\\int_{\\mathbb{R}} |zP(z)|^2|z|^{2\\alpha+2\\kappa+\\gamma-2} dz\\Big)^{1\/2} \\|D\\|_{C^{0,\\alpha}_\\kappa}.\n\\end{split}\n\\]\nNow as the exponent $2\\alpha+2\\kappa+\\gamma-2\\in(0,2)$, \nwe can use H\\\"older inequality to bound the integral above by the integral over $|z|^2|P(z)|^2$ and $|z|^4|P(z)|^2$, \nwhich in turn gives the $L^2$-norm of $\\widehat{P}'$ and $\\widehat{P}''$.\nWe obtain\n\\[\\begin{split} \n\\int_{\\mathbb{R}} |zP(z)|^2|z|^{2\\alpha+2\\kappa+\\gamma-2} dz\n&\\leqslant \\Big( \\int_{\\mathbb{R}} |zP(z)|^2 dz \\Big)^{2-\\alpha-\\kappa-\\frac{\\gamma}2} \n\\Big( \\int_{\\mathbb{R}} |z^2P(z)|^2 dz \\Big)^{\\alpha+\\kappa+\\frac{\\gamma}2-1 }\\\\\n&\\leqslant \\|\\widehat{P}'\\|_{L^2}^{2(2-\\alpha-\\kappa-\\frac{\\gamma}2)} \\|\\widehat{P}''\\|_{L^2}^{2(\\alpha+\\kappa+\\frac{\\gamma}2-1) },\n\\end{split}\\]\nwhich implies the claim.\n\\end{proof}\n\n\n\\subsection{Applications of Fourier estimates}\n\n\nNow we rephrase the bounds of the previous subsection to bound operators given by a Fourier multiplier, as for example\nin the statement of the exchange Lemmas~\\ref{lem:lemma1}, \\ref{lem:lemma2}, and \\ref{lem:lemma3}. \nAnother example we have in mind are bounds on the semigroup generated by the Swift-Hohenberg operator which is presented later in Corollary \\ref{cor:SGbound}.\n\nIn the first step we use regularity of the kernel to bound the operator.\n\\begin{lemma}\n\\label{lem:ours}\n\tLet $m>\\frac{1}{2}$ and $\\mathcal{H}\\cdot=H\\star\\cdot$ be an operator such that the Fourier transform\n\t$\\hat{H}$ of the kernel $H$ is in $H^m(\\mathbb{R})$.\n\tFor any $\\kappa \\in (0, m -\\frac12)$ there is a constant such that \n\t for all $u\\in C^0_\\kappa$ we have \n\t\\[ \n\t\\|\\mathcal{H} u\\|_{C^0_\\kappa} \\leqslant C \\|\\hat{H}\\|_{H^m(\\mathbb{R})} \\| u\\|_{C^0_\\kappa} \\;.\n\t\\]\n\\end{lemma}\n\n\\begin{proof} By the definition of the convolution\n\t\\begin{equation*}\n\t\\begin{split}\n\t\\mathcal{H} u (x) &= \\int_{\\mathbb{R}} H(x-y) u(y) dy = C\\int_{\\mathbb{R}} H(z) u(x-z) dz \\\\\n\t&= \\int_{\\mathbb{R}} H(z) (1+(x-z)^2)^{\\kappa\/2} w_\\kappa(x-z) u(x-z) dz.\n\t\\end{split}\n\t\\end{equation*}\n\tNow we use\n\t\\[1+(x-z)^2 \\leqslant 2(1+z^2)(1+x^2)\\]\n\tto obtain\n\t\\begin{equation*}\n\\begin{split}\n\t|\\mathcal{H} u (x)|\n\t&= C(1+x^2)^{\\kappa\/2} \\int_{\\mathbb{R}}|H(z)| (1+z^2)^{\\kappa\/2} dz \\| u\\|_{C^0_\\kappa}\\\\\n\t& \\leqslant C(1+x^2)^{\\kappa\/2} \\Big( \\int_{\\mathbb{R}}(1+z^2)^{-m+\\kappa} dz\\Big)^{\\frac{1}{2}} \\Big( \\int_{\\mathbb{R}}|H(z)|^2 (1+z^2)^{m} dz\\Big)^{\\frac{1}{2}} \\| u\\|_{C^0_\\kappa}.\n\\end{split}\n\t\\end{equation*}\n\t%\n\tWe finish the proof by noting that $ m -\\kappa >\\frac12$ by assumption, and that by Plancherel theorem\n\t\\[\n\t\\int_{\\mathbb{R}}|H(z)|^2 (1+z^2)^{m} dz =\\|\\hat{H}\\|_{H^m(\\mathbb{R})} \\;.\\qedhere\n\t\\]\n\\end{proof}\nIn order for the previous Lemma to be useful in our case, we have to control the $H^m$-norm of the kernel. \nThis is straightforward for the Swift-Hohenberg semigroup \nif we add a smooth projection on bounded Fourier domains. \n\\begin{lemma}\\label{lem:thisone}\n\tFix $m\\in[0,1)$ and $\\ell\\in\\mathbb{Z}$. Consider $\\widehat{P} :\n\t\\mathbbm{R} \\rightarrow [0, 1]$ smooth with $\\textup{supp} (\\widehat{P}) \\subset [\\ell - 2 \\delta, \\ell + 2\n\t\\delta]$ for some $0<\\delta < 1 \/ 2$. Then it holds that\n\t\\[\n\t\t\\sup_{t \\in [0, T_0 \\varepsilon^{-2}]} \\|\\widehat{P}e^{\\lambda_\\nu t}\\|_{H^m}\\leqslant C \\varepsilon^{-\\max\\{0,m-\\frac12\\}}\n\t\\]\n\twhere $\\lambda_\\nu(k)=-(1-k^2)^2+\\nu\\varepsilon^2= - (1 - k)^2 (1+ k)^2 + \\nu \\varepsilon^2$ is the Fourier-symbol of the operator ${\\mathcal{L}}_\\nu$.\n\\end{lemma}\n\n\\begin{corollary}\n\\label{cor:SGbound}\nConsider the Fourier-projection $\\mathcal{P}=P\\star\\cdot$ with $\\widehat{P}$ as in the lemma above,\nthen we obtain in case $\\kappa \\in (0, \\frac14)$ with $m=\\frac12+\\kappa$ that \n\n\t\\[\n\t\t\\|e^{\\mathcal{L}_\\nu t}\\mathcal{P}u\\|_{C^0_\\kappa}\\leqslant C\\|\\widehat{P}e^{\\lambda_\\nu t}\\|_{H^m}\\|u\\|_{C^0_\\kappa}\n\t\t\\leqslant C \\epsilon^{-\\kappa} \\|u\\|_{C^0_\\kappa}\\;.\n\t\\]\n\tfor all $t\\in[0,T_0\\varepsilon^{-2}]$.\n\\end{corollary}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:thisone}]\nFor the proof we only focus on the most complicated case $\\ell=1$, i.e. with $\\textup{supp}(P)\\subset[1-2\\delta,1+2\\delta]$. The \ncase $\\ell=-1$ is almost verbatim and for $|\\ell|\\not=1$ the proof is actually much simpler, \nas $\\lambda_\\nu \\leqslant -c <0$ is strictly negative there and we obtain exponentially small terms.\n\n A straightforward calculation shows\n\t\\begin{equation*}\n\t\\begin{split}\n\t\t\\| \\widehat{P} e^{\\lambda_\\nu t} \\|^2_{H^1} & = \\| \\widehat{P} (\\cdot - 1)\n\t\te^{\\lambda_\\nu (\\cdot - 1) t} \\|^2_{H^1 (- 2 \\delta, 2 \\delta)}\\\\\n\t\n\t\t& \\leqslant C\n\t\t\\int_{- 2 \\delta}^{2 \\delta} e^{- 2 (2 - k)^2 k^2 t} d k \\\\\n\t\t& \\qquad + C \\int_{- 2 \\delta}^{2 \\delta} 4 (2 (k - 2) k^2 + (k - 2)^2 k)^2 t^2\n\t\te^{- 2 (2 - k)^2 k^2 t} d k\\\\\n\t\n\t\t& \\leqslant C \\int_{- 2 \\delta}^{2 \\delta}\n\t\te^{- C_{\\delta} k^2 t} d k + C \\int_{- 2 \\delta}^{2 \\delta} (k^2 + k)^2\n\t\tt^2 e^{- C_{\\delta} k^2 t} d k\\\\\n\t\n\t\t& \\leqslant C + C \\int_{- 2 \\delta}^{2 \\delta} k^2\n\t\tt^2 e^{- C_{\\delta} k^2 t} d k.\n\t\\end{split}\n\t\\end{equation*}\n\tNow we have to consider two cases, depending on $t$. First if $t \\leqslant 1$, then\n\talso the second integral can be bound by a constant $C$. If $t > 1$, then we can\n\tcontinue with the substitution $l = \\sqrt{t} k$, which gives\n\t$d l = \\sqrt{t} d k$, and we derive\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\\| \\widehat{P} e^{\\lambda_\\nu t} \\|^2_{H^1} & \\leqslant C + C \\int_{- 2\n\t\t\t\\delta \\sqrt{t}}^{2 \\delta \\sqrt{t}} l^2 \\sqrt{t} e^{- Cl^2} d l\n\t \\leqslant C + C \\sqrt{t} \\int_{- \\infty}^{\\infty} l^2 e^{- Cl^2} d l\\\\\n\t\t& \\leqslant C \\sqrt{t} \\;.\n\t\t\\end{split}\n\t\\end{equation*}\n\tThus\n\t\\[ \\| \\widehat{P} e^{\\lambda_\\nu t} \\|_{H^1}^2 \\leqslant \\left\\{\n\t\\begin{array}{ll}\n\tC \\sqrt{t} \\;, & t \\geqslant 1\\\\\n\tC \\;, & t \\leqslant 1\n\t\\end{array} \\right. \\;.\\]\n\tIn a similar way, we can consider the bounds in $L^2$:\n\t\\begin{equation*}\n\t\t\\| \\widehat{P} e^{\\lambda_\\nu t} \\|^2_{L^2} \\leqslant C \\int_{- 2\n\t\t\t\\delta}^{2 \\delta} e^{- C_{\\delta} k^2 t} d k \n\t\t = C \\int_{- 2 \\delta \\sqrt{t}}^{2 \\delta \\sqrt{t}} \\frac{1}{\\sqrt{t}}\n\t\te^{- C_{\\delta} l^2} d l \n\t\t \\leqslant \\left\\{\n\t\\begin{array}{ll}\n\tC t^{-1\/2} \\;, & t \\geqslant 1\\\\\n\tC \\;,& t \\leqslant 1\n\t\\end{array} \\right. \n\t\t\\;.\n\t\\end{equation*}\n\tWe finally get to the bounds in $H^m$ by interpolation:\n\t\\begin{equation*}\n\t\t\\| \\widehat{P} e^{\\lambda_\\nu t} \\|_{H^m} \n\t\t \\leqslant C \\| \\widehat{P}e^{\\lambda_\\nu t} \\|^{1 - m}_{L^2} \\| \\widehat{P} e^{\\lambda_\\nu t} \\|^m_{H^1}\n\t\t \\leqslant \\left\\{ \\begin{array}{ll}\n\t\t\tC\\;,& t \\leqslant 1\\\\\n\t\t\tCt^{- \\frac{1}{4}+\\frac{m}{2}}\\;,& t \\geqslant 1\n\t\t\\end{array} \\right\\} \\leq C \\varepsilon^{-\\max\\{0,m-\\frac12\\}}\n\t\\end{equation*}\n\tfor all $t\\leqslant T_0\\varepsilon^{-2}$.\n\\end{proof}\n\n\n\\subsection{Proof of Exchange Lemma II}\n\nFor the proof of Lemma~\\ref{lem:lemma3} we write the differences of semigroups as convolution operators.\n\nFirst we define a smooth Fourier-multiplier that cuts out regions around $\\pm1$ in Fourier space, \nwhere the eigenvalues of the Swift-Hohenberg operator are close to $0$.\nFix a small $\\delta>0$ independent of $0<\\varepsilon\\ll 1$ and consider a smooth function $\\widehat{P}:\\mathbb{R}\\to[0,1]$ such that \n$\\textup{supp}(\\widehat{P})=[-1-2\\delta, -1+2\\delta]\\cup[1-2\\delta, 1+2\\delta]$\nand $\\widehat{P}=1$ on $[-1-\\delta, -1+\\delta]\\cup[1-\\delta, 1+\\delta]$.\nWe define $\\hat{Q}=1-\\widehat{P}^2$ and let $\\mathcal{Q}=I-\\mathcal{P}^2$.\n\n\nNow we obtain\n\\[ \ne^{T \\varepsilon^2 \\mathcal{L}_{\\nu}} [D (\\varepsilon \\cdot) \\mathrm{e}_3] \n=\ne^{T \\varepsilon^2 \\mathcal{L}_{\\nu}} \\mathcal{P} [\\mathcal{P}D\n(\\varepsilon \\cdot) \\mathrm{e}_3] + e^{T \\varepsilon^2 \\mathcal{L}_{\\nu}}\n\\mathcal{Q} [D (\\varepsilon \\cdot) \\mathrm{e}_3]\n\\]\nand we bound separately the two terms. For the first term we use the semigroup\nestimate from Corollary \\ref{cor:SGbound} (with $\\ell=\\pm1$), \nusing the $H^\\alpha$-estimate on the kernel and Lemma \\ref{lem:guidos}.\nNote that for the application we need to split the estimate\ninto two terms: one concentrated around $1$ and the other around $-1$.\nWe obtain\n\\[ \n\\| e^{T \\varepsilon^2 \\mathcal{L}_{\\nu}} \\mathcal{P} [\\mathcal{P}D\n(\\varepsilon \\cdot) \\mathrm{e}_3] \\|_{C_{\\kappa}^0}\n\\leqslant C \\epsilon^{-\\kappa}\n\\| \\mathcal{P}D\n(\\varepsilon \\cdot) \\mathrm{e}_3 \\|_{C_{\\kappa}^0}\n\\leqslant C\n\\varepsilon^{\\alpha-\\kappa} \\| D \\|_{C_{\\kappa}^{0, \\alpha}}\\;.\n\\]\nFor the second term we need some more work. We start by writing\n\\[ \ne^{T \\varepsilon^2 \\mathcal{L}_{\\nu}} \\mathcal{Q} [D (\\varepsilon \\cdot)\n\\mathrm{e}_3] = (\\mathcal{H}_T D) (\\varepsilon \\cdot) \\mathrm{e}_3\n\\]\nand denoting the kernel of $\\mathcal{H}_T=H_T\\star$ by\n\\[ \n{\\hat{H}_T} (k) = e^{\\nu T} e^{- T \\varepsilon^{- 2} (4 + k \\varepsilon)^2 (2 + k\n\t\\varepsilon)^2} \\hat{Q} (3 + \\varepsilon k) \\;.\n\\]\nIn view of Lemma \\ref{lem:ours} we only need to bound the $H^1$-norm of the kernel ${\\hat{H}_T}$.\nTherefore, we split the $H^1$-norm into two different areas in Fourier space\n\\[ \\| {\\hat{H}_T} \\|^2_{H^1 (\\mathbbm{R})} \\leqslant 2 \\| {\\hat{H}_T} \\|^2_{H^1 \\left( \\left[ -\n\t\\frac{c}{\\varepsilon}, \\frac{c}{\\varepsilon} \\right] \\right)} + 2 \\| {\\hat{H}_T}\n\\|^2_{H^1 \\left( \\left[ - \\frac{c}{\\varepsilon}, \\frac{c}{\\varepsilon}\n\t\\right]^C \\right)}\\;.\n\t\\]\nNote first that both $\\hat{Q}$ and $\\hat{Q}'$ are bounded smooth functions independent of $\\varepsilon$.\nThen we use in the first term that $\\varepsilon | k | \\leqslant C$ and that $| T | $ is bounded.\n\nThus ${\\hat{H}_T}$ is uniformly bounded on $[ -\\frac{c}{\\varepsilon}, \\frac{c}{\\varepsilon}]$ by $ C e^{- TC_0 \\varepsilon^{- 2}}$ \nwhere $- C_0$ is the level\nwhere we cut out the two bumps around $- 2 \/ \\varepsilon$ and $- 4 \/\\varepsilon$. Note that $\\hat{Q}$ is identically zero there.\nWith similar arguments we show that the derivative ${\\hat{H}_T}'$ is uniformly bounded by\n$ C \\varepsilon (1+T\\varepsilon^{-2}) e^{- TC_0 \\varepsilon^{- 2}}$. Thus\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\| {\\hat{H}_T} \\|^2_{H^1 \\left( \\left[ - \\frac{c}{\\varepsilon}, \\frac{c}{\\varepsilon}\n\t\t\\right] \\right)}\n\t\t& \\leqslant C \\int_{-\\frac{c}{\\varepsilon}}^{\\frac{c}{\\varepsilon}} (1 + \\varepsilon^2+ T^2 \\varepsilon^{-2}) e^{- 2TC_0 \\varepsilon^{- 2}} dk\\\\\n\t\t& \\leqslant C \\varepsilon^{- 1} (1 + T^2 \\varepsilon^{- 2}) e^{-2 TC_0\n\t\t\\varepsilon^{- 2}}\\\\\n\t\t& \\leqslant C \\varepsilon^{- 1} e^{- TC_0 \\varepsilon^{- 2}}\\\\\n\t\n\t\t& \\leqslant C \\varepsilon^{4 \\gamma - 1} T^{- 2 \\gamma}\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we used first that $xe^{-x} \\leqslant 1$ and then that $e^{-x} \\leqslant C_\\gamma x^{-\\gamma}$.\nThe final estimate is not necessary at this point, but it is still sufficient for our purposes, \nas other terms in the estimate are bounded by this weaker estimate. \n\n\nNow we have to consider the case $\\varepsilon | k | > c$ \nwhen we are away from the bumps. In this case, by adjusting $c$ we can use that $\\hat{Q}$ is a constant.\nMoreover, the bound for negative and positive $k$ is the same, so we restrict ourselves to the case with $k>c\/\\varepsilon$. \n\\begin{equation*}\n\t\\begin{split}\n\t\t\\| {\\hat{H}_T} \\|^2_{H^1 \\left( \\left[ \\frac{c}{\\varepsilon}, + \\infty \\right)\n\t\t\\right)} & \\leqslant \\int_{\\frac{c}{\\varepsilon}}^{\\infty} e^{2 T\n\t\t\\varepsilon^{- 2} (4 + k \\varepsilon)^2 (2 + k \\varepsilon)^2} dk \\\\\n\t\t&\\quad + \\int_{\\frac{c}{\\varepsilon}}^{\\infty} e^{2 T \\varepsilon^{- 2} (4 +\n\t\tk \\varepsilon)^2 (2 + k \\varepsilon)^2} (T \\varepsilon^{- 2} (4 + \\varepsilon k) (2+ \\varepsilon k) (3 +\\varepsilon k) \\varepsilon)^2 dk\\\\\n\n\t& \\leqslant \\frac{1}{\\varepsilon} \\int_c^{\\infty} e^{2\n\t\tT \\varepsilon^{- 2} (4 + k)^2 (2 + k )^2} dk \\\\\n\t& \\quad + \\frac{C}{\\varepsilon} \\int_c^{\\infty} e^{2 T \\varepsilon^{- 2} (4\n\t\t+ k )^2 (2 + k )^2} (T \\varepsilon^{- 1} (4 + k) \n\t(2 + k) (3 + k))^2 dk \\;.\n\t\\end{split}\n\\end{equation*}\nNow we use that $(4 + k)^2 (2 + k)^2 \\geqslant k^4$ and $(4 + k) (2 + k) (3\n+ k) \\leqslant C (k^3 + 1) \\leqslant Ck^3$, for $|h|> \\frac{C}{\\varepsilon}$ with an $\\varepsilon$-independent constant $C$, so that\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\| {\\hat{H}_T} \\|^2_{H^1 \\left( \\left[ \\frac{c}{\\varepsilon}, + \\infty \\right)\n\t\t\\right)} & \\leqslant \\frac{C}{\\varepsilon} \\int_c^{\\infty} e^{- CT\n\t\t\\varepsilon^{- 2} k^4} dk + CT^2 \\varepsilon^{- 3} \\int_c^{\\infty} k^3 e^{-\n\t\tCT \\varepsilon^{- 2} k^4} dk\\\\\n\t\n\t\t& =\n\t\t\\frac{C}{\\varepsilon} (T \\varepsilon^{- 2})^{- \\frac{1}{4}} \\int_{c (T\n\t\t\\varepsilon^{- 2})^{\\frac{1}{4}}}^{\\infty} e^{- Ck^4} dk \\\\\n\t\t& \\quad+ CT^2 \\varepsilon^{- 3} (T \\varepsilon^{- 2})^{- \\frac{1}{4}} \n\t\t\\int_{c (T \\varepsilon^{- 2})^{\\frac{1}{4}}}^{\\infty} k^3 (T \\varepsilon^{-\n\t\t2})^{- \\frac{3}{4}} e^{- Ck^4} dk\\\\\n\t\t& \\leqslant \\frac{C}{\\varepsilon} (T \\varepsilon^{- 2})^{- \\frac{1}{4}} \n\t\t\\int_{c (T \\varepsilon^{- 2})^{\\frac{1}{4}}}^{\\infty} e^{- Ck^4} dk + T\n\t\t\\varepsilon \\int_{0}^{\\infty} k^3 e^{-\n\t\tCk^4} dk \\\\\n\t\t& \\leqslant \\frac{C}{\\varepsilon} (T \\varepsilon^{- 2})^{- \\frac{1}{4}} \n\t\\int_{c (T \\varepsilon^{- 2})^{\\frac{1}{4}}}^{\\infty} e^{- Ck^4} dk + C\\varepsilon \\;.\n\t\\end{split}\n\\end{equation*}\nFor the remaining term we use that for $\\alpha \\geqslant 0$ \n\\[ \n\t\\sup_{z>0}\\Big\\{z^{\\alpha} \\int_z^{\\infty} e^{- ck^4} dk\\Big\\} <\\infty,\n\\]\nto obtain for $\\gamma= (1+\\alpha)\/8 \\geqslant 1\/8 $ \n\\[\n\\| {\\hat{H}_T} \\|^2_{H^1 \\left( \\left[ \\frac{c}{\\varepsilon}, + \\infty \\right)\n\t\t\\right)} \n\t\t \\leqslant C \\varepsilon^{4\\gamma - 1} T^{- 2 \\gamma}+ C\\varepsilon \\;.\n\\]\nNote finally, that for $\\gamma<1\/2$ and bounded $T$ \nwe have $\\varepsilon \\leqslant C \\varepsilon^{4\\gamma - 1} T^{- 2 \\gamma}$ and \nwe can neglect the $ C\\varepsilon$ in the estimate above.\n\n\n\\subsection{Proof of Exchange Lemma I}\n\n\n\nThe proof of the Exchange Lemma I stated in Lemma~\\ref{lem:lemma2} is similar to the one for the Exchange Lemma II in Lemma~\\ref{lem:lemma3}, but requires additional arguments.\nWe start again by smoothly projecting in Fourier-space, but now in $k=1$ and $k=3$.\n\nFix a small $\\delta>0$ and consider for $\\ell\\in\\mathbb{Z}$ a smooth function $\\hat{P_\\ell}:\\mathbb{R}\\to[0,1]$ such that \n$\\textup{supp}(\\hat{P_\\ell})=[-\\ell-2\\delta, -\\ell+2\\delta]\\cup[\\ell-2\\delta, \\ell+2\\delta]$\nand $\\widehat{P}_\\ell=1$ on $[-\\ell-\\delta, -\\ell+\\delta]\\cup[\\ell-\\delta, \\ell+\\delta]$.\n\nNow we can rewrite:\n\\begin{multline*}\n\te^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] - (e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1 = \\mathcal{P}_3^2 e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] + \\mathcal{P}_1^2 e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1]\\\\ \n\t- \\mathcal{P}_1^2 (e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1 +(1-\\mathcal{P}_1^2-\\mathcal{P}_3^2)e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] -(1-\\mathcal{P}_1^2)(e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1.\n\\end{multline*}\nNow the first term on the right hand side is bounded the same way as the second term in the proof of the Exchange Lemma II (Lemma~\\ref{lem:lemma3}) in the previous section.\nAlso the last two terms can be controlled in a similar way as the first term in the proof of Lemma~\\ref{lem:lemma3}.\nWe only need the semigroup\nestimate from Corollary \\ref{cor:SGbound} (now for $\\ell=\\pm1$ and $\\ell=\\pm3$) and Lemma \\ref{lem:guidos}.\n\n\nLet us focus on the missing two terms:\n\\[\n\t\\mathcal{P}_1^2 e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] - \\mathcal{P}_1^2 (e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1 =: \\mathcal{H}D,\n\\]\nwith $\\mathcal{H}\\cdot=H\\star\\cdot$ such that $\\textup{supp}(\\hat{H})\\subset (-2\\delta\/\\varepsilon, 2\\delta\/\\varepsilon)$ and \n\\[\n\\begin{split}\n\t\\hat{H} & = \\widehat{P}(\\varepsilon\\cdot)[e^{T\\varepsilon^{-2}\\lambda_\\nu(1+l\\varepsilon)}-e^{-4l^2T+\\nu T}]\\\\\n\t& = \\widehat{P}(\\varepsilon\\cdot)e^{-4Tl^{2}+\\nu T}[e^{-4Tl^{3}\\varepsilon-l^4T\\varepsilon^2}-1].\n\\end{split}\n\\]\nIn view of Lemma \\ref{lem:ours} it is enough to show that $\\|\\hat{H}\\|_{H^1(\\mathbb{R})}$ is small. Thus we need the derivative\n\\begin{multline*}\n\\frac{d}{dl}\\hat{H} = \\widehat{P}^\\prime(\\varepsilon\\cdot)e^{-4Tl^{2}+\\nu T}[e^{-4Tl^{3}\\varepsilon-l^4T\\varepsilon^2}-1]-8Tl \\widehat{P}(\\varepsilon\\cdot)e^{-4Tl^{2}+\\nu T}[e^{-4Tl^{3}\\varepsilon-l^4T\\varepsilon^2}-1]\\\\\n-4T\\varepsilon l^2(3+4\\varepsilon l) \\widehat{P}(\\varepsilon\\cdot)e^{-4Tl^{2}+\\nu T}[e^{-4Tl^{3}\\varepsilon-l^4T\\varepsilon^2}-1].\n\\end{multline*}\nNow we collect the common exponential term in the parenthesis, and then we write the Taylor expansion in $l=0$.\n To get the estimate in $H^1$ we bound both $\\hat{H}$ and $\\frac{d}{dl}\\hat{H}$ in $L^2$.\n \nActually we can do better and provide point-wise estimates (and not just $L^2$). First of all we observe that by means of Taylor expansion and triangular inequality\n\\[\n |1-e^z|\\leqslant |z|e^{|z|},\n\\]\nand in our case $z = -4Tl^{3}\\varepsilon-l^4T\\varepsilon^2$, with $|z|\\leqslant 8T\\delta l^2 + 4\\delta^2Tl^2\\leqslant 2Tl^2$ \nif $|\\ell| \\leqslant \\delta\/\\varepsilon$ for some small fixed $\\delta\\leqslant 1\/2$.\n\nIf we consider $\\hat{H}$ we have the following bound:\n\\begin{equation*}\n\t\\begin{split}\n\t\t|\\hat{H}| & \\leqslant C_{T_0}e^{-2Tl^2}\\left(|4Tl^3\\varepsilon|+ |l^4T\\varepsilon^2|\\right)\\\\\n\t\t& \\leqslant C_{T_0}T^{-1\/2}e^{-Tl^2}\\left(4\\varepsilon+ l\\varepsilon^2\\right)\\\\\n\t\t& \\leqslant C_{T_0}T^{-1\/2}e^{-Tl^2}\\left(4+2\\delta\\right)\\varepsilon\\;,\n\t\\end{split}\n\\end{equation*}\nwhere we used the inequality\n\\[\n\te^{-Tl^2}\\left(Tl^2\\right)^{3\/2}\\leqslant C.\n\\]\n\nIn the same way we can bound the derivative of $\\hat{H}$:\n\\[\n\t\\hat{H}^\\prime = \\left(\\varepsilon\\widehat{P}^\\prime - 8Tl\\widehat{P}-4T\\varepsilon l^2(3+4\\varepsilon l)\\widehat{P}\\right)e^{-4Tl^{2}+\\nu T}[e^{-4Tl^{3}\\varepsilon-l^4T\\varepsilon^2}-1],\n\\]\nwhere we have almost $\\hat{H}$ with a different prefactor that we can bound by using the previous one on $\\hat{H}$ and the fact that $\\varepsilon l\\leqslant \\delta$ such that\n\\[\n\t|\\hat{H}^\\prime|\\leqslant C(1+Tl)T^{-1\/2}e^{-Tl^2}\\varepsilon.\n\\]\nNow we use that $\\sqrt{T}le^{-1\/2\\cdot Tl^2}\\leqslant C$, so\n\\[\n\t|\\hat{H}^\\prime|\\leqslant C(1+\\sqrt{T})T^{-1\/2}e^{-1\/2\\cdot Tl^2}\\varepsilon\\;.\n\\]\nThus using $T\\leqslant T_0$\n\\[\n\t\\begin{split}\n\t\t\\|\\hat{H}\\|_{H^1} & \\leqslant CT^{-1\/2}\\varepsilon\\|^{-1\/2\\cdot Tl^2}\\|_{L^2}\\\\\n\t\t& \\leqslant C T^{-3\/4}\\varepsilon \\;.\n\t\\end{split}\n\\]\n \n \n\n\\subsection{Proof of Exchange Lemma IC}\n\n\n\nThe idea behind this proof is almost the same as before, but the proof itself is technically slightly different, relies on Corollary~\\ref{cor:new}, and does not need $L^2$-estimates on the kernel.\n\nWe start again by smoothly projecting in Fourier-space, but onto the modes $k=\\pm1$ and $k=\\pm3$.\n\nFix a small $\\delta>0$ and consider a smooth function $\\widehat{P}:\\mathbb{R}\\to[0,1]$ such that \n$\\textup{supp}(\\widehat{P})=[-2\\delta, 2\\delta]$\nand $\\widehat{P}=1$ on $[-\\delta, \\delta]$.\nDefine now for $\\ell\\in\\mathbb{Z}$ the function \n\\[\\widehat{P}_\\ell:\\mathbb{R}\\to[0,1] \\text{ defined by }\\widehat{P}_\\ell(k)=\\widehat{P}(k-\\ell).\\]\nNow we can rewrite:\n\\begin{equation}\n\\label{e:ELIC}\n\\begin{split}\n\te^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] - (e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1 \n\t = & \\mathcal{P}_3^2 e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] \\\\\n\t& + \\mathcal{P}_1 e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] - \\mathcal{P}_1 (e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1 \\\\\n\t& +(1-\\mathcal{P}_1-\\mathcal{P}_3^2)e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] \\\\ \n\t&- (1-\\mathcal{P}_1)(e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1.\n\\end{split}\n\\end{equation}\n\\subsubsection{First term}\nNow the first term is bounded the same way as the second term in the proof the Exchange Lemma II (Lemma~\\ref{lem:lemma3}).\nWe only need the semigroup\nestimate from Corollary \\ref{cor:SGbound} and Lemma \\ref{lem:guidos} to obtain.\n\\[ \n\\| e^{T \\varepsilon^2 \\mathcal{L}_{\\nu}} \\mathcal{P}_3^2 [D(\\varepsilon \\cdot) \\mathrm{e}_1] \\|_{C_{\\kappa}^0}\n\\leqslant C\n\\varepsilon^{\\alpha-\\kappa} \\| D \\|_{C_{\\kappa}^{0, \\alpha}}\\;.\n\\]\n\\subsubsection{Second term}\nWe can write the second term in view of Lemma \\ref{lem:guidoext}:\n\\[\n \\mathcal{P}_1 e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] - \\mathcal{P}_1 (e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1 = \\mathcal{H} [D(\\varepsilon \\cdot)] \\cdot e_1\n\\]\nwith convolution operator $ \\mathcal{H}_T\\cdot = H_T\\star\\cdot$ with Fourier transform\n\\[\n\\begin{split}\n \\hat{H}_T(k)& = \\widehat{P}(k)[ e^{-T \\varepsilon^{-2}k^2( k+2)^2} - e^{-4k^2 \\varepsilon^{-2} T}]e^{\\nu T}\n \\\\& = \\widehat{P}(k)[ e^{-T\\varepsilon^{-2}(4k^3+k^4)} - 1]e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T},\n \\end{split}\n\\]\nwhere $\\hat{H}_T(0)=0$.\nNow we bound the $L^2$-norms of $\\hat{H}_T$, $\\hat{H}^\\prime_T$, and $\\hat{H}''_T$, and apply the results in Lemma~\\ref{lem:guidoext}.\nWe can get the following point-wise bound, \nusing the support of $\\widehat{P}$ together with mean-value theorem and $\\delta<1\/2$\n\\begin{equation}\n\\label{e:*}\n\\begin{split}\n |\\hat{H}_T(k)|& \\leqslant C |\\widehat{P}(k) T\\varepsilon^{-2}|4k^3+k^4| e^{-k^2 \\varepsilon^{-2} T}\n \\\\& \\leqslant C |\\widehat{P}(k)| T\\varepsilon^{-2}|k|^3 e^{-k^2 \\varepsilon^{-2} T},\n \\end{split}\n\\end{equation}\nfor all $\\gamma \\geqslant0$.\n\nWe use that for $a>0$ and $\\xi>0$\n\\[\n\\begin{split}\n\\int_0^{2\\delta} k^a e^{-\\xi k^2} dk & \n = \\int_0^{2\\delta\\sqrt{\\xi}} \\xi^{-1\/2-a\/2} k^a e^{- k^2} dk \n \\\\& \\leqslant C \\min\\{ \\xi^{-1-a}\\ , \\ 1 \\}^{1\/2}.\n \\end{split}\n\\]\nThus for the $L^2$-norm we integrate the squared inequality \\eqref{e:*} \nand use the previous estimate with $a=6$ and $\\xi=T\\varepsilon^{-2}$ to obtain\n\n\\[\n\\|\\hat{H}_T\\|_{L^2} \\leqslant C (T\\varepsilon^{-2})\\min\\{ (T\\varepsilon^{-2})^{-7}\\ , \\ 1 \\}^{1\/4} \\leqslant C \\;.\n\\]\n\nFor the first derivative\n\\[\n\\begin{split}\n \\hat{H}'_T (k)\n =& \\widehat{P}'(k)[ e^{-4T\\varepsilon^{-2}(4k^3+k^4)} - 1]e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T} \\\\\n & + \\widehat{P}(k)[ (-T\\varepsilon^{-2}(12k^2+4k^3)) e^{-T \\varepsilon^{-2}(4k^3+k^4)} e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T} \\\\\n & + \\widehat{P}(k)[ e^{-T\\varepsilon^{-2}(4k^3+k^4)} - 1] (-8k \\varepsilon^{-2} T) e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T}.\n \\end{split}\n\\]\nAs before, \n\\[\n\\begin{split}\n |\\hat{H}'_T (k)|\n \\leqslant & C |\\widehat{P}'(k)| T\\varepsilon^{-2} |k|^3 e^{-4k^2 \\varepsilon^{-2} T} \n + C |\\widehat{P}(k)| [ (T\\varepsilon^{-2}) k^2 + (T\\varepsilon^{-2})^2 k^4 ]e^{-k^2 \\varepsilon^{-2} T} \\\\\n \\leqslant & C_\\delta |\\widehat{P}'(k)| T\\varepsilon^{-2} e^{- \\delta^2 \\varepsilon^{-2} T} \n + C |\\widehat{P}(k)| (T\\varepsilon^{-2}) k^2 e^{-k^2 \\varepsilon^{-2} T} . \n \\end{split}\n\\]\nThus for the $L^2$-norm\n\\[\n\\|\\hat{H}'_T\\|_{L^2} \\leqslant C+ C (T\\varepsilon^{-2})\\min\\{ (T\\varepsilon^{-2})^{-5}\\ ; \\ 1 \\}^{1\/4} \\leqslant C \\;.\n\\]\n\nFor the second derivative we obtain similarly\n\\[\n\\begin{split}\n |\\hat{H}''_T (k)|\n \\leqslant & \n C_\\delta |\\widehat{P}'' (k)| T\\varepsilon^{-2} e^{- \\delta^2 \\varepsilon^{-2} T} \\\\\n & + C_\\delta |\\widehat{P}'(k)| [T\\varepsilon^{-2} + (T\\varepsilon^{-2})^2] e^{- \\delta \\varepsilon^{-2} T} \\\\\n &+ C |\\widehat{P}(k)| [ (T\\varepsilon^{-2}) |k| + (T\\varepsilon^{-2})^2 |k|^3 + (T\\varepsilon^{-2})^3 |k|^5] e^{-k^2 \\varepsilon^{-2} T} ,\n \\end{split}\n\\]\nso for the $L^2$-norm\n\\[\n\\|\\hat{H}''_T\\|_{L^2} \\leqslant C+ C (T\\varepsilon^{-2})\\min\\{ (T\\varepsilon^{-2})^{-3}\\ , \\ 1 \\}^{1\/4} \\leqslant C \\varepsilon^{-1\/2} \\;.\n\\]\nNow Lemma \\ref{lem:guidoext} yields: \n \\[\n\\| \\mathcal{P}_1 e^{t{\\mathcal{L}}_\\nu}[D(\\varepsilon \\cdot){\\mathrm{e}}_1] - \\mathcal{P}_1 (e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1\\|_{C^0_\\kappa}\n\\leqslant C\\varepsilon^\\alpha (1+\\varepsilon^{-1\/2}) \\|D\\|_{C^{0,\\alpha}_\\kappa}.\n \\]\n\n %\n \\subsubsection{Final two terms}\n Let us now turn to the last two terms in~\\eqref{e:ELIC} where we need Corollary \\ref{cor:new}.\nBoth are bounded in a similar way. We focus only on the last one.\nFor the other one, we cut out a small part in the middle and then bound the infinite rest as done here. \nRecall that the argument is slightly asymmetric, as we only have a $\\mathcal{P}_1$ but a $\\mathcal{P}_3^2.$\n\nWe have\n\\[ \n(1-\\mathcal{P}_1)(e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1\n= \\mathcal{H} [D(\\varepsilon \\cdot)] \\cdot e_1,\n\\]\nwith convolution operator $ \\mathcal{H}_T = H_T\\star$ with Fourier transform\n\\[\n \\hat{H}_T(k) = \\hat{Q}(k) e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T},\n\\]\nwhere $\\hat{H}_T(0)=0$ and we defined here $\\hat{Q}(k)=1-\\widehat{P}(k)$, which is slightly different $\\hat{Q}$ as defined before, \n\nbut it has the same properties. It is smooth, has support outside of $[-\\delta,\\delta]$, and is constant outside $[-2\\delta,2\\delta]$.\nThe bounded support is a key point in the argument for this Exchange Lemma, because the $L^2$-norm is not small uniformly in $T$, \nso we need to use Corollary~\\ref{cor:new} instead of Lemma~\\ref{lem:guidoext}.\n\nNow \n\n\\begin{align*}\n \\hat{H}_T'(k) &= \\hat{Q}'(k) e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T} - \\hat{Q}(k)T\\varepsilon^{-2}8k e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T},\\\\\n \\hat{H}_T''(k) &= \\hat{Q}''(k) e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T} -16\\hat{Q}'(k)T\\varepsilon^{-2}8k e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T} \n \\\\&\\qquad+ \\hat{Q}(k) [ (8T\\varepsilon^{-2}k )^2 -T\\varepsilon^{-2}8 ] e^{-4k^2 \\varepsilon^{-2} T}e^{\\nu T}. \n \\end{align*}\n\nNow we use that on the support of $Q'$ we have $|k|\\in[\\delta,2\\delta]$ and the bounds already used many times before, to derive\n\\[\n\\begin{split}\n |\\hat{H}_T'(k)| & \\leqslant C|\\hat{Q}'(k)| e^{-4\\delta^2 \\varepsilon^{-2} T} + C|\\hat{Q}(k)| T\\varepsilon^{-2} k^2 e^{-4k^2 \\varepsilon^{-2} T},\\\\\n |\\hat{H}_T''(k)| & \\leqslant C|\\hat{Q}''(k)| e^{-4\\delta^2 \\varepsilon^{-2} T} +C|\\hat{Q}'(k)| \\varepsilon^{-2}T e^{-4 \\delta^2 \\varepsilon^{-2} T} \\\\ &\\qquad+ C |\\hat{Q}(k)| T\\varepsilon^{-2}(1+T\\varepsilon^{-2}k^2) e^{-2k^2 \\varepsilon^{-2} T}. \n \\end{split}\n\\] \nThus we can write\n\\[\n\\begin{split}\n \\|\\hat{H}_T'\\|^2_{L^2} \n & \\leqslant C + C T^2\\varepsilon^{-4} \\int_{\\delta}^{\\infty} k^2e^{-8k^2 \\varepsilon^{-2} T}dk \\\\\n & \\leqslant C + C T^{1\/2}\\varepsilon^{-1} \\int_{\\delta T^{1\/2} \\varepsilon^{-1} }^{\\infty} k^2e^{-8k^2}dk \\leqslant C,\n \\end{split}\\]\n and similarly\n \\[\n \\|\\hat{H}_T''(k)\\|_{L^2} \n \\leqslant C .\n\\] \nUsing Corollary \\ref{cor:new} we obtain\n\\[\n\\| (1-\\mathcal{P}_1)(e^{\\Delta_\\nu T}D)(\\varepsilon \\cdot){\\mathrm{e}}_1\\|_{C_\\kappa^0} \\leqslant C \\varepsilon^\\alpha \\|D\\|_{ C_\\kappa^{0,\\alpha}},\n\\]\nwhich concludes the proof.\n\n\\section{Approximation}\n\\label{sec:app}\n\nIn this section we present the proof our main approximation result using the bound on the residual derived in the sections above.\nAs the result should hold for very long times of order $\\epsilon^{-2}$ we need to rely on the sign of the cubic nonlinearity \nand energy-time estimates. But as the Swift-Hohenberg operator does not allow for straightforward $L^p$-estimates, \nwe have to restrict the final result to $L^2$-spaces. \n\nLet us recall the main setting: $A$ is a mild solution of the amplitude equation~\\eqref{e:GL}\nand assume that there is a $\\varrho>2$ such that for all $p>0$ one has $A(0)\\in W^{1,p}_{\\varrho}$,\nand $u$ is a solution of the Swift-Hohenberg equation~\\eqref{e:SH}.\n\nIn order to prove our main result,\nwe need to bound the error\n\\[\n R(t) =u(t)-u_A(t) \n\\]\nbetween $u$ and the approximation $u_A$ defined in \\eqref{def:uA}. Using the definition \nof the residual from \\eqref{e:residual} and the mild formulation for the Swift-Hohenberg equation, we obtain\n\\begin{equation}\n \\label{e:derR}\n R(t) = {\\mathrm{e}}^{t{\\mathcal{L}}_\\nu}R(0) + \\int_0^t {\\mathrm{e}}^{(t-s){\\mathcal{L}}_\\nu} [u_A^3-(u_A+R)^3] ds + \\text{Res}(u_A)(t)\\;.\n\\end{equation}\nAs the residual $\\text{Res}$ is not differentiable in time, \nwe cannot proceed with $L^2$-energy estimates as in the deterministic case, but the proof is still very similar.\n\nSubstituting $D = R - \\text{Res}$, we obtain first (note that $\\text{Res}(0)=0$)\n\\[\nD(t) = {\\mathrm{e}}^{t{\\mathcal{L}}_\\nu}R(0) + \\int_0^t {\\mathrm{e}}^{(t-s){\\mathcal{L}}_\\nu} [u_A^3-(D+u_A+\\text{Res})^3] ds\n\\]\nand thus\n\\[\n\\partial_t D = {\\mathcal{L}}_\\nu D -(D+u_A+\\text{Res}(u_A))^3 - u_A^3 \\;.\n\\]\nNow we can use $L^2_{\\varrho,\\varepsilon}$-energy estimates \n\\[\n \\frac12\\partial_t \\| D\\|^2_{L^2_{\\varrho,\\varepsilon}} \n= \\langle {\\mathcal{L}}_\\nu D, D \\rangle_{L^2_{\\varrho,\\varepsilon}} \n- \\int_{\\mathbb{R}}{ w_{\\varrho,\\varepsilon}} D[ (D+u_A+\\text{Res}(u_A))^3 - u_A^3] \\;dx \\;.\n\\]\nWe choose the weight (see Definition \\ref{def:weight}) for some $\\varrho>1$ as\n\\[\n w_{\\varrho,\\varepsilon}(x) := \\frac1{(1+|\\varepsilon x|^2)^{\\varrho\/2}},\n\\]\nwhich is integrable with $\\| w_{\\varrho,\\varepsilon}\\|_{L^1}=C\\varepsilon^{-1}$. \nAlso recall that by Lemma \\ref{lem:spec}\n\\[\\langle {\\mathcal{L}}_\\nu D, D \\rangle_{L^2_{\\varrho_\\varepsilon}} \\leqslant C \\varepsilon^2 \\| D\\|^2_{L^2_{\\varrho,\\varepsilon}} \\;.\n\\]\nFor the nonlinearity we use a straightforward modification of the standard dissipativity result for the cubic in $L^2$-spaces, \nwhich states that \n\\[ \\langle (-(D+u_A)^3+D^3, D \\rangle_{L^2_\\rho} \\leq 0 \\;.\n\\]\nBut here we have the additional term $\\text{Res}(u_A)$ that we need to take care of.\nUsing Young's inequality several times, we obtain\n\\begin{multline*}\n - [ (D+u_A+\\text{Res}(u_A))^3 - u_A^3] \\cdot D \\\\\n \\begin{aligned}\n \t= {} & - [ (D+\\text{Res})^3+ 3 u_A(D+\\text{Res})^2+ 3u_A^2(D+\\text{Res}) ] \\cdot D \\\\\n \t= {} & - D^4 - 3 D^2 \\text{Res}^2 - 3u_A^2D^2 \\\\\n \t& - 3 D^3 \\text{Res} - 3 D^3 u_A -6 D^2u_A\\text{Res} \n \t- D \\text{Res}^3- 3D u_A^2 \\text{Res} - 3 D u_A \\text{Res}^2 \\\\\n \t\\leqslant {}& C u_A^2 \\text{Res}^2 + C \\text{Res}^4 \\;.\n \\end{aligned}\n\\end{multline*}\nThe critical terms in the estimates above are:\n\\[\n6 D^2u_A\\text{Res} \\leqslant \\delta D^2u_A^2 + \\delta D^4 + C_\\delta \\text{Res}^4 \n\\]\nand with $\\delta=8\/15$ we have\n\\[\n3D^3u_A \\leqslant \\frac32 \\delta D^4 + \\frac3{2\\delta} D^2u_A^2 = \\frac45 D^4 + \\frac{45}{16} D^2u_A^2\\;.\n\\]\nIn summary we obtain\n\\[\n\\partial_t \\| D\\|^2_{L^2_{\\varrho,\\varepsilon}} \n\\leqslant \nC \\varepsilon^2 \\| D\\|^2_{L^2_{\\varrho,\\varepsilon}} \n+ C \\| u_A\\|^2_{L^4_{\\varrho,\\varepsilon}} \\|\\text{Res}\\|^2_{L^4_{\\varrho,\\varepsilon}} \n+ C \\|\\text{Res}\\|^4_{L^4_{\\varrho,\\varepsilon}} \\;.\n\\]\nThus by Gronwall's inequality or comparison principle for ODEs, we obtain directly\n\\[\n\\| D(t) \\|^2_{L^2_{\\varrho,\\varepsilon}} \n\\leqslant \ne^{C t\\varepsilon^2} \\| R(0) \\|^2_{L^2_{\\varrho,\\varepsilon}} \n+ \\int_0^t e^{C (t-s)\\varepsilon^2} \\|\\text{Res}\\|^2_{L^4_{\\varrho,\\varepsilon}} (\\|\\text{Res}\\|^2_{L^4_{\\varrho,\\varepsilon}} + \\| u_A\\|^2_{L^4_{\\varrho,\\varepsilon}}) ds\n\\]\nand finally we established the following result.\n\\begin{lemma}\nLet $A$ and $u$ be given as in the beginning of the section.\nFor the error $R$ given in \\eqref{e:derR} we obtain \n\\[ \\sup_{[0,T_0\\varepsilon^{-2}]} \\| R-\\text{\\rm Res}\\|_{L^2_{\\varrho,\\varepsilon}} \n\\leqslant C \\| R(0) \\|_{L^2_{\\varrho,\\varepsilon}} \n+ C\\varepsilon^{-1} \\sup_{[0,T_0\\varepsilon^{-2}]} \\Big[ \\|\\text{\\rm Res}\\|_{L^4_{\\varrho,\\varepsilon}} (\\|\\text{\\rm Res}\\|_{L^4_{\\varrho,\\varepsilon}} + \\| u_A\\|_{L^4_{\\varrho,\\varepsilon}}) \\Big]\\;.\n\\]\n\\end{lemma}\nBy assumption, $A(0)\\in W^{1,p}_\\varrho$ for all $p>0$, so by Corollary \\ref{cor:maxregA} we have\n\\[\n\t\\mathbb{E}\\sup_{[0,T]}\\|A\\|^p_{C^0_\\kappa}\\leqslant C_p\\qquad \\forall p>1, \\forall \\kappa>0\n\\]\nwhere $p$ is ``large'' and $\\kappa$ is ``small''.\n\nThen, by the Chebychev inequality, we have, in the sense of Definition~\\ref{def:0},\n\\[\n\t\\sup_{[0,T]}\\|A\\|_{C^0_\\kappa}=\\mathcal{O}(\\varepsilon^{-\\delta}), \\qquad \\forall \\delta>0,\n\\]\nand thus\n\\[\n\t\\sup_{[0,T_0\\varepsilon^{-2}]}\\|u_A\\|_{C^0_{\\varrho,\\varepsilon}}=\\mathcal{O}(\\varepsilon^{1-\\delta}).\n\\]\n\nNote that due to the $\\varepsilon$ scaling in the weight we have \n\\begin{lemma}\n Let $A(0)\\in W^{1,p}_\\varrho$ for all $p>0$. Then for all $p>1$, $\\varrho>1$ and $\\delta>0$ \n\\[\\sup_{[0,T_0\\varepsilon^{-2}]}\\|u_A\\|_{L^p_{\\varrho,\\varepsilon}} = \\mathcal{O} ( \\varepsilon^{1 - 1\/p-\\delta} ).\n\\]\n\\end{lemma}\n\n\\begin{proof}\n\tThe claim follows from the simple scaling argument below, which is based on a substitution:\n\t\\begin{equation*}\n\t\t\\|u_A\\|_{L^p_{\\varrho,\\varepsilon}} \\leqslant \\varepsilon \\|A\\|_{L^p_{\\varrho,\\varepsilon}} =\\varepsilon^{1-1\/p}\\|A\\|_{L^p_{\\varrho}},\n\t\\end{equation*}\n\tand we can conclude by noting that $\\|A\\|_{L^p_{\\varrho}}=\\mathcal{O}(1)$, with the meaning given in Definition~\\ref{def:0}.\n\\end{proof}\nBy the result on the residual in Theorem~\\ref{thm:res} we have, for all small $\\kappa>0$,\n\\[\n\t\\sup_{[0,T_0\\varepsilon^{-2}]}\\|\\text{Res}(u_A)\\|_{C^0_\\kappa}=\\mathcal{O}(\\varepsilon^{3\/2-2\\kappa}),\n\\]\nthus\n\\[\n\t\\sup_{[0,T_0\\varepsilon^{-2}]}\\|\\text{Res}(u_A)\\|_{L^p_{\\varrho,\\varepsilon}}=\\mathcal{O}(\\varepsilon^{3\/2-1\/p-2\\kappa}).\n\\]\n\nIn conclusion,\n\\[\n\t\\begin{split}\n\t\\sup_{[0,T_0\\varepsilon^{-2}]}\\|R\\|_{L^2_{\\varrho,\\varepsilon}} \n\t& \\leqslant \\sup_{[0,T_0\\varepsilon^{-2}]}\\|R-\\text{Res}(u_A)\\|_{L^2_{\\varrho,\\varepsilon}}\n\t+\\sup_{[0,T_0\\varepsilon^{-2}]}\\|\\text{Res}(u_A)\\|_{L^2_{\\varrho,\\varepsilon}}\\\\\n\t&\\leqslant C\\|R(0)\\|_{L^2_{\\varrho,\\varepsilon}} + \\mathcal{O}(\\varepsilon^{1-\\delta-2\\kappa}),\n\t\\end{split}\n\\]\nwhere we used once more Definition~\\ref{def:0} for the $\\mathcal{O}(\\varepsilon^{1-\\delta-\\kappa})$ term.\nThus we finished the estimate on the error. Setting $2\\kappa=\\delta$, we established:\n\\begin{theorem}\\label{thm:final} Let \n\t $A$ be a solution of the amplitude equation \\eqref{e:GL}\n\t on $[0,T_0]$ such that there is a $\\varrho>2$ so that $A(0)\\in W^{1,p}_\\varrho$ for all $p>1$. \n\t Let $u$ be the solution to the Swift Hohenberg equation~\\eqref{e:SH} and $u_A$ the approximation built through $A$, which is defined in \\eqref{def:uA}.\n\t \n\tThen for all $\\delta>0$, $q>0$ there exists a constant $C_{q,\\delta}$ such that\n\t\\[\n\t\t\\mathbb{P}(\\sup_{[0,T_0\\varepsilon^{-2}]}\\|u-u_A\\|_{L^2_{\\varrho,\\varepsilon}}\\leqslant C\\|u(0)-u_A(0)\\|_{L^2_{\\varrho,\\varepsilon}} + C\\varepsilon^{1-2\\delta})\\geqslant 1-C_{q,\\delta}\\varepsilon^q,\n\t\\]\n\twhere the weight $ w_{\\varrho,\\varepsilon}(x) = (1+|\\varepsilon x|^2)^{-\\varrho\/2}$ \n\t(see Definition \\ref{def:weight}) for some $\\varrho>1$.\n\\end{theorem}\n\n\n\n\n\n\\paragraph{Acknowledgments}\nL.A.B. and D.B. were supported by DFG-funding BL535-9\/2 ``Mehrskalenanalyse stochastischer partieller Differentialgleichungen (SPDEs)'', and would also like to thank the M.O.P.S. program for providing a continuous support during the development of this research.\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum walks (QWs) are dynamical tools that are used to control the motion of a quantum particle in space and time. Due to their potential applications in quantum information and physical sciences, they have been at the focus of much research work since its first introduction \\cite{aharonov}. Owing to their quantum mechanical resources, i.e., quantum superposition, quantum interference, and entanglement, QWs hold the promise to develop new algorithms for computations on quantum computers \\cite{deutsch1992rapid,divincenzo1995quantum,farhi1998quantum, kempe,ambainis2003quantum,computation1,computation2, chandrashekar2010discrete,venegas2012quantum}. In Physics QWs provide a versatile platform to simulate various physical phenomena, e.g., topological phases \\cite{Kitagawa2010, Kitagawa2012, Kitagawa2012a, Asboth2012, Asboth2013, Asboth2014, Tarasinski2014, Asboth2015, Obuse2015, Cedzich2016BECorspnds, Groh2016, Xiao2017, Zhan2017, Sajid2019}, Anderson localization \\cite{disorder1, Ahlbrecht2011, Ahlbrecht2011a, disorder2, disorder3}, Bloch Oscillations \\cite{Genske2013, Cedzich2013, Arnault2020}, molecular binding \\cite{Ahlbrecht2012, Lahini2012, Krapivsky2015}, and Hofstadter spectrum \\cite{SajidThesis, Cedzich2020}, to name just a few. Due to their broad spectrum of applications, QWs have been realized in experiments using different physical systems, e.g., neutral atoms trapped in optical lattices \\cite{karski, robens}, trapped ions on a line \\cite{exp2, Xue2009, zahringer}, photons in free space \\cite{broome, schreiber}, correlated photons on continuously evanescently coupled waveguides \\cite{exp1}, and integrated photonics \\cite{sansoni}.\n\nQWs exhibit different features from their classical counterparts owing to their quantum mechanical resources.\nIn a quantum walk (QW) a quantum particle, which can exist in a superposition of several quantum states, moves to explore several paths simultaneously. \nQuantum interference takes place when different trajectories cross each other. As a result the probability distribution of a QW is strikingly different from a classical random walk. For example, the variance of a quantum walk grows quadratically faster with the number of steps of the walk compared to the linear growth of a classical random walk \\cite{kempe,venegas2012quantum}.\nQWs have shown speedups in comparison to their classical counterparts and, hence, are useful tools to design new fast algorithms for computation on a quantum computer \\cite{ambainis2003quantum,search1}, and to simulate and control certain quantum computational tasks \\cite{chandrashekar2010discrete}.\nThis has sparked a great interest to engineer different types of QWs and investigate their properties in different settings in the context of controlling and manipulating a desired quantum state for quantum computation and simulation.\nIn this vein, QWs have been investigated on a large scale which has resulted in the exploration of various types of walks. This includes QWs with decoherence \\cite{decoherent,Romanelli2005,Kendon,Zhang2013,albertinDecoherence2014}, QWs with time-dependent coin \\cite{Banuls2006,Xue2015,Cedzich2016,panahiyan,Katayama2020}, QWs with position dependent coin \\cite{Wojcik2004,Uzma2020}, and QWs with different types of phase defects \\cite{zhang2014one,Zhang2014TW0,Farooq2020}.\n\nIn this paper we engineer QW protocols (one standard QW protocol and one split-step protocol) which imprint a time and spin-dependent phase shift (TSDPS) to the wave function of a quantum particle undergoing a QW on a one-dimensional (1D) lattice. We get inspiration from the previous works \\cite{Xue2015,Cedzich2016,panahiyan,Katayama2020,zhang2014one,Zhang2014TW0,Farooq2020,Cedzich2019} where a desired time evolution of a QW is achieved either by manipulating the coin parameter by making it time or position dependent, or by introducing different types of spatial phase defects into the wavefunction of the quantum particle. Our engineered QWs share common features, e.g., complete revivals and partial revivals in the probability distribution of QWs which are investigated in \\cite{Xue2015,Cedzich2016,Katayama2020,Farooq2020}. We use the phase factor ($\\phi$) as a control knob to control and manipulate the evolution of a QW. By numerically computing the probability distribution $P(x,n)$ and the standard deviation $\\sigma(n)$ of the walk, and the probability of the particle to return to its initial position $P(x=x_{\\text{i}},n)$ as a function of the number of steps of the walk, we show revivals in the evolution of QWs with TSDPS.\nFor rational values of the phase factor, i.e., $\\phi\/2\\pi=p\/q$, where $p$ and $q$ are mutually coprime integers, periodic revivals occur in the QW driven by the standard protocol where the period of revivals depends on the denominator $q$. In this case the quantum particle takes a finite excursion in the lattice and then comes back to its initial position. \nFor an irrational $\\phi\/2\\pi$, the QW shows partial revivals with irregular periods, and the walker remains localized in a small region of the lattice, hence, showing no transport. The periods of partial revivals in this case are not regular due to the incommensurability of the $\\phi\/2\\pi$ with the step size of the QW. \nIn the case of a QW walk driven by the split-step protocol, partial revivals occurs for rational values of $\\phi\/2\\pi$ with periods of revivals double of the standard QW.\n\nIn experiments, imperfections are inevitable and, hence, imprinting an exact rational or irrational $\\phi\/2\\pi$ is challenging. From this perspective we investigate the robustness of revivals against imperfections in the imprinted phase factor. We show with our numerical results that in the presence of a linear random noise (random fluctuations) in the imprinted phase factor, signatures of revivals persist for smaller values of the noise (fluctuations) parameter in the case of $\\phi\/2\\pi=p\/q$. Increasing the strength of the noise parameter results in suppression of revivals for both rational and irrational $\\phi\/2\\pi$. In this case the quantum particle starts to spread out showing quantum transport.\n\nThe rest of the paper is organized as follows. In Sec.~\\ref{QW_TSDPS} we introduce our system, the standard QW protocol, coin operator, and shift operator with TSDPS. We present our numerical results for revivals and partial revivals in the evolution of the QW driven by the standard protocol for rational and irrational values of $\\phi\/2\\pi$, respectively. In Sec.~\\ref{SPLIT_STEP} we introduce the split-step protocol with TSDPS, and investigate revivals in the probability distribution for rational values of $\\phi\/2\\pi$. Section~\\ref{QW_TSDPS_Noise} presents the effects of random fluctuations in the phase factor on revivals in the probability distribution of the QW driven by the standard protocol. We summarize our results and conclude with a brief outlook in Sec.~\\ref{conc}.\n\n\\section{Quantum Walks with a Time and Spin-dependent Phase Shift} \\label{QW_TSDPS}\nWe consider a single quantum particle (also called a walker) with two internal degrees of freedom undergoing a QW on a 1D lattice. The internal states of the particle (also called spin states due to its analogy with a spinor particle) are represented by basis vectors $\\{\\ket{s}:\\ s\\in \\{\\uparrow, \\downarrow\\}\\}$ which span a two-dimensional Hilbert space $\\mathcal{H}^s$. The position states of the walker are represented by the lattice coordinates $ x$ with basis vectors $\\{ \\ket{x}: x \\in \\mathbb{Z} \\}$ spanning a 1D Hilbert space $\\mathcal{H}^x$. The quantum particle undergoing a quantum walk resides in a Hilbert space that is the tensor product of the two Hilbert spaces, i.e., $\\mathcal{H}^s \\otimes \\mathcal{H}^x$. For simplicity, we use dimensionless units by assuming the lattice constant and the time duration of a single-step of the QW to be equal to 1.\n\nThe 1D standard protocol of the QW with TSDPS consists of a set of unitary operators. We call this protocol as the ``walk operator'' which is defined as,\n\\begin{align}\n\\label{eq:_1}\n\\hat{W}_{\\phi}(n) = \\hat{S}_x(\\phi n) \\, \\hat{C},\n\\end{align}\nhere $\\hat{C}$ is known as a coin operator \nand $\\hat{S}_x(\\phi n)$ as a spin-dependent shift operator. The coin operator acts on the internal states of the walker and rotates its spin state in the two-dimensional Hilbert space $\\mathcal{H}^s$. In this work we employ the so-called Hadamard coin which is defined as,\n\\begin{align}\n\\label{eq:_coin}\n\\hat{C} = \\frac{1}{\\sqrt{2}} \\begin{pmatrix} 1 & \\ \\ \\ \\ 1 \\\\ 1 & \\ \\ -1 \\end{pmatrix} \\otimes \\ket{x}\\bra{x}.\n\\end{align}\nThe spin-dependent shift operator $\\hat{S}_x(\\phi n)$ translates the walker by one lattice site to the right or to the left depending on its internal state, and imprints a step and spin-dependent phase shift $\\phi$ to the walker's wavefunction. The shift operator is defined as,\n\\begin{align} \\label{eq:shift-x}\n \\hat{S}_x (\\phi n) = & \\sum_{x} \n \\Big[ \\exp\\big[i \\phi n\\big] \\ket{\\uparrow}\\bra{\\uparrow} \\otimes \\ket{x+1}\\bra{x} \\nonumber \\\\ & \n + \\exp\\big[ - i \\phi n\\big] \\ket{\\downarrow}\\bra{\\downarrow} \\otimes \\ket{x-1}\\bra{x} \\Big],\n\\end{align} \nwhich shifts the walker in the spin-up (spin-down) state to the right (left). By applying the walk operator ( given in eq. (\\ref{eq:_1})) to an initial state $(\\ket{\\Psi_{\\text{i}}})$ of the walker one time constitutes a single step of the QW.\nThe evolution of the walk results by applying the walk operator to the initial state of the walker repeatedly for a large number of times. After certain $n$ number of steps of the walk (where $n\\in \\mathbb{N}$) the final state of the walker can be written as,\n\\begin{equation} \\label{eq:final-state}\n\\ket{\\Psi_n(\\phi)}=\\hat{W}_{\\phi}(n) \\ket{\\Psi_{\\text{i}}}.\n\\end{equation}\nFor a fixed value of $\\phi$ the factor $n$ ensures the time dependence of the phase factor that is imprinted to the wavefunction of the walker at each step of the walk. The spatial probability distribution $P(x,n)$ is obtained by tracing out the coin degrees of freedom, i.e.,\n\\begin{equation} \\label{eq:prob-dist}\nP(x,n)=\\sum_{s=0,1} \\braket{\\Psi_n(\\phi)|\\Psi_n(\\phi)}.\n\\end{equation}\nFor rational values of the phase factor, i.e., $\\phi\/2\\pi=p\/q$, the walk operator $\\hat{W}_{\\phi}(n)$ is periodic, i.e., $\\hat{W}_{\\phi}(n+q*r) = \\hat{W}_{\\phi}(n)$ for some $r\\in\\mathbb{N}$. As a result the evolution of the walk (and hence, $P(x,n)$) is also periodic \\cite{Cedzich2016}, i.e., $P(x,n+q*r) =P(x,n)$. To investigate revivals in the evolution of the QW we compute the return probability $P(x=x_{\\text{i}},n)$ of the walker to its initial position $x_{\\text{i}}$, and the standard deviation $\\sigma(n)$ of the walk. After certain $n$ number of steps of the QW, $P(x=x_{\\text{i}},n)$ and $\\sigma(n)$ are obtained using the following expressions,\n\\begin{equation} \\label{eq:prob-init-state}\nP(x=x_{\\text{i}},n)=\\sum_{s=0,1} \\braket{\\Psi_{\\text{i}}|\\Psi_n(\\phi)},\n\\end{equation}\n\\begin{equation} \\label{eq:standard-deviation}\n\\sigma(n)= \\sqrt{ - ^2}.\n\\end{equation}\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=170mm]{TDPD2PiBy200S_100_49ym.eps} \\\\\n \\includegraphics[width=170mm]{TDPD2PiBy200S_100_49AntSym1.eps} \\\\\n \\includegraphics[width=170mm]{Simga_RtrnProb_2Piby200_100_49B.eps}\n\\caption{(First two rows). Probability distribution of the QW with TSDPS in the position-time plane for rational values of $\\phi\/2\\pi$ (red, white, and blue colors indicate the maximum ($1.0$), moderate ($\\sim0.1$), and minimum ($\\sim0.$) probabilities, respectively).\nFirst row: The evolution of the walk with a quantum particle initially prepared in the initial state $\\Psi_{i1}$. The values of $\\phi\/2\\pi$ are indicated in the inset.\nSecond row: The evolution of the walk with a quantum particle initially prepared in the initial state $\\Psi_{i2}$. The values of $\\phi\/2\\pi$ remain same in a given column.\nThe difference in the probability distribution for the two initial states $\\Psi_{i1}$ and $\\Psi_{i2}$ can be noticed by a careful inspection.\nThird row: The evolution of the return probability of the walker to its initial position (solid curve), and the standard deviation (dotted curve) of the walk upto $200$ steps of the walk. \nThe behaviors of $P(x=x_{\\text{i}}, n)$ and $\\sigma(n)$ are same for both $\\Psi_{i1}$ and $\\Psi_{i2}$, and hence, the results are shown only for the first choice of the initial state $\\Psi_{i1}$.\n(a) For $\\phi\/2\\pi=1\/200$ and (b) $\\phi\/2\\pi=1\/100$ complete revivals occur after $n=q=200$ and $n=q=100$ steps of the walk, respectively. The return probability is equal to $1$ as it can be clearly seen from the corresponding subfigures (g) and (h) in the third row. (c) For $\\phi\/2\\pi=1\/49$ there occurs a partial revival when $n$ is an odd integral multiple of $q=49$ (i.e., $P(x=x_{\\text{i}}, n)<1$), and a complete revival occurs when $n$ is an even integral multiple of $q=49$. The occurrence of revivals, their periodicities, and the boundedness of the standard deviation are independent of the initial state of the walker.}\n\\label{fig:a}\n\\end{figure*}\nHere $$ represents the average or expected value of the walker position $x$. Like $P(x,n)$, for $\\phi\/2\\pi=p\/q$ the return probability and the standard deviation of the walk are also periodic, i.e., $P(x=x_{\\text{i}},n+q*r)$=$P(x=x_{\\text{i}},n)$ and $\\sigma(n+q*r)=\\sigma(n)$. If initially a walker spreads from its initial position $x=x_{\\text{i}}$, and after certain $n$ number of steps of the walk the return probability $P(x=x_{\\text{i}}, n)$ becomes equal to $1$ (correspondingly $\\sigma(n)$ becomes zero) we call this a complete revival. Similarly, if the walker returns to its initial position $x=x_{\\text{i}}$ but with $P(x=x_{\\text{i}},n)<1$ (correspondingly $\\sigma(n)>0$), we call it a partial revival.\nIn the following sections we will investigate revivals in the probability distribution of a QW with TSDPS driven by the standard protocol (given in Eq.~(\\ref{eq:_1})), and in a QW with TSDPS driven by the split-step protocol (given in Eq.~(\\ref{eq:_SS_W_operator})) for various values of $\\phi\/2\\pi$.\n\n\n\\subsection{Standard protocol with TSDPS for rational phase factor} \\label{QW_TSDPS_Rational}\nIn this section we investigate the evolution of a QW driven by the standard protocol Eq.~(\\ref{eq:_1}) for rational values of the phase factors, i.e., $\\phi\/2\\pi=p\/q$, where $p$ and $q$ are mutually coprime integers. \nWe consider a walker initially residing at the origin of a 1D lattice, and consider two possibilities for the internal state of the walker, i.e., an equal superposition of the two spin states, and a spin up state only. The two initial states of the walker can be written as,\n\\begin{align} \\label{eq:initial-state}\n\\begin{aligned}\n\\ket{\\Psi_{i1}} &=\\frac{1}{2} \\Big( \\ket{\\uparrow} + i \\ket{\\downarrow} \\Big) \\otimes \\ket{0}, \\\\\n\\ket{\\Psi_{i2}} & = \\ket{\\uparrow} \\otimes \\ket{0}.\n\\end{aligned}\n\\end{align}\nWe evolve these initial states by periodically applying the walk operator given in Eq.~(\\ref{eq:_1}) for a large number of steps. To demonstrate revivals in the walk we calculate the probability distribution ($P(x,n)$) of the walk in the position-time plane, the probability of the walker to return to its initial position ($P(x=x_{\\text{i}}, n)$), and the standard deviation ($\\sigma(n)$) of the walk using Eqs.~(\\ref{eq:prob-dist}), (\\ref{eq:prob-init-state}), and (\\ref{eq:standard-deviation}), respectively.\n\nIn Fig.~\\ref{fig:a}, we show our numerically simulated results of the evolution of $P(x,n)$ in the position-time plane for $n=200$ steps of the walk for $\\phi\/2\\pi=1\/200$ (Fig.~\\ref{fig:a}(a)), $\\phi\/2\\pi=1\/100$ (Fig.~\\ref{fig:a}(b)), and $\\phi\/2\\pi=1\/49$ (Fig.~\\ref{fig:a}(c)). The evolution shown in Figs.~\\ref{fig:a}(a)-(c) is computed with the initial state $\\psi_{i1}$ as defined in Eq.~\\ref{eq:initial-state}. Revivals in the probability distribution are apparent in all three cases with period of revival equal to $n=q$ for even values of $q$ (Fig.~\\ref{fig:a}(a) and (b)) and $n=2q$ for odd values of $q$ ((Fig.~\\ref{fig:a}(c)). The difference in the periods of the revivals for the even and odd values of $q$ is due to the fact that after odd number of steps of the walk the probability of the walker at its initial position is zero \\cite{Cedzich2016}. In the case of odd $q$ the probability distribution shows partial revivals when $n$ is an odd multiple of $q$, and complete revivals when $n$ is an even multiple of $q$. In all three cases, i.e., Fig.~\\ref{fig:a}(a)-(c), the walker initially takes an excursion in a finite region of the 1D lattice when $n0$ (let us call it lower branch of the $P(x,n)$) throughout the evolution of the walk. In the case of $\\psi_{i2}$ higher probabilities alternate between the upper ($x<0$) and the lower ($x>0$) branches of the probability distribution, i.e., $P(x,n)$ remains higher in a single branch for certain range of $n$ and then switches to the other branch. \n\nIn order to give a clear demonstration of complete and partial revivals in the QW with TSDPS, in Figs.~\\ref{fig:a}(g)-(i) we show the evolution of the return probabilities $P(x=x_{\\text{i}}, n)$ of the walker to its initial position, and the standard deviation $\\sigma(n)$ of the walk. Both the initial states $\\psi_{i1}$ and $\\psi_{i2}$ show identical behavior of $P(x=x_{\\text{i}}, n)$ and $\\sigma(n)$, we therefore show these results only for the first choice of the initial state, i.e., $\\psi_{i1}$. The values of the phase factor remain same in a given column. For $\\phi\/2\\pi=1\/200$ and $\\phi\/2\\pi=1\/100$ complete revivals are clearly visible, i.e., $P(x=x_{\\text{i}}, n)=1$ when $n$ is an integral multiple of $q$. In the case of $\\phi\/2\\pi=1\/49$, there are partial revivals when $n$ is an odd integral multiple of $q$ as $P(x=x_{\\text{i}}, n)<1$, and complete revivals occur when $n$ is an even integral multiple of $q$ as $P(x=x_{\\text{i}}, n)=1$. The extent of the region in which the walker remains bounded during its evolution can be estimated from the standard deviation as these are directly related. For a smaller value of the phase factor $\\phi\/2\\pi$ (e.g., $\\phi\/2\\pi=1\/200$) the walker has a large excursion, i.e., $\\sigma(n) \\sim 26 $, and a smaller excursion ($\\sigma(n) \\sim 7 $) for a larger phase factor (e.g., $\\phi\/2\\pi=1\/49$).\nIn general, the TSDPS inhibits the ballistic expansion of the QW and confine the walker to a finite region of the lattice.\nThe extension of the bounded region decreases with the increase in $\\phi\/2\\pi$. This behaviour is apparent from both the probability distribution and the standard deviation of the walk.\n\\begin{figure*}[t]\n \\centering \n \\includegraphics[width=57mm]{TDPD_Golden_Ratio_Sym2.eps}\n \\includegraphics[width=57mm]{RtrnProb_Golden_Ratio.eps} \n \\includegraphics[width=57mm]{Sigma_Golden_Ratio.eps} \\\\ \n\t\\includegraphics[width=57mm]{TDPD_Golden_Ratio_AntSym1.eps} \n \\caption{Evolution of the QW with TSDPS and with the irrational phase factor $\\phi\/2\\pi = (\\sqrt{5}-1)\/2$ (red, white, and blue colors indicate maximum ($1.$), moderate ($\\sim0.1$), and minimum ($\\sim0.$) probabilities, respectively). (a) and (d) show the probability distribution $P(x, n)$ of the walk in the position-time plane for $n=200$ and with symmetric (asymmetric) initial state $\\Psi_{i1}$ ($\\Psi_{i2}$) given in Eq.~(\\ref{eq:initial-state}). (b) and (c) respectively show the long time evolution of the return probability $P(x=x_{\\text{i}}, n)$ and the standard deviation of the walker with the initial state $\\Psi_{i1}$ for $n=1000$. From (a) and (d) it is clear that the probability distribution is no more periodic for the irrational value of $\\phi\/2\\pi$. The walker is completely localized in a finite region of the 1D lattice, which is also apparent from (c) which shows that $\\sigma(n)$ has a finite upper bound, i.e., $\\sigma<3$. The return probability in (b) shows a number of partial revivals with unpredictable periods in the long time evolution of the walk. $P(x=x_{\\text{i}}, n)$ of the walker is bounded away from 0, i.e., $P(x=x_{\\text{i}}, n) \\ge 0.32$. \n By carefully inspecting (a) and (d) one can notice that the probability distributions for the two choices of the initial states are not exactly similar. However, the return probability and the standard deviation show similar behavior for the two choices of the initial states. We have, therefore, shown $P(x=x_{\\text{i}}, n)$ and $\\sigma(n)$ for the first choice of the initial state, i.e., $\\Psi_{i1}$. }\n\\label{fig:b}\n\\end{figure*}\n\\subsection{Standard protocol with TSDPS for an irrational phase factor} \\label{QW_TSDPS_Irrational}\nWe now consider the case of an irrational phase factor $\\phi\/2\\pi$. The well known approximate irrational number is the Golden ratio $(\\sqrt{5} - 1)\/2$. The walk operator (given in Eq.~(\\ref{eq:_1}) with the phase factor equal to the Golden ratio is no more periodic as $(\\sqrt{5} - 1)\/2$ is incommensurate with the number of steps of the walk. However, the behaviour of the probability distribution of the walker still shows a number of partial revivals in its long time evolution. In Fig.~\\ref{fig:b} we show the evolution of the probability distribution $P(x,n)$ for $n=200$ for both choices of the initial states ($\\Psi_{i1}$ and $\\Psi_{i2}$), the evolution of the return probability of the walker with initial state $\\Psi_{i1}$ to its initial position $P(x=x_{\\text{i}}, n)$ for $n=1000$, and the standard deviation $\\sigma(n)$ of the walker with initial state $\\Psi_{i1}$ for $n=1000$. Figure~\\ref{fig:b}(a) shows that the walker remains completely localized in a small region of the lattice throughout its evolution. Figure~\\ref{fig:b}(b) shows a number of revivals with unpredictable periods in the long time evolution of the walk. These revivals are not strictly complete as the return probability $P(x=x_{\\text{i}}, n)$ is not exactly equal to unity for all $n>1$. Note that $P(x=x_{\\text{i}}, n)$ is bounded away from $0$, i.e., $P(x=x_{\\text{i}}, n) \\ge 0.32$ for all $n$ showing that there is a significant probability of the walker to remain at its initial position during the evolution of the walk. The reason for this behavior is that the initial state of the walker has a significant overlap with a bound state \\cite{Cedzich2013}. Figure~\\ref{fig:b}(c) shows the evolution of the standard deviation of the walk which has an upper bound, i.e., $\\sigma(n)<3$ for all values of $n$. This shows that the walker takes a very short excursion during the evolution of the walk and remains localized in a small region of the 1D lattice. Figure~\\ref{fig:b}(d) shows the same evolution of the walk as in the Fig.~\\ref{fig:b}(a) but with the second choice of the initial state, i.e., $\\Psi_{i2}$. By a careful inspection of Fig.~\\ref{fig:b}(a) and Fig.~\\ref{fig:b}(d) one can notice that the probability distributions for the two initial states are not similar. However, the return probability, the standard deviation, and the localization behaviour of the walk are same for the two initial states. We, therefore, show the return probability and the standard deviation for the first initial state, i.e., $\\Psi_{i1}$ only. \n\\section{Split-Step Protocol with TSDPS for a rational phase factor} \\label{SPLIT_STEP}\nWe now investigate revivals in a QW with a TSDPS driven by a split-step protocol. We consider only rational values of the phase factor, i.e., $\\phi\/2\\pi=p\/q$. The walk operator for this protocol is defined as,\n\\begin{align}\n\\label{eq:_SS_W_operator}\n\\hat{W}_{ss, \\phi}(n) = \\hat{S}_{x}^{\\downarrow}(\\phi n) \\, \\hat{C}_2 \\, \\hat{S}_{x}^{\\uparrow}(\\phi n) \\, \\hat{C}_1,\n\\end{align}\n\\begin{figure*}[t]\n \\centering \n \\includegraphics[width=71mm]{TDPD2PiBy100_Split_Step_400Steps.eps}\n \\includegraphics[width=75mm]{Simga_RtrnProb_2Piby100_Split_Step_400Steps.eps} \\\\\n \\includegraphics[width=71mm]{TDPD2PiBy49_Split_Step_400Steps.eps}\n \\includegraphics[width=75mm]{Simga_RtrnProb_2Piby49_Split_Step_400Steps.eps}\n\\caption{Evolution of the split-step QW with TSDPS with rational phase factor $\\phi\/2\\pi$ (red, white, and blue colors indicate maximum ($1.$), moderate ($\\sim 0.1$), and minimum ($\\sim 0.$) probabilities, respectively). (a) and (c) show the Probability distribution $P(x, n)$ of the walk in the position-time plane for $\\phi\/2\\pi=1\/100$ and $\\phi\/2\\pi=1\/49$, respectively. (b) and (d) show the evolution of the return probability $P(x=x_{\\text{i}}, n)$ (indicated by the blue solid curve) and $\\sigma(n)$ (indicated by the orange dotted curve) corresponding to (a) and (c), respectively. \nIt is clear from the probability distribution that the walker remains bounded in a finite region of the lattice, as it was the case in the QW with TSDPS driven by the standard protocol. The return probability remains smaller than unity in both (b) and (d), showing partial revivals. The periods of the partial revivals are double of the standard protocol case (compare with Fig.~(\\ref{fig:a})). At each period of revivals the return probability decreases and the standard deviation increases with the number of steps of the walk. The standard deviation has an upper bound due to the fact that the walker remains bounded during the evolution of the walk.}\n\\label{fig:Split_Step}\n\\end{figure*}\nwhich consists of two coin operators $\\hat{C}_1$ and $\\hat{C}_2$, and two shift operators $\\hat{S}_{x}^{\\uparrow}(\\phi n)$ and $\\hat{S}_{x}^{\\downarrow}(\\phi n)$. We consider both the coin operators to be the Hadamard coins as defined in Eq.~(\\ref{eq:_coin}).\nThe shift operator $\\hat{S}_{x}^{\\uparrow} (\\phi n)$ shifts only the spin-up state of the walker to the right by a unit length and imprints a TSDPS to its wavefunction leaving the spin-down state unchanged. Mathematically, it is defined as,\n\\begin{align} \\label{eq:shift-x-up}\n \\hat{S}_{x}^{\\uparrow} (\\phi n) = & \\sum_{x}\n \\Big[ \\exp\\big[i \\phi n\\big] \\ket{\\uparrow}\\bra{\\uparrow} \\otimes \\ket{x+1}\\bra{x} \\nonumber \\\\ & \n + \\ket{\\downarrow}\\bra{\\downarrow} \\otimes \\ket{x}\\bra{x}\\Big].\n\\end{align} \nSimilarly, the operator $\\hat{S}_{x}^{\\downarrow}(\\phi n)$ shifts only the spin-down state of the walker to the left by a unit length and imprints a TSDPS to its wavefunction leaving the spin-up state unchanged. It is defined as,\n\\begin{align} \\label{eq:shift-x-up}\n \\hat{S}_{x}^{\\downarrow} (\\phi n) = & \\sum_{x}\n \\Big[ \\ket{\\uparrow}\\bra{\\uparrow} \\otimes \\ket{x}\\bra{x} \\nonumber \\\\ & \n + \\exp\\big[- i \\phi n\\big] \\ket{\\downarrow}\\bra{\\downarrow} \\otimes \\ket{x-1}\\bra{x}\\Big].\n\\end{align} \nTo investigate revivals in the QW driven by the split-step protocol with TSDPS, we consider a walker in an initial state $\\Psi_{i1}$ defined in Eq.~(\\ref{eq:initial-state}), and evolve it through the walk operator defined in Eq.~(\\ref{eq:_SS_W_operator}) for a large number of steps. We numerically compute the probability distribution $P(x,n)$, the return probability $P(x=x_{\\text{i}},n)$ to the initial position, and the standard deviation $\\sigma(n)$ of the walker similar to the QW driven by the standard protocol.\n\nIn Fig.~(\\ref{fig:Split_Step}) we show our numerical results for $P(x,n)$, $P(x=x_{\\text{i}},n)$, and $\\sigma(n)$ of the split-step QW with TSDPS. The evolution is carried out with the initial state $\\Psi_{i1}$ for $n=400$ steps of the walk. Figures~\\ref{fig:Split_Step}(a) and (c) show $P(x,n)$ in the position-time plane for $\\phi\/2\\pi=1\/100$ and $\\phi\/2\\pi=1\/49$, respectively. The probability distributions show periodic behavior. The walker spreads initially for $n 0$, $\\alpha_{i,r+1} = \\cdots =\n\\alpha_{in} = 0$ and $r=1,\\ldots,n$ where $r=r(i)$.\nWe set\n\\[\nU_{i \\eta_i} = [s_{i1},\\ldots,s_{i\\eta_i}],\\qquad V_{i \\eta_i} = [d_{i1},\\ldots, d_{i\\eta_i}]\\qquad\n\\mbox{and}\\qquad \\Sigma_{i \\eta_i} = \\mbox{diag}(\\alpha_{i1},\\ldots,\\alpha_{i\\eta_i}),\n\\]\nwhere $U_{i\\eta_i}\\in {\\mathbb R}^{m\\times \\eta_i}$, $V_{i\\eta_i}\\in {\\mathbb R}^{n\\times \\eta_i}$ and $\\Sigma_{i\\eta_i}\\in {\\mathbb R}^{\\eta_i \\times \\eta_i}$.\n Now we def\\\/ine $K_{i\\eta_i} \\in{\\mathbb R}^{m\\times n}$ and ${\\mathcal K}_{i\\eta_i}:L^{2}(\\Omega,{\\mathbb R}^{n})\\rightarrow L^{2}(\\Omega,{\\mathbb R}^{m})$ by\n\\begin{gather}\n\\label{trsvd}\nK_{i\\eta_i} = U_{i\\eta_i}\\Sigma_{i \\eta_i}V_{i\\eta_i}^{T}\\qquad\\mbox{and}\\qquad [{\\mathcal K}_{i\\eta_i}({\\boldsymbol w}_i)](\\omega) = K_{i\\eta_i}[{\\boldsymbol w}_i(\\omega)],\n\\end{gather}\nrespectively, for any ${\\boldsymbol w}_i\\in L^{2}(\\Omega,{\\mathbb R}^{n})$.\n\n\\begin{theorem}\n\\label{sol1}\nLet ${\\boldsymbol v}_1,\\ldots,{\\boldsymbol v}_p$ be determined by Lemma~{\\rm \\ref{lemma1}}.\nThen the vector $f^0$ and operators ${\\mathcal F}_1^0, \\ldots, {\\mathcal F}_p^0$,\nsatisfying \\eqref{min1}--\\eqref{con1}, are determined by\n\\begin{gather}\n\\label{sol-f0}\nf^0 = E[{\\boldsymbol x}] - \\sum_{k=1}^p F^0_k E[{\\boldsymbol v}_k]\\qquad \\mbox{and}\\qquad\n{\\mathcal F}_1^0 = {\\mathcal K}_{1\\eta_1}, \\quad \\ldots, \\quad {\\mathcal F}_p^0 = {\\mathcal K}_{p\\eta_p}.\n\\end{gather}\nThe accuracy associated with transform ${\\mathcal T}_p^0$, determined by \\eqref{th1}\n and \\eqref{sol-f0}, is given by\n\\begin{gather}\n\\label{er1}\nE[\\|{\\boldsymbol x} - {\\mathcal T}_p^0({\\boldsymbol y})\\|^2] =\\|{\\mathbb E}_{xx}^{1\/2}\\|^2 -\\sum_{k=1}^p \\sum_{j=1}^{\\eta_k} \\alpha^2_{kj} .\n\\end{gather}\n\\end{theorem}\n\n\\begin{proof} The functional $J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p)$ is written as\n\\begin{gather}\nJ(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p) = \\mbox{tr}\\Bigg[E_{xx}\n- E[{\\boldsymbol x}]f^T - \\sum_{i=1}^p E_{xv_i}F_i^T - f E[{\\boldsymbol x}^T] + ff^T\n+ f\\sum_{i=1}^p E[{\\boldsymbol v}_i^T]F_i^T\\nonumber \\\\\n\\phantom{J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p) =}{}\n- \\sum_{i=1}^p F_i E_{v_ix}\n+ \\sum_{i=1}^p F_iE[{\\boldsymbol v}_i]f^T + E\\Bigg(\\sum_{i=1}^p {\\mathcal F}_i\n({\\boldsymbol v}_i) \\Bigg[\\sum_{k=1}^p {\\mathcal F}_i ({\\boldsymbol v}_i)\\Bigg]^T\\Bigg)\\Bigg]. \\label{jqa}\n\\end{gather}\nWe remind (see Section \\ref{summ}) that here and below, $F_i$\nis def\\\/ined by $[{\\mathcal F}_i({\\boldsymbol v}_i)](\\omega) = F_i[{\\boldsymbol v}_i(\\omega)]$\nso that, for example, $E[{\\mathcal F}_k({\\boldsymbol v}_k){\\boldsymbol x}_k^T] = F_k E_{v_k x_k}$.\nIn other words, the right hand side in (\\ref{jqa}) is a function of $f$, ${\\mathcal F}_1, \\ldots ,{\\mathcal F}_p$.\n\nLet us show that $J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p)$ can be represented as\n\\begin{gather}\n\\label{j012}\nJ(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p) = J_0 + J_1 + J_2,\n\\end{gather}\nwhere\n\\begin{gather}\n\\label{j0}\nJ_0 = \\|{\\mathbb E}_{xx}^{1\/2}\\|^2 -\\sum_{i=1}^p \\|{\\mathbb E}_{xv_i}\\|^2,\n\\\\\n\\label{j12}\nJ_1 = \\|f - E[{\\boldsymbol x}] + \\sum_{i=1}^p F_i E[{\\boldsymbol v}_i]\\|^2\\qquad\\mbox{and}\\qquad\nJ_2 =\\sum_{i=1}^p \\|F_i - {\\mathbb E}_{xv_i}\\|^2.\n\\end{gather}\nIndeed, $J_1$ and $J_2$ are rewritten as follows\n\\begin{gather}\nJ_1 = \\mbox{tr} \\Bigg(ff^T - fE[{\\boldsymbol x}^T] + \\sum_{i=1}^p f E[{\\boldsymbol v}_i^T]F_i + E[{\\boldsymbol x}] E[{\\boldsymbol x}^T]\n - E[{\\boldsymbol x}]f^T - \\sum_{i=1}^p E[{\\boldsymbol x}] E[{\\boldsymbol v}_i^T] F_i^T \\nonumber\\\\\n\\phantom{J_1 =}{} + \\sum_{i=1}^p F_i E[{\\boldsymbol v}_i]f^T\n- \\sum_{i=1}^p F_i E[{\\boldsymbol v}_i] E[{\\boldsymbol x}^T] + \\sum_{i=1}^p F_i E[{\\boldsymbol v}_i] \\sum_{k=1}^p E[{\\boldsymbol v}_k^T] F_k^T\\Bigg)\\label{j11}\n\\end{gather}\nand\n\\begin{gather}\n\\label{j2}\nJ_2 = \\sum_{i=1}^p \\mbox{tr}\\, (F_i - {\\mathbb E}_{xv_i})(F_i^T - {\\mathbb E}_{v_i x})\n= \\sum_{i=1}^p \\mbox{tr} \\,(F_i F_i^T - F_i {\\mathbb E}_{v_i x} - {\\mathbb E}_{xv_i}F_i^T + {\\mathbb E}_{xv_i} {\\mathbb E}_{v_ix}).\n\\end{gather}\nIn (\\ref{j2}), $\\sum\\limits_{i=1}^p \\mbox{tr}\\, (F_i F_i^T)$ can be represented in the form\n\\begin{gather}\n\\label{aea}\n\\sum_{i=1}^p \\mbox{tr} \\,(F_i F_i^T) = \\mbox{tr}\\Bigg[E\\Bigg(\\sum_{i=1}^p F_i {\\boldsymbol v}_i \\sum_{k=1}^p {\\boldsymbol v}_k^T F_k^T\\Bigg)\n\\Bigg] - \\mbox{tr}\\Bigg(\\sum_{i=1}^p F_i E[{\\boldsymbol v}_i] \\sum_{k=1}^p E[{\\boldsymbol v}_k^T] F_k^T\\Bigg)\n\\end{gather}\nbecause\n\\begin{gather}\n\\label{v-ort}\nE[{\\boldsymbol v}_i {\\boldsymbol v}_k^T] - E[{\\boldsymbol v}_i] E[ {\\boldsymbol v}_k^T] = \\left \\{ \\begin{array}{@{}cc}\n{\\mathbb O}, & i\\neq k,\\\\\n I, & i=k \\end{array} \\right.\n\\end{gather}\n due to the orthonormality of vectors ${\\boldsymbol v}_1, \\ldots, {\\boldsymbol v}_p$.\n\nThen\n\\begin{gather}\nJ_0 + J_1 + J_2 = \\mbox{tr}(E_{xx} - E[{\\boldsymbol x}]E[{\\boldsymbol x}^T]) - \\sum_{i=1}^p \\mbox{tr} [{\\mathbb E}_{xv_i}{\\mathbb E}_{v_ix}]\\label{jj012}\\\\\n\\phantom{J_0 + J_1 + J_2 =}{} + \\mbox{tr} \\Bigg(ff^T - fE[{\\boldsymbol x}^T] + \\sum_{i=1}^p f E[{\\boldsymbol v}_i^T]F_i\n+ E[{\\boldsymbol x}] E[{\\boldsymbol x}^T] - E[{\\boldsymbol x}]f^T \\nonumber\\\\\n\\phantom{J_0 + J_1 + J_2 =}{} - \\sum_{i=1}^p E[{\\boldsymbol x}] E[{\\boldsymbol v}_i^T] F_i^T + \\sum_{i=1}^p F_i E[{\\boldsymbol v}_i]f^T\n - \\sum_{i=1}^p F_i E[{\\boldsymbol v}_i] E[{\\boldsymbol x}^T] \\nonumber\\\\\n\\phantom{J_0 + J_1 + J_2 =}{} + \\sum_{i=1}^p F_i E[{\\boldsymbol v}_i] \\sum_{k=1}^p E[{\\boldsymbol v}_k^T] F_k^T\\Bigg)\n + \\mbox{tr}\\Bigg[E\\Bigg(\\sum_{i=1}^p F_i {\\boldsymbol v}_i \\sum_{k=1}^p {\\boldsymbol v}_k^T F_k^T\\Bigg)\\Bigg] \\nonumber\\\\\n\\phantom{J_0 + J_1 + J_2 =}{} - \\mbox{tr}\\Bigg(\\sum_{i=1}^p F_i E[{\\boldsymbol v}_i] \\sum_{k=1}^p E[{\\boldsymbol v}_k^T] F_k^T\\Bigg)\\nonumber\\\\\n\\phantom{J_0 + J_1 + J_2 =}{} - \\sum_{i=1}^p \\mbox{tr} (F_i E_{v_i x} - F_i E[{\\boldsymbol v}_i] E[{\\boldsymbol x}^T] + E_{xv_i}F_i^T-E[{\\boldsymbol x}]E[{\\boldsymbol v}_i^T]F_i^T - {\\mathbb E}_{xv_i} {\\mathbb E}_{v_ix}) \\nonumber\\\\\n\\phantom{J_0 + J_1 + J_2}{} = J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p).\\label{nj012}\n\\end{gather}\nHence, (\\ref{j012}) is true. Therefore,\n\\begin{gather}\n J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p) =\n\\|{\\mathbb E}_{xx}^{1\/2}\\|^2 - \\sum_{k=1}^p \\|{\\mathbb E}_{xv_k}\\|^2\n+ \\|f - E[{\\boldsymbol x}] + \\sum_{k=1}^p F_k E[{\\boldsymbol v}_k]\\|^2 \\nonumber \\\\\n\\phantom{J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p) =}{}+ \\sum_{k=1}^p \\|F_k - {\\mathbb E}_{xv_k}\\|^2.\\label{proof2}\n\\end{gather}\nIt follows from (\\ref{proof2}) that the constrained minimum\n(\\ref{min1})--(\\ref{con1}) is achieved if $f=f^0$ with $f^0$ given by (\\ref{sol-f0}),\nand if $F_k^0$ is such that\n\\begin{gather}\n\\label{min-k}\nJ_k(F_k^0) = \\min_{F_k} J_k(F_k) \\qquad \\mbox{subject to} \\quad \\mbox{rank}\\, (F_k) = \\eta_k,\n\\end{gather}\nwhere $J_k(F_k) = \\|F_k - {\\mathbb E}_{xv_k}\\|^2$.\nThe solution to (\\ref{min-k}) is given \\cite{gol1} by\n\\begin{gather}\n\\label{proof3}\nF_k^0 = K_{k\\eta_k}.\n\\end{gather}\nThen\n\\[\nE[\\|{\\boldsymbol x} - {\\mathcal T}_p^0({\\boldsymbol y})\\|^2]\n=\\|{\\mathbb E}_{xx}^{1\/2}\\|^2 -\\sum_{k=1}^p (\\|{\\mathbb E}_{xv_k}\\|^2 - \\|K_{k\\eta_k} - {\\mathbb E}_{xv_k}\\|^2).\n\\]\nHere \\cite{gol1},\n\\begin{gather}\n\\label{kr}\n \\|{\\mathbb E}_{xv_k}\\|^2 = \\sum_{j=1}^{r} \\alpha^2_{kj}\\qquad\\mbox{and}\\qquad \\|K_{k\\eta_k} - {\\mathbb E}_{xv_k}\\|^2 =\n \\sum_{j=\\eta_k +1}^{r} \\alpha^2_{kj}\n\\end{gather}\nwith $r=r(k)$. Thus, (\\ref{er1}) is true. The theorem is proved.\n\\end{proof}\n\n\\begin{corollary}\\label{corollary1}\nLet ${\\boldsymbol v}_1,\\ldots,{\\boldsymbol v}_p$ be determined by Lemma~{\\rm \\ref{lemma1}}.\nThen the vector $\\hat{f}$ and operators $\\hat{\\mathcal F}_1, \\ldots, \\hat{\\mathcal F}_p$\nsatisfying the unconstrained problem \\eqref{min1}, are determined by\n\\begin{gather}\n\\label{sol-q}\n\\hat{f} = E[{\\boldsymbol x}] - \\sum_{k=1}^p \\hat{F}_k E[{\\boldsymbol v}_k]\\qquad \\mbox{and}\\qquad\n\\hat{\\mathcal F}_1 = {\\mathcal E}_{x v_1}, \\quad \\ldots, \\quad \\hat{\\mathcal F}_p = {\\mathcal E}_{x v_p}\n\\end{gather}\nwith $\\hat{\\mathcal F}_k$ such that $[\\hat{\\mathcal F}_k({\\boldsymbol v}_k)](\\omega) = \\hat{F}_k{\\boldsymbol v}_k(\\omega)$ where\n$\\hat{F}_k\\in{\\mathbb R}^{n\\times m}$ and $k=1,\\ldots,p$.\n\nThe accuracy associated with transform $\\hat{{\\mathcal T}}_p$ given by\n\\begin{gather}\n\\label{fr1}\n\\hat{{\\mathcal T}}_p({\\boldsymbol y})= \\hat{f} + \\sum _{k=1}^p \\hat{\\mathcal F}_k({\\boldsymbol v}_k)\n\\end{gather}\nis such that\n\\begin{gather}\n\\label{er2}\nE[\\|{\\boldsymbol x} - \\hat{{\\mathcal T}}_p({\\boldsymbol y})\\|^2] =\\|{\\mathbb E}_{xx}^{1\/2}\\|^2 -\\sum_{k=1}^p \\|{\\mathbb E}_{xv_k}\\|^2 .\n\\end{gather}\n\\end{corollary}\n\n\\begin{proof} The proof follows directly from (\\ref{proof2}).\n\\end{proof}\n\n\n\\subsubsection[The case when matrix ${\\mathbb E}_{v_k v_k}$ is not invertible for $k=1,\\ldots,p$]{The case\nwhen matrix $\\boldsymbol{{\\mathbb E}_{v_k v_k}}$ is not invertible for $\\boldsymbol{k=1,\\ldots,p}$}\n\\label{det-fk}\n\nWe write $A_k\\in {\\mathbb R}^{m\\times n}$ for an arbitrary matrix, and\ndef\\\/ine operators ${\\cal A}_k: L^2(\\Omega,{\\mathbb R}^{n})\\rightarrow L^2(\\Omega,{\\mathbb R}^{m})$ and\n${\\mathcal E}_{v_k v_k}, {\\mathcal E}^{\\dag}_{v_k v_k}, ({\\mathcal E}^{1\/2}_{v_k v_k})^{\\dag} :L^{2}(\\Omega,{\\mathbb R}^{n})\\rightarrow\nL^{2}(\\Omega,{\\mathbb R}^{n})$ similarly to those in (\\ref{qk}) and (\\ref{ge}).\n\n\nFor the case under consideration (matrix ${\\mathbb E}_{v_k v_k}$ is not invertible),\nwe introduce the SVD of ${\\mathbb E}_{xv_k}({\\mathbb E}^{1\/2}_{v_k v_k})^{\\dag}$,\n\\begin{gather}\n\\label{svd2}\n U_k \\Sigma_k V^{T}_k = {\\mathbb E}_{xv_k}({\\mathbb E}^{1\/2}_{v_k v_k})^{\\dag},\n\\end{gather}\n where, as above,\n$U_k\\in {\\mathbb R}^{m\\times n}$, $V_k\\in {\\mathbb R}^{n\\times n}$\nare orthogonal and $\\Sigma_k\\in {\\mathbb R}^{n\\times n}$ is diagonal,\n\\begin{gather}\n\\label{sqd3}\nU_k = [s_{k1},\\ldots,s_{kn}],\\qquad V_k =[d_{k1},\\ldots,d_{kn}] \\qquad\\mbox{and}\n\\qquad \\Sigma_k = \\mbox{diag}(\\beta_{k1},\\ldots,\\beta_{kn})\n\\end{gather}\n with $\\beta_{k1} \\geq\n\\cdots \\geq \\beta_{kr} > 0$, $\\beta_{k,r+1} = \\cdots = \\beta_{kn} = 0$, $r=1,\\ldots,n$ and\n$r=r(k)$.\n\nLet us set\n\\begin{gather}\nU_{k \\eta_k} = [s_{k1},\\ldots,s_{k\\eta_k}],\\qquad V_{k\\eta_k} = [d_{k1},\\ldots, d_{k\\eta_k}]\\qquad\n\\mbox{and}\\nonumber\\\\\n\\Sigma_{k\\eta_k} = \\mbox{diag}\\,(\\beta_{k1},\\ldots,\\beta_{k\\eta_k}),\\label{tr-svd}\n\\end{gather}\nwhere $U_{k\\eta_k}\\in {\\mathbb R}^{m\\times \\eta_k}$, $V_{k\\eta_k}\\in\n{\\mathbb R}^{n\\times \\eta_k}$ and $\\Sigma_{k\\eta_k}\\in {\\mathbb R}^{\\eta_k \\times \\eta_k}$.\n Now we def\\\/ine $G_{k\\eta_k} \\in{\\mathbb R}^{m\\times n}$ and ${\\mathcal G}_{k\\eta_k}:L^{2}\n (\\Omega,{\\mathbb R}^{n})\\rightarrow L^{2}(\\Omega,{\\mathbb R}^{m})$ by\n\\begin{gather}\n\\label{trsvd2}\nG_{k\\eta_k} = U_{k\\eta_k}\\Sigma_{k\\eta_k}V_{k\\eta_k}^{T}\\qquad\\mbox{and}\\qquad [{\\mathcal G}_{k\\eta_k}({\\boldsymbol w}_k)](\\omega) = G_{k\\eta_k}[{\\boldsymbol w}_k(\\omega)],\n\\end{gather}\nrespectively, for any ${\\boldsymbol w}_k\\in L^{2}(\\Omega,{\\mathbb R}^{n})$.\n\nAs noted before, we write ${\\cal I}$ for the identity operator.\n\n\\begin{theorem}\n\\label{sol2}\nLet ${\\boldsymbol v}_1,\\ldots,{\\boldsymbol v}_p$ be determined by Lemma {\\rm \\ref{ort3}}.\nThen $f^0$ and ${\\mathcal F}_1^0, \\ldots, {\\mathcal F}_p^0$,\n satisfying \\eqref{min1}--\\eqref{con1}, are determined by\n\\begin{gather}\n\\label{sol-f02}\nf^0 = E[{\\boldsymbol x}] - \\sum_{k=1}^p F^0_k E[{\\boldsymbol v}_k]\n\\end{gather}\nand\n\\begin{gather}\n\\label{sol-f12}\n{\\mathcal F}^0_1 = {\\mathcal G}_{1\\eta_1}({\\mathcal E}^{1\/2}_{v_1 v_1})^{ \\dag}\n+ {\\cal A}_1[{\\cal I} - {\\mathcal E}_{v_1 v_1}^{1\/2}({\\mathcal E}^{1\/2}_{v_1 v_1})^{\\dag}],\\\\\n\\cdots \\cdots \\cdots\\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots \\cdots\\nonumber \\\\\n\\label{sol-fp2}\n{\\mathcal F}^0_p = {\\mathcal G}_{p\\eta_p}({\\mathcal E}^{1\/2}_{v_p v_p})^{ \\dag}\n+ {\\cal A}_p[{\\cal I} - {\\mathcal E}_{v_p v_p}^{1\/2}({\\mathcal E}^{1\/2}_{v_p v_p})^{\\dag}],\n\\end{gather}\nwhere for $k=1, \\ldots,p,$ ${\\cal A}_k$ is any linear operator such that $\\mbox{\\rm rank}\\,\n {\\mathcal F}^0_k \\leq \\eta_k$\\footnote{In particular, ${\\cal A}_k$ can be chosen as the zero operator.}.\n\nThe accuracy associated with transform ${\\mathcal T}_p^0$ given by \\eqref{th1}\nand \\eqref{sol-f02}--\\eqref{sol-fp2} is such that\n\\begin{gather}\n\\label{er12}\nE[\\|{\\boldsymbol x} - {\\mathcal T}_p^0({\\boldsymbol y})\\|^2] =\\|{\\mathbb E}_{xx}^{1\/2}\\|^2 -\\sum_{k=1}^p \\sum_{j=1}^{\\eta_k} \\beta^2_{kj}.\n\\end{gather}\n\\end{theorem}\n\n\n\\begin{proof} For ${\\boldsymbol v}_1,\\ldots,{\\boldsymbol v}_p$ determined by Lemma \\ref{ort3}, $J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p)$\nis represented by (\\ref{jqa}) as well.\nLet us consider $J_0$, $J_1$ and $J_2$ given by\n\\begin{gather}\n\\label{j02}\nJ_0 = \\|{\\mathbb E}_{xx}^{1\/2}\\|^2 -\\sum_{k=1}^p \\|{\\mathbb E}_{xv_k}({\\mathbb E}_{v_k v_k}^{1\/2})^{\\dag} \\|^2,\n\\\\\n\\label{j122}\nJ_1 = \\|f - E[{\\boldsymbol x}] + \\sum_{k=1}^p F_k E[{\\boldsymbol v}_k]\\|^2\\qquad\\mbox{and}\\qquad\nJ_2 =\\sum_{k=1}^p \\|F_k{\\mathbb E}_{v_kv_k}^{1\/2} - {\\mathbb E}_{xv_k}({\\mathbb E}_{v_kv_k}^{1\/2})^{\\dag}\\|^2.\n\\end{gather}\nTo show that\n\\begin{gather}\n\\label{jff}\nJ(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p) = J_0 + J_1 + J_2\n\\end{gather}\nwith $J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p)$ def\\\/ined by (\\ref{jqa}), we use the relationships (see \\cite{tor1})\n\\begin{gather}\n\\label{eee1}\n{\\mathbb E}_{xv_k}{\\mathbb E}_{v_k v_k}^\\dag {\\mathbb E}_{v_k v_k}\n= {\\mathbb E}_{xv_k} \\qquad\\mbox{and}\\qquad {\\mathbb E}_{v_k v_k}^{\\dag}{\\mathbb E}_{v_k v_k}^{ 1\/2}\n= ({\\mathbb E}_{v_k v_k}^{1\/2})^{\\dag}\n\\end{gather}\nThen\n\\begin{gather}\nJ_1 = \\mbox{tr} \\Bigg(ff^T - fE[{\\boldsymbol x}^T] + \\sum_{k=1}^p f E[{\\boldsymbol v}_k^T]F_k + E[{\\boldsymbol x}] E[{\\boldsymbol x}^T]\n - E[{\\boldsymbol x}]f^T - \\sum_{k=1}^p E[{\\boldsymbol x}] E[{\\boldsymbol v}_k^T] F_k^T\\nonumber\\\\\n \\phantom{J_1 =}{} + \\sum_{k=1}^p F_k E[{\\boldsymbol v}_k]f^T\n- \\sum_{k=1}^p F_k E[{\\boldsymbol v}_k] E[{\\boldsymbol x}^T] + \\sum_{k=1}^p F_k E[{\\boldsymbol v}_k] \\sum_{i=1}^p E[{\\boldsymbol v}_i^T] F_i^T\\Bigg)\\label{j112}\n\\end{gather}\nand\n\\begin{gather}\nJ_2 = \\sum_{k=1}^p \\mbox{tr}\n(F_k - {\\mathbb E}_{xv_k}{\\mathbb E}_{v_kv_k}^\\dag){\\mathbb E}_{v_kv_k}(F_k^T\n- {\\mathbb E}_{v_kv_k}^\\dag{\\mathbb E}_{v_k x})\\nonumber\\\\\n\\phantom{J_2 }{} = \\sum_{k=1}^p \\mbox{tr} (F_k {\\mathbb E}_{v_kv_k} F_k^T\n- F_k {\\mathbb E}_{v_k x} - {\\mathbb E}_{xv_k}F_k^T + {\\mathbb E}_{xv_k}{\\mathbb E}_{v_kv_k}^\\dag {\\mathbb E}_{v_kx}),\n\\label{j22}\n\\end{gather}\nwhere\n\\begin{gather}\n\\label{aea2}\n\\sum_{k=1}^p \\mbox{tr} (F_k {\\mathbb E}_{v_kv_k} F_k^T)= \\mbox{tr}\\Bigg[E\\Bigg(\\sum_{k=1}^p F_k\n{\\boldsymbol v}_k \\sum_{i=1}^p {\\boldsymbol v}_i^T F_i^T\\Bigg)\\Bigg] -\\mbox{tr}\\Bigg(\\sum_{k=1}^p F_k E[{\\boldsymbol v}_k] \\sum_{i=1}^p E[{\\boldsymbol v}_i^T] F_i^T\\Bigg)\n\\end{gather}\nbecause\n\\begin{gather}\n\\label{v-ort2}\nE[{\\boldsymbol v}_i {\\boldsymbol v}_k^T] - E[{\\boldsymbol v}_i] E[ {\\boldsymbol v}_k^T] = {\\mathbb O}\\qquad\\mbox{for}\\quad i\\neq k\n\\end{gather}\n due to orthogonality of the vectors ${\\boldsymbol v}_1, \\ldots, {\\boldsymbol v}_s$.\nOn the basis of (\\ref{eee1})--(\\ref{aea2}) and similarly to (\\ref{jj012})--(\\ref{nj012}),\nwe establish that (\\ref{jff}) is true.\n Hence,\n\\begin{gather}\n J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p) =\n\\|{\\mathbb E}_{xx}^{1\/2}\\|^2 -\n\\sum_{k=1}^p \\|{\\mathbb E}_{xv_k}({\\mathbb E}_{v_k v_k}^{1\/2})^{\\dag} \\|^2\n+ \\|f - E[{\\boldsymbol x}] \\nonumber \\\\\n\\label{proof22}\n\\phantom{J(f,{\\mathcal F}_1, \\ldots ,{\\mathcal F}_p) =}{}\n+ \\sum_{k=1}^p F_k E[{\\boldsymbol v}_k]\\|^2+ \\sum_{k=1}^p \\|F_k{\\mathbb E}_{v_k v_k}^{1\/2} - {\\mathbb E}_{xv_k}({\\mathbb E}_{v_k v_k}^{1\/2})^{\\dag}\\|^2.\n\\end{gather}\nIt follows from the last two terms in (\\ref{proof22}) that\nthe constrained minimum (\\ref{min1})--(\\ref{con1}) is achieved\nif $f=f^0$ with $f^0$ given by (\\ref{sol-f02}), and $F_k^0$ is such that\n\\begin{gather}\n\\label{min-k2}\nJ_k(F_k^0) = \\min_{F_k} J_k(F_k) \\qquad \\mbox{subject to} \\quad \\mbox{rank} \\,(F_k) = \\eta_k,\n\\end{gather}\nwhere $J_k(F_k) = \\|F_k{\\mathbb E}_{v_k v_k}^{1\/2} - {\\mathbb E}_{xv_k}({\\mathbb E}_{v_k v_k}^{1\/2})^{\\dag}\\|^2$.\nThe constrained minimum (\\ref{min1})--(\\ref{con1})\nis achieved if $f=f^0$ is def\\\/ined by (\\ref{sol-f02}), and if \\cite{gol1}\n\\begin{gather}\n\\label{me}\nF_k{\\mathbb E}_{v_k v_k}^{1\/2} = G_{\\eta_k}.\n\\end{gather}\nThe matrix equation (\\ref{me}) has the general solution \\cite{ben1}\n\\begin{gather}\n\\label{proof312}\nF_k = F^0_k =G_{\\eta_k}({\\mathbb E}_{v_k v_k}^{1\/2})^\\dag\n+ A_k[I - {\\mathbb E}_{v_k v_k}^{1\/2} ({\\mathbb E}_{v_k v_k}^{1\/2})^\\dag]\n\\end{gather}\nif and only if\n\\begin{gather}\n\\label{proof42}\nG_{\\eta_k}({\\mathbb E}_{v_k v_k}^{1\/2})^\\dag {\\mathbb E}_{v_k v_k}^{1\/2}= G_{\\eta_k}.\n\\end{gather}\n The latter is satisf\\\/ied on the basis of the following\n derivation\\footnote{Note that the matrix\n$I - {\\mathbb E}_{v_k v_k}^{1\/2} ({\\mathbb E}_{v_k v_k}^{1\/2})^\\dag$\nis simply a projection onto the null space of ${\\mathbb E}_{v_k v_k}$\n and can be replaced by $I - {\\mathbb E}_{v_k v_k} ({\\mathbb E}_{v_k v_k})^\\dag$.}.\n\nAs an extension of the technique presented in the proving Lemmata~1\nand~2 in \\cite{tor1}, it can be shown that for any matrices $Q_1, Q_2\\in{\\mathbb R}^{m\\times n}$,\n\\begin{gather}\n\\label{proof52}\n{\\cal N}(Q_1)\\subseteq {\\cal N}(Q_2)\\quad \\Rightarrow \\quad Q_2(I- Q_1^\\dag Q_1) = \\mathbb O,\n\\end{gather}\n where ${\\cal N}(Q_i)$ is the null space of $Q_i$ for $i=1,2$. In regard of the equation under consideration,\n\\begin{gather}\n\\label{proof62}\n{\\cal N}([{\\mathbb E}_{v_k v_k}^{1\/2}]^\\dag)\n\\subseteq {\\cal N}({\\mathbb E}_{x v_k}[{\\mathbb E}_{v_k v_k}^{1\/2}]^\\dag).\n\\end{gather}\n The def\\\/inition of $G_{\\eta_k}$ implies that\n\\[\n{\\cal N}({\\mathbb E}_{x v_k}[{\\mathbb E}_{v_k v_k}^{1\/2}]^\\dag)\n\\subseteq {\\cal N}(G_{\\eta_k})\\qquad\\mbox{and then}\\qquad {\\cal N}([{\\mathbb E}_{v_k v_k}^{1\/2}]^\\dag)\n \\subseteq {\\cal N}(G_{\\eta_k}).\n\\]\n On the basis of (\\ref{proof52}), the latter implies\n $G_{\\eta_k}[I-({\\mathbb E}_{v_k v_k}^{1\/2})^\\dag {\\mathbb E}_{v_k v_k}^{1\/2}]={\\mathbb O}$,\n i.e.\\ (\\ref{proof42}) is true. Hence, (\\ref{proof312}) and (\\ref{sol-f12})--(\\ref{sol-fp2}) are true as well.\n\nNext, similar to (\\ref{kr}),\n\\begin{gather}\n\\label{g-e}\n \\|{\\mathbb E}_{xv_k}({\\mathbb E}_{v_kv_k}^{1\/2})^\\dag\\|^2 -\\|G_{\\eta_k}\n - {\\mathbb E}_{xv_k}({\\mathbb E}_{v_kv_k}^{1\/2})^\\dag\\|^2 =\n \\sum_{j=1}^{\\eta_k} \\beta^2_{kj}.\n\\end{gather}\nThen (\\ref{er12}) follows from (\\ref{proof22}), (\\ref{proof312}), (\\ref{sol-f02}) and (\\ref{g-e}).\n\\end{proof}\n\n\\begin{remark}\\label{remark4}\nThe known reduced-rank transforms based on the Volterra polynomial structure\n\\cite{yam2,tor3,tor4} require the computation of a covariance matrix similar\nto ${\\mathbb E}_{vv}$, where ${\\boldsymbol v} = [{\\boldsymbol v}_1,\\ldots,{\\boldsymbol v}_p]^T$,\nbut for $p=N$ where $N$ is large (see Sections \\ref{intr} and \\ref{summ}).\nThe relationships (\\ref{jj012})--(\\ref{min-k}) and (\\ref{j112})--(\\ref{min-k2})\nillustrate the nature of the proposed method and its dif\\\/ference from the\ntechniques in \\cite{yam2,tor3,tor4}: due to the structure (\\ref{t3})\nof the transform ${\\mathcal T}_p$, the procedure for f\\\/inding $f^0$,\n${\\mathcal F}_1^0$, $\\ldots$, ${\\mathcal F}_p^0$ avoids direct computation of ${\\mathbb E}_{vv}$\nwhich could be troublesome due to large $N$. If operators ${\\mathcal Q}_1,\\ldots,{\\mathcal Q}_p$\nare orthonormal, as in Theorem~\\ref{sol1}, then (\\ref{v-ort}) is true and\nthe covariance matrix ${\\mathbb E}_{vv}$ is reduced to the identity.\nIf operators ${\\mathcal Q}_1,\\ldots,{\\mathcal Q}_p$ are orthogonal,\nas in Theorem \\ref{sol2}, then (\\ref{v-ort2}) holds and the covariance matrix ${\\mathbb E}_{vv}$\nis reduced to a block-diagonal form with non-zero blocks ${\\mathbb E}_{v_1v_1}, \\ldots, {\\mathbb E}_{v_pv_p}$\nso that\n\\[\n {\\mathbb E}_{vv}=\\left [ \\begin{array}{cccc}\n{\\mathbb E}_{v_1v_1} & {\\mathbb O} & \\ldots & {\\mathbb O} \\\\\n{\\mathbb O} & {\\mathbb E}_{v_2v_2} & \\ldots & {\\mathbb O} \\\\\n\\ldots & \\ldots & \\ldots & \\ldots \\\\\n{\\mathbb O} & {\\mathbb O} & \\ldots & {\\mathbb E}_{v_p v_p}\n\\end{array} \\right ]\n\\]\n with ${\\mathbb O}$ denoting the zero block. As a result,\n the procedure for f\\\/inding $f^0$, ${\\mathcal F}_1^0, \\ldots, {\\mathcal F}_p^0$\n is reduced to $p$ separate rank-constrained problems (\\ref{min-k}) or (\\ref{min-k2}).\n Unlike the methods in \\cite{yam2,tor3,tor4}, the operators ${\\mathcal F}^0_1,\\ldots,{\\mathcal F}^p_0$\n are determined with {\\em much smaller} $m\\times n$ and {$n\\times n$ matrices}\n given by the simple formulae (\\ref{sol-f0}) and (\\ref{sol-f02})--(\\ref{sol-fp2}).\n This implies a reduction in computational work compared with that required by the approach\n in \\cite{tor3,tor4,how2}.\n\\end{remark}\n\n\\begin{corollary}\\label{corollary2} Let ${\\boldsymbol v}_1,\\ldots,{\\boldsymbol v}_p$ be determined by Lemma {\\rm\\ref{ort3}}.\nThen the vector $\\bar{f}$ and operators $\\bar{\\mathcal F}_1, \\ldots, \\bar{\\mathcal F}_p$,\nsatisfying the unconstrained minimum \\eqref{min1}, are determined by\n\\begin{gather}\n\\label{sol-fc0}\n\\bar{f} = E[{\\boldsymbol x}] - \\sum_{k=1}^p \\bar{F}_k E[{\\boldsymbol v}_k]\n\\end{gather}\nand\n\\begin{gather}\n\\label{sol-fc2}\n\\bar{\\mathcal F}_1 = {\\mathcal E}_{xv_1}{\\mathcal E}_{v_1 v_1}^{ \\dag}\n+ {\\cal A}_1[{\\cal I} - {\\mathcal E}_{v_1 v_1}{\\mathcal E}_{v_1 v_1}^{\\dag}],\\\\\n\\cdots \\cdots\\cdots \\cdots\\cdots \\cdots\\cdots \\cdots\\cdots \\cdots\\cdots \\cdots \\nonumber \\\\\n\\label{sol-fpc2}\n\\bar{\\mathcal F}_p = {\\mathcal E}_{xv_p}{\\mathcal E}_{v_p v_p}^{ \\dag}\n+ {\\cal A}_p[{\\cal I} - {\\mathcal E}_{v_p v_p}{\\mathcal E}_{v_p v_p}^{\\dag}].\n\\end{gather}\nThe associated accuracy for transform $\\bar{{\\mathcal T}}_p$, defined by\n\\[\n\\bar{{\\mathcal T}}_p({\\boldsymbol y}) = \\bar{f} + \\sum _{k=1}^p \\bar{\\mathcal F}_k({\\boldsymbol v}_k),\n\\]\n is given by\n\\begin{gather}\n\\label{er-c2}\nE[\\|{\\boldsymbol x} - \\bar{{\\mathcal T}}_p({\\boldsymbol y})\\|^2] =\\|{\\mathbb E}_{xx}^{1\/2}\\|^2\n-\\sum_{k=1}^p \\|{\\mathbb E}_{xv_k}({\\mathbb E}^{1\/2}_{v_k v_k})^{ \\dag}\\|^2.\n\\end{gather}\n\\end{corollary}\n\n\\begin{proof} It follows from (\\ref{proof22}) that the unconstrained minimum (\\ref{min1})\nis achieved if $f$ is def\\\/ined by~(\\ref{sol-fc0}) and if ${F}_k$ satisf\\\/ies the equation\n$\nF_k{\\mathbb E}_{v_k v_k}^{1\/2} - {\\mathbb E}_{xv_k}({\\mathbb E}_{v_k v_k}^{1\/2})^{\\dag} = {\\mathbb O}\n$\nfor each $k=1,\\ldots,p$.\nSimilar to (\\ref{me})--(\\ref{proof312}), its general solution is given by\n\\[\nF_k = \\bar{F}_k ={\\mathbb E}_{xv_k}{\\mathbb E}_{v_k v_k}^{\\dag}\n+ A_k[I - {\\mathbb E}_{v_k v_k}{\\mathbb E}_{v_k v_k}^{\\dag}]\n\\]\nbecause ${\\mathbb E}_{v_k v_k}^{1\/2}({\\mathbb E}_{v_k v_k}^{1\/2})^{\\dag}\n ={\\mathbb E}_{v_k v_k} {\\mathbb E}_{v_k v_k}^{\\dag}$.\nWe def\\\/ine $\\bar{\\mathcal F}_k$ by $[\\bar{\\mathcal F}_k({\\boldsymbol w}_k)](\\omega)\n= \\bar{F}_k[{\\boldsymbol w}_k(\\omega)]$ for all $k=1,\\ldots,p$, and then (\\ref{sol-fc2})--(\\ref{sol-fpc2})\nare true. The relation (\\ref{er-c2}) follows from (\\ref{proof22}) and (\\ref{sol-fc0})--(\\ref{sol-fpc2}).\n\\end{proof}\n\n\\begin{remark} The dif\\\/ference between the transforms given by Theorems \\ref{sol1} and \\ref{sol2} is that\n${\\mathcal F}^0_k$ by~(\\ref{sol-f0}) (Theorem~\\ref{sol1}) does not contain a factor\nassociated with $({\\mathbb E}_{v_kv_k}^{1\/2})^\\dag$ for all $k=1,\\ldots.p$.\nA similar observation is true for Corollaries \\ref{corollary1} and \\ref{corollary2}.\n\\end{remark}\n\n\\begin{remark}\nThe transforms given by Theorems \\ref{sol1} and \\ref{sol2} are not unique due to arbitrary operators\n${\\mathcal A}_1, \\ldots, {\\mathcal A}_p$. A natural particular choice\nis ${\\mathcal A}_1 = \\cdots = {\\mathcal A}_p = \\mathbb O$.\n\\end{remark}\n\n\n\\subsubsection[Compression procedure by ${\\mathcal T}^0_p$]{Compression procedure by $\\boldsymbol{{\\mathcal T}^0_p}$}\n\\label{sec5.2.3}\n\n\nLet us consider transform ${\\mathcal T}^0_p$ given by (\\ref{th1}),\n(\\ref{sol-f02})--(\\ref{sol-fp2}) with $A_k={\\mathbb O}$ for $k=1,\\ldots,p$ where $A_k$\nis the matrix given in (\\ref{proof312}).\nWe write $[{\\mathcal T}^0_p({\\boldsymbol y})](\\omega) = T^0_p(y)$\nwith $T^0_p:{\\mathbb R}^n\\rightarrow {\\mathbb R}^m$.\n\nLet\n\\[\nB^{(1)}_k = S_{k\\eta_k}V_{k\\eta_k}D_{k\\eta_k}^{T}\\qquad\\mbox{and}\n\\qquad B^{(2)}_k = D_{k\\eta_k}^{T}({\\mathbb E}^{1\/2}_{v_k v_k})^{\\dag}\n\\]\nso that $B^{(1)}_k\\in{\\mathbb R}^{m\\times \\eta_k}$\nand $B^{(2)}_k\\in{\\mathbb R}^{\\eta_k\\times n}$. Here, $\\eta_1$, $\\ldots$, $\\eta_p$\nare determined by (\\ref{con1}). Then\n\\[\nT^0_p(y) = f + \\sum_{k=1}^p B^{(1)}_k B^{(2)}_k v_k,\n\\]\nwhere $v_k = {\\boldsymbol v}_k(\\omega)$ and $B^{(2)}_k v_k\\in{\\mathbb R}^{\\eta_k}$ for $k=1,\\ldots,p$\nwith $\\eta_1 + \\cdots + \\eta_p d_0>> M^{p,q+1} @>d_0>> M^{p+1,q+1} @>d_0>> \\\\\n @. @AAd_1A @AAd_1A @. \\\\\n @>d_0>> M^{p,q} @>d_0>> M^{p+1,q} @>d_0>> \\\\\n @. @AAd_1A @AAd_1A @. \n\\end{CD}\n\\end{equation*}\nThe associated \\emph{total complex} is defined as $\\mathrm{Tot}^n M\\colonequals\\oplus_{p+q=n}M^{p,q}$, with total differential $d\\colonequals d_0+d_1$. A double complex allows for two filtrations, namely,\n\\begin{equation}\nF_{\\rm I}^p(\\mathrm{Tot}^n\\ M) = \\oplus_{r\\geqslant p}M^{r,n-r}\\;, \\qquad F_{\\rm II}^p(\\mathrm{Tot}^n\\ M) = \\oplus_{r\\geqslant p}M^{n-r,r}~.\n\\end{equation}\nThese filtrations are bounded in each dimension if for each $n$ only a finite number of $M^{p,q}$ with $n=p+q$ are non-zero.\n\nCorrespondingly, we can consider two spectral sequences converging to $H^*(\\mathrm{Tot}\\ M, d)$ with as first terms\n\\begin{eqnarray}\n&_{\\rm I} E^{p,q}_1 \\cong H^{p,q}(M,d_1)\\;, \\qquad & _{\\rm I} E^{p,q}_2 \\cong H^{p,q}(H^{*,*}(M,d_1),d_0)\\\\\n&_{\\rm II} E^{p,q}_1 \\cong H^{p,q}(M,d_0)\\;, \\qquad & _{\\rm II} E^{p,q}_2 \\cong H^{p,q}(H^{*,*}(M,d_0),d_1)~.\n\\end{eqnarray}\nNote that here one can show that the first term of the spectral sequence is equal to the one mentioned in the more general case above. Higher differentials $d_{r+1}$ for $r\\geqslant 1$ are defined by $d_{r+1}x=d_1 y$ where $y$ is defined by $d_0 y=d_r x$. Such a $y$ can be proven to always exist, so that the higher differentials are always well-defined.\n\n\\paragraph{Example} As a simple example of the utility of spectral sequences, let us reproduce a proof of the K\\\"unneth formula \\cite{deBoer:1992sy}. Consider a differential graded algebra $(\\Ab,d)$, \\ie, a graded algebra endowed with a differential $d$ of degree one satisfying the Leibniz rule. Let it have two graded subalgebras $\\Ab_1$ and $\\Ab_2$ which are respected by the differential, \\ie, $d\\Ab_i\\subseteq \\Ab_i$. Let us assume the multiplication map $m:\\Ab_1\\otimes\\Ab_2\\rightarrow \\Ab$ is an isomorphism of vector spaces. Then one can define the double complex $(M^{p,q}; d_0, d_1)$ by \n\\begin{equation}\nM^{p,q}\\colonequals m(\\Ab_1^p\\otimes\\Ab_2^q)~,\\quad d_0(a_1a_2)=d(a_1)a_2~,\\quad d_1(a_1a_2)=(-1)^{\\mathrm{deg}(a_1)}a_1 d(a_2)~.\n\\end{equation} \nAssume that this double complex is bounded in each dimension; then one can make use of the spectral sequence for the double complex as described above. One finds for the first couple of levels \n\\begin{equation}\nE^{p,q}_1\\cong m(\\Ab_1^p\\otimes H^q(\\Ab_2,d))~,\\qquad E^{p,q}_2 \\cong m(H^p(\\Ab_1,d)\\otimes H^q(\\Ab_2,d))~.\n\\end{equation} \nHigher differentials all manifestly vanish, so the spectral sequence terminates. At the level of vector spaces, the above-stated theorem implies that $H^*(\\Ab,d) \\cong m(H^*(\\Ab_1,d)\\otimes H^*(\\Ab_2,d))$. This statement can be extended to an isomorphism of algebras because $a_1a_2$ is a representative of an element in $H^*(\\Ab,d)$.\n\\subsection{Argyres-Seiberg duality}\n\\label{subapp:Argyres_Seiberg}\n\nFirst we describe in detail the check of Argyres-Seiberg duality at the level of chiral algebras described in Section \\ref{subsubsec:t3_chiral_algebra}. The first duality frame is that of SQCD, the chiral algebra of which was described in \\cite{Beem:2013sza}. There the generators of the chiral algebra were found to include a $\\widehat{\\mf{su}(6)}_{-3}\\times\\widehat{\\mf{u}(1)}$ affine current algebra with currents $J_i^j$ and $J$, along with baryonic and anti-baryonic operators $\\{b_{ijk},\\tilde b^{ijk}\\}$ of dimension $\\Delta=\\frac32$. The singular OPEs for these generators are as follows,\n{\\allowdisplaybreaks\n\\begin{equation}\n\\label{eq:scqcdopes}\n\\begin{alignedat}{3}\n&& J_i^j(z) J_k^l(0) \t &~~\\sim~& &-\\frac{3 (\\delta_i^l \\delta^j_k - \\text{trace})}{z^2} ~+~ \\frac{\\delta_k^j J_i^l(z)-\\delta_i^l J_k^j(z)}{z}~,\\\\\n&& J(z) J(0) \t\t\t &~~\\sim~& &-\\frac{18}{z^2}~,\\\\\n&& J_i^j(z) b_{k_1 k_2 k_3}(0) &~~\\sim~& &\\phantom{-}\\frac{3 \\delta_{[k_1|}^j b_{i |k_2 k_3]}(0) - \\frac{1}{2} \\delta_i^j b_{k_1 k_2 k_3}(0)}{z}~,\\\\\n&& J(z) b_{k_1 k_2 k_3}(0) &~~\\sim~& &\\phantom{-}\\frac{3b_{k_1 k_2 k_3}(0)}{z}~,\\\\\n&& J(z) b^{k_1 k_2 k_3}(0) &~~\\sim~& &-\\frac{3b^{k_1 k_2 k_3}(0)}{z}~,\\\\\n&& b_{i_1i_2i_3}(z)\\tilde b^{j_1j_2j_3}(0) &~~\\sim~& &\\phantom{-}\\frac{36\\,\\delta_{[i_1}^{[j_1} \\delta_{\\phantom{[}\\!i_2}^{\\phantom{[}\\!j_2} \\delta_{i_3]}^{j_3]}}{z^3} - \\frac{36\\, \\delta_{[i_1}^{[j_1} \\delta_{\\phantom{[}\\!i_2}^{\\phantom{[}\\!j_2}\\hat J_{i_3]}^{j_3]}(0)}{z^2} \\\\\n&& &~~\\phantom{\\sim}~& &\\phantom{-}\\qquad +\\frac{18\\, \t\\delta_{[i_1}^{[j_1} \t \\hat J_{\\phantom{[}\\!i_2}^{\\phantom{[}\\!j_2} \t \\hat J_{i_3]}^{j_3]}(0) - 18\\,\\delta_{[i_1}^{[j_1} \\delta_{\\phantom{[}\\!i_2}^{\\phantom{[}\\!j_2} \\partial \\hat J_{i_3]}^{j_3]}(0)}{z}~.\n\\end{alignedat}\n\\end{equation}}%\nAntisymmetrizations are performed with weight one, and lower (upper) indices $i,j,\\ldots$ transform in the fundamental (antifundamental) representation of $\\mf{su}(6)$. In the last line we have introduced the $\\mf{u}(6)$ current $\\hat J^i_j \\colonequals J^i_j + \\frac{1}{6} \\delta^i_j J$. \nIt was conjectured in \\cite{Beem:2013sza} that the SQCD chiral algebra is a $\\WW$-algebra with just these generators. This proposal passed a few simple checks. All the generators of the Hall-Littlewood chiral ring have been accounted for and the OPE closes. There is no additional stress tensor as a generator because the Sugawara stress tensor of the $\\mf{u}(6)$ current algebra turns out to do the job (this again implies a relation in the Higgs branch chiral ring of SQCD). The spectrum of the chiral algebra generated by these operators also correctly reproduces the low-order expansion of the superconformal index.\n\nOur aim in the remainder of this appendix is to reproduce this chiral algebra from the `exceptional side' of the duality using our proposal that the chiral algebra $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ is the current algebra $(\\,\\widehat{\\mf{e}}_6\\,)_{-3}$. The two free hypermultiplets contribute symplectic bosons $q_{\\alpha}$ and $\\tilde q^{\\alpha}$ with $\\alpha=1,2$ with singular OPE given by\n\\begin{equation}\n\\label{eq:app_symplectic_boson_OPE}\nq_\\alpha(z) \\tilde q^\\beta (0) \\sim \\frac{\\delta_\\alpha^\\beta}{z}~.\n\\end{equation}\nThe $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra should be re-expressed in terms of an $\\mf{su}(6)\\times\\mf{su}(2)$ maximal subalgebra, in terms of which the affine currents split as\n\\begin{equation}\n\\label{eq:e6_as_relabeling}\n\\{J_{A=1,\\ldots,78}\\} ~~\\Longrightarrow~~ \\{X^i_j,~ Y^{[ijk]}_\\alpha,~ Z_\\alpha^\\beta\\}~.\n\\end{equation}\nThe operators $X^i_j$ and $Z_\\alpha^\\beta$ are the affine currents of $\\mf{su}(6)$ and $\\mf{su}(2)$, respectively, with $X^i_i = Z_\\alpha^\\alpha = 0$. The additional operators $Y^{ijk}_\\alpha$ transform in the $(\\mathbf{20},\\mathbf{2})$ of $\\mf{su}(6)\\times\\mf{su}(2)$. The nonvanishing OPEs amongst these operators are simply a rewriting of the $\\wh{\\mf{e}}_6$ current algebra,\n\\begin{eqnarray}\n\\label{eq:e6_ope_decomposition}\nX_i^j(z) X_k^l(0) &~\\sim~& - \\frac{3 (\\delta_i^l \\delta^j_k - \\text{trace})}{z^2} + \\frac{\\delta_k^j X_i^l(0) - \\delta_i^l X_k^j(0)}{z} \\nn\\\\\nZ_\\alpha^\\beta(z) Z_\\gamma^\\delta(0) &~\\sim~& - \\frac{3 (\\delta_\\alpha^\\delta \\delta^\\beta_\\gamma - \\text{trace})}{z^2} + \\frac{\\delta_\\gamma^\\beta Z_\\alpha^\\partial(0) - \\delta_\\alpha^\\delta Z_\\gamma^\\beta(0)}{z}\\nn\\\\\nX_i^j(z) Y^{klm}_\\alpha (0) &~\\sim~& - \\frac{3 \\delta_i^{[k} Y_\\alpha^{lm]j}(0)}{z} - \\text{trace}\\\\\nZ_\\alpha^\\beta(z) Y^{ijk}_\\gamma (0) &~\\sim~& \\phantom{-}\\frac{\\delta_\\gamma^\\beta Y^{ijk}_{\\alpha} (0)}{z} - \\text{trace}\\nn\\\\\nY^{ijk}_\\alpha (z) Y^{lmn}_\\beta (0) &~\\sim~& \\phantom{-}\\frac{\\e_{\\alpha \\beta} \\epsilon^{ijklmn}}{z^2} +\\frac{\\e^{ijklmn} \\e_{\\alpha \\gamma} Z^\\gamma_\\beta(0) - 3 \\e_{\\alpha \\beta} \\e^{[ijklm|p} X^{|n]}_p(0)}{z}~.\\nn\n\\end{eqnarray}\nGluing introduces a dimension $(1,0)$ ghost system in the adjoint of $\\mf{su}(2)$ and restricting to the appropriate cohomology of the following BRST operator,\n\\begin{equation}\n\\label{eq:AS_BRST}\nJ_{\\rm BRST}=~c^\\alpha_\\beta (Z_\\alpha^\\beta - q_\\alpha \\tilde q^\\beta) -\\frac{1}{2} (\\delta_{\\alpha_1}^{\\alpha_6}\\delta_{\\alpha_3}^{\\alpha_2}\\delta_{\\alpha_5}^{\\alpha_4}-\\delta_{\\alpha_1}^{\\alpha_4}\\delta_{\\alpha_3}^{\\alpha_6}\\delta_{\\alpha_5}^{\\alpha_2}) c_{\\alpha_2}^{\\alpha_1}b_{\\alpha_4}^{\\alpha_3}c_{\\alpha_6}^{\\alpha_5}.\n\\end{equation}\nThe cohomology can be constructed level by level using the \\texttt{OPEdefs} package for \\texttt{Mathematica} \\cite{Thielemans:1991uw}. Up to dimension $h=\\frac32$, we find the following operators,\n\\begin{equation}\n\\label{eq:AS_generators}\nX^i_j~,\\qquad\nq_\\alpha\\tilde q^\\alpha~,\\qquad\n\\e_{ijklmn}\\tilde q^\\alpha Y^{lmn}_\\alpha~,\\qquad\n\\e^{\\alpha\\beta}q_\\alpha Y_\\beta^{ijk}~.\n\\end{equation}\nUp to normalizations, these can naturally be identified with the generators of the SQCD chiral algebra,\n\\begin{equation}\n\\label{eq:argyresseibergmatch}\nX_i^j \\simeq J_i^j~, \\qquad \n3 q_\\alpha \\tilde q^\\alpha \\simeq J~, \\qquad \n\\frac{1}{6} \\e_{ijk l m n} \\tilde q^\\alpha Y^{l m n}_\\alpha \\simeq b_{ijk}~, \\qquad\n \\e^{\\alpha \\beta} q_\\alpha Y_\\beta^{i j k} \\simeq \\tilde b^{ijk}~.\n\\end{equation}\nThe equations relating chiral algebra generators in the two duality frames are the same as the ones obtained in \\cite{Gaiotto:2008nz}, with the operators being viewed as generators of the Higgs branch chiral ring. In that work, establishing them at the level of the Higgs branch required a detailed understanding of the chiral ring relations on both sides. By contrast, to establish equivalence of the chiral algebras we need to check that the above operators have the same singular OPEs. Relations in the chiral ring will then show up automatically as null states.\n\nWith the \\texttt{OPEdefs} package we have also computed the OPEs of the composite operators in \\eqref{eq:AS_generators} and found perfect agreement with \\eqref{eq:scqcdopes}. Most of the OPEs are reproduced in a fairly trivial fashion. However, the simple pole in the baryon-antibaryon OPE can only be matched by realizing that there is a null state at level two in the $(\\,\\widehat{\\mf{e}}_6\\,)_{-3}$ algebra given by\n\\begin{equation}\n\\label{eq:t3nullstate}\nY^{ijk}_\\alpha Y^{abc}_\\beta \\e^{\\alpha \\beta} \\e_{abc l m n} + 108 \\partial X^{[i}_{[l} \\delta^j_m \\delta^{k]}_{n]} + 108 X^{[i}_{[l} X^j_m \\delta^{k]}_{n]} + \\frac{1}{72} Z_\\alpha^\\beta Z_\\beta^\\alpha \\delta^{[i}_{[l} \\delta^j_m \\delta^{k]}_{n]}~.\n\\end{equation}\n\nThus we have shown that using our proposal for the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra in the Argyres-Seiberg duality problem, one at least produces a self-contained $\\WW$-algebra that matches between the two sides of the duality. It would be nice to prove that this $\\WW$-algebra is the full chiral algebra. Indeed, if one could demonstrate this fact for the SQCD side of the duality, it seems likely that it wouldn't be too hard to prove that there can be no additional generators in the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra beyond the affine currents.\n\\subsection{Schur indices}\n\\label{subapp:cylinder_cap_index}\n\nAlthough they are only formally defined (there is no true four-dimensional SCFT associated to the cylinder and cap geometries), the reduction rules for the Schur index allow us to define an index for these geometries that must behave appropriately under gluing. Let us determine these indices.\n\n\\paragraph{Cylinder}\nUsing the general results given in Eqns. \\eqref{eq:SchurindexUVcurve} and \\eqref{eq:psi_max}, the index of the two-punctured sphere theory can be determined immediately\n\\begin{align}\\label{eq:indexcylinder}\n\\begin{split}\n\\II_\\text{cylinder}\\left(q;\\textbf{a},\\mathbf{b}\\right) &~=~ K_{\\mathrm{max.}}(\\textbf{a};q) K_{\\mathrm{max.}}(\\mathbf{b};q) \\sum_{\\mf{R}} \\chi_{\\mf{R}}(\\textbf{a}) \\chi_{\\mf{R}}(\\mathbf{b})\\\\\n&~=~{\\rm PE}\\left[ \\frac{q}{1-q} \\left(\\chi_{\\mathrm{adj}}(\\textbf{a}) + \\chi_{\\mathrm{adj}}(\\mathbf{b})\\right)\\right] \\sum_{\\mf{R}} \\chi_{\\mf{R}}(\\textbf{a}) \\chi_{\\mf{R}}(\\mathbf{b})~.\n\\end{split}\n\\end{align}\nUpon using the relation $\\sum_{\\mf{R}} \\chi_{\\mf{R}}(\\textbf{a}) \\chi_{\\mf{R}}(\\mathbf{b}) = \\delta(\\textbf{a},\\mathbf{b}^{-1})$, where the delta function is defined with respect to the Haar measure, we can rewrite this index as\n\\begin{equation}\n\\II_\\text{cylinder}\\left(q;\\textbf{a},\\mathbf{b}\\right) = {\\rm PE}\\left[ \\frac{2q}{1-q} \\chi_{\\mathrm{adj}}(\\textbf{a}) \\right] \\delta(\\textbf{a},\\mathbf{b}^{-1}) = I_V^{-1}(\\textbf{a};q)\\ \\delta(\\textbf{a},\\mathbf{b}^{-1})~,\n\\end{equation}\nwhere $I_V$ is the vector multiplet index \\eqref{eq:vector_multiplet_index}. This makes it clear that when the gluing prescription for the index given in Eqn. \\eqref{eq:indexgluing} is applied, the index $\\II_{\\TT}(q; \\textbf{a},\\ldots)$ of a generic theory $\\TT$ containing a maximal puncture with fugacities $\\textbf{a}$ remains the same after gluing a cylinder to that maximal puncture\n\\begin{equation}\n\\int [d\\textbf{a}] \\Delta(\\textbf{a}) I_V(q;\\textbf{a})\\ \\II_{\\TT}(q; \\textbf{a},\\ldots)\\ \\II_\\text{cylinder}\\left(q;\\textbf{a}^{-1},\\mathbf{b}\\right) = \\II_{\\TT}(q; \\mathbf{b},\\ldots)~.\n\\end{equation}\nHere $[d\\textbf{a}] = \\prod_{j=1}^{\\text{rank} \\mf{g}}\\frac{da_j}{2\\pi i a_j}$ and $\\Delta(\\textbf{a})$ is the Haar measure.\n\nReturning to expression \\eqref{eq:indexcylinder}, we wish to rewrite the sum over representations. Let us therefore consider the regularized sum\n\\begin{equation}\n\\label{eq:regsumreps}\n\\sum_{\\mf{R}} u^{|\\mf{R}|}\\chi_{\\mf{R}}(\\textbf{a}) \\chi_{\\mf{R}}(\\mathbf{b}) = {\\rm PE}\\left[ u\\ \\chi_\\mf{f}(\\textbf{a})\\chi_\\mf{f}(\\mathbf{b}) - u^{n} \\right]\\;,\n\\end{equation}\nwhere $|\\mf{R}|$ denotes the number of boxes in the Young diagram corresponding to the representation $\\mf{R}$ of $\\mf{g}=\\mf{su}(n).$ For $\\mf{g}=\\mf{su}(2)$ we have checked this equality exactly by performing the geometric sums and for $\\mf{su}(3),$ $\\mf{su}(4)$ and $\\mf{su}(5)$ in a series expansion in $u.$ In the limit $u\\rightarrow 1$ one can verify that the right hand side behaves as a $\\delta$-function with respect to the Haar measure, as expected. Consequently, the cylinder index can then be rewritten in a particularly useful form,\n\\begin{equation}\n\\II_\\text{cylinder}\\left(q;\\textbf{a},\\mathbf{b}\\right)={\\rm PE}\\left[ \\frac{q}{1-q} \\left(\\chi_{\\mathrm{adj}}(\\textbf{a}) +\\chi_{\\mathrm{adj}}(\\mathbf{b})\\right) + \\chi_\\mf{f}(\\textbf{a})\\chi_\\mf{f}(\\mathbf{b}) - 1\\right]~.\n\\end{equation}\nBy using $\\chi_{\\mathrm{adj}}(\\textbf{a}) = \\chi_{\\mf{f}}(\\textbf{a})\\chi_{\\mf{f}}(\\textbf{a}^{-1})-1 $ and the $\\delta$-function constraint, one can finally rewrite the index as\n\\begin{align}\\begin{split}\n\\II_\\text{cylinder}\\left(q;\\textbf{a},\\mathbf{b}\\right)&~=~{\\rm PE}\\left[ \\frac{q}{1-q} \\left(\\chi_{\\mathrm{adj}}(\\mathbf{b}) + \\left(\\chi_{\\mf{f}}(\\textbf{a})\\chi_{\\mf{f}}(\\mathbf{b})-1 \\right)\\right) + \\chi_\\mf{f}(\\textbf{a})\\chi_\\mf{f}(\\mathbf{b}) - 1\\right]~,\\\\\n&~=~ {\\rm PE}\\left[ \\frac{q}{1-q} \\chi_{\\mathrm{adj}}(\\mathbf{b}) + \\frac{1}{1-q}\\left(\\chi_{\\mf{f}}(\\textbf{a})\\chi_{\\mf{f}}(\\mathbf{b})-1 \\right)\\right]~.\n\\end{split}\\end{align}\nNote that this looks like the partition function of a finitely generated chiral algebra satisfying a single relation. Namely, it appears that the chiral algebra has one set of dimension one currents transforming in the adjoint of $\\mf{su}(n)$, in addition to a bifundamental field $g_{ab}$ of dimension zero subject to a dimension zero constraint in the singlet representation. Going further, using this interpretation of the index and reintroducing the fugacity $u$ as in \\eqref{eq:regsumreps}, we see that $u$ counts the power of the bifundamental generators in an operator, and the constraint should then involve $n$ bifundamental fields. A natural form for such a relation (after proper rescaling of the generators) is the following,\n\\begin{equation}\n\\label{eq:cylinderconstraint}\n\\frac{1}{n!}\\epsilon^{a_1a_2\\ldots a_n}\\epsilon^{b_1b_2\\ldots b_n} g_{a_1b_1}g_{a_2b_2}\\ldots g_{a_nb_n} = 1.\n\\end{equation}\nInterpreting $g_{ab}$ as a matrix, this is a unit determinant condition. This picture, guessed on the basis of the superconformal index, will be borne out in the qDS analysis below.\n\n\\paragraph{Cap} \nA similarly heuristic analysis is possible for the theory associated to a decorated cap, which is obtained by further partially closing a puncture in the cylinder theory. The index of the decorated cap theory takes the form\n\\begin{align}\n\\II_{\\text{cap}({\\Lambda})}\\left(q;\\textbf{a},\\mathbf{b}_{{\\Lambda}}\\right) &~=~ K_{\\mathrm{max.}}(\\textbf{a};q) K_{{\\Lambda}}(\\mathbf{b}_{{\\Lambda}},q) \\sum_{\\mf{R}} \\chi_{\\mf{R}}(\\textbf{a}) \\chi_{\\mf{R}}(\\mathrm{fug}_{{\\Lambda}}(\\mathbf{b}_{{\\Lambda}};q))~,\\nn\\\\\n&~=~{\\rm PE}\\left[ \\frac{q}{1-q} \\chi_{\\mathrm{adj}}(\\textbf{a}) + \\sum_j \\frac{q^{j+1}}{1-q} \\mbox{Tr}_{\\RR_j^{(\\mathrm{adj})}}(\\mathbf{b}_{{\\Lambda}})\\right] \\sum_{\\mf{R}} \\chi_{\\mf{R}}(\\textbf{a}) \\chi_{\\mf{R}}(\\mathrm{fug}_{{\\Lambda}}(\\mathbf{b}_{{\\Lambda}};q))~,\\\\\n&~=~I_V^{-1\/2}(\\textbf{a};q)\\ K_{{\\Lambda}}(\\mathbf{b}_{{\\Lambda}},q)\\ \\delta(\\textbf{a}^{-1},\\mathrm{fug}_{{\\Lambda}}(\\mathbf{b}_{{\\Lambda}};q))~.\\nn\n\\end{align}\nAgain it is clear how gluing this index reduces the flavor symmetry of the puncture. Using \\eqref{eq:indexgluing} and the general expression for a class $\\SS$ index \\eqref{eq:SchurindexUVcurve} for some theory $\\TT$ of genus $g$ and containing $s$ punctures, of which the first is maximal with corresponding flavor fugacities $\\textbf{a}$, one obtains by gluing the cap to this maximal puncture\n\\begin{multline}\n\\int [d\\textbf{a}] \\Delta(\\textbf{a}) I_V(\\textbf{a};q)\\ \\II_{\\text{cap}({\\Lambda})}\\left(q;\\textbf{a}^{-1},\\mathbf{b}_{{\\Lambda}}\\right)\\ \\sum_{\\mf{R}} C_\\mf{R}(q)^{2g-2+s} K_{\\mathrm{max.}}(\\textbf{a};q) \\chi_{\\mf{R}}(\\textbf{a}) \\prod_{i=2}^s \\psi_{\\mf{R}}^{\\Lambda_i}({\\bf x}_{\\Lambda_i} ;q) \\\\ = \\sum_{\\mf{R}} C_\\mf{R}(q)^{2g-2+s} K_{{\\Lambda}}(\\mathbf{b}_{{\\Lambda}},q) \\chi_{\\mf{R}}(\\mathrm{fug}_{{\\Lambda}}(\\mathbf{b}_{{\\Lambda}};q)) \\prod_{i=2}^s \\psi_{\\mf{R}}^{\\Lambda_i}({\\bf x}_{\\Lambda_i} ;q)~,\n\\end{multline}\nwhere we have again used that $K_{\\mathrm{max.}}(\\textbf{a};q) = I_V^{-1\/2}(\\textbf{a};q).$\n\nAs in the previous paragraph we can rewrite the index in a suggestive fashion,\n\\begin{align}\n\\II_{\\text{cap}({\\Lambda})}\\left(q;\\textbf{a},\\mathbf{b}_{\\Lambda}\\right)\n&~=~ {\\rm PE}\\left[ \\sum_j \\frac{q^{j+1}}{1-q} \\mbox{Tr}_{\\RR_j^{(\\mathrm{adj})}}(\\mathbf{b}_{{\\Lambda}}) + \\frac{1}{1-q}\\left(\\chi_{\\mf{f}}(\\textbf{a})\\chi_{\\mf{f}}(\\mathrm{fug}_{{\\Lambda}}(\\mathbf{b}_{{\\Lambda}};q))-1 \\right)\\right]~.\n\\end{align}\nA natural interpretation of this index is as that of a chiral algebra with generators given by currents $J_{\\bar\\alpha}$ for $T_{\\bar\\alpha} \\in \\ker( ad_{{\\Lambda}(t_+)})$ with dimensions shifted by their ${\\Lambda}(t_0)$ weight. Moreover, for each $\\mf{su}(2)$ irrep in the decomposition \\eqref{eq:generaldecomposition} of the fundamental representation $\\mf{f}$ there are an additional $2j+1$ generators transforming in representation $\\mf{f}\\otimes\\RR_j^{(\\mf{f})}$ with dimensions $-j,-j+1,\\ldots,j$. The latter generators satisfy a singlet relation of dimension zero.\n\\subsection{QDS argument}\n\\label{subapp:cylinder_cap_qDS}\n\nNow that we have some intuition for the kinds of chiral algebras to expect, let us study the cylinder theory for $\\mf{g}=\\mf{su}(3)$ by fully closing a puncture in the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ theory. Full closure is achieved via the principal embedding $\\rho:\\mf{su}(2)\\rightarrow \\mf{g}$, which is can be specified explicitly in components as \n\\begin{equation}\n\\rho(t_-) = 2(T_2^{\\ 1} + T_3^{\\ 2})~,\\qquad \\rho(t_0) = T_1^{\\ 1}-T_3^{\\ 3}~,\\qquad \\rho(t_+)=T_1^{\\ 2} + T_2^{\\ 3}~.\n\\end{equation}\nThe grading by $\\rho(t_0)$ is integral, with the negatively graded generators being $T_3^{\\ 1}$ with grade minus two and $T_2^{\\ 1},T_3^{\\ 2}$ with grade minus one. We should then impose the constraints\n\\begin{equation}\n\\label{eq:cylinderconstraints}\n\\left(J^{(1)}\\right)_2^{\\ 1} + \\left(J^{(1)}\\right)_3^{\\ 2} = 1\\;,\\qquad \\left(J^{(1)}\\right)_2^{\\ 1} - \\left(J^{(1)}\\right)_3^{\\ 2} = 0\\;, \\qquad \\left(J^{(1)}\\right)_3^{\\ 1}=0~.\n\\end{equation}\nUpon introducing three $(b,c)$-ghost systems -- $(b_2^{\\ 1},c_1^{\\ 2})$, $(b_3^{\\ 2},c_2^{\\ 3})$, and $(b_3^{\\ 1},c_1^{\\ 3})$ -- these first-class constraints are implemented by a BRST procedure via the current\n\\begin{equation}\nd(z) = (J^{(1)})_2^{\\ 1} c_1^{\\ 2}(z) + (J^{(1)})_3^{\\ 2}c_2^{\\ 3}(z) + (J^{(1)})_3^{\\ 1} c_1^{\\ 3}(z) - \\frac{1}{2} (c_1^{\\ 2}+c_2^{\\ 3})(z) - b_3^{\\ 1} c_1^{\\ 2}c_2^{\\ 3}(z)~.\n\\end{equation}\n\n\\begin{table}[t!]\n\\begin{center}\n\\begin{tabular}{c|l}\n\\hline\\hline\n\\text{dimension} & \\text{generators}\\\\\n\\hline\n0 & $\\WW_{3bc}, \\tilde{ \\WW}^{1bc} $\\\\\n1 & $\\WW_{2bc}, \\tilde{ \\WW}^{2bc}, ( \\JJ^{2})_{b_1}^{\\ b_2}, (\\JJ^{3})_{c_1}^{\\ c_2}$\\\\\n2 & $\\hat{ \\JJ}_{\\text{sum}}, \\WW_{1bc}, \\tilde{ \\WW}^{3bc}$\\\\\n3 &$ (\\hat{ \\JJ}^{1})_1^{\\ 3}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab:generatorscylinder}(Redundant) generators of the cylinder theory for $\\mf{g}=\\mf{su}(3)$.}\n\\end{table}\n\nThis cohomological problem is partly solved by following the same approach as that advocated in Subsec. \\ref{subsubsec:qDSspecseq}. The redundant generators of the reduced algebra are the tic-tac-toed versions of the currents $ (\\hat J^{1})_1^{\\ 3}$ and $\\hat J_{\\text{sum}} \\equiv (\\hat J^{1})_1^{\\ 2} + (\\hat J^{1})_2^{\\ 3},$ as well as of the generators $\\{(J^{2})_{b_1}^{\\ b_2},\\,(J^{3})_{c_1}^{\\ c_2},\\,W_{abc},\\,\\tilde W^{abc}\\}$. These currents can be seen arranged according to their dimensions in Table \\ref{tab:generatorscylinder}.\n\nThe explicit form of the tic-tac-toed generators of dimensions zero and one are fairly simple,\n\\begin{align}\n\\WW_{3bc}\t\t\t\t&~\\colonequals~ W_{3bc}~,\\\\\n\\tilde{\\WW}^{1bc}\t\t&~\\colonequals~ \\tilde W^{1bc}~,\\\\\n\\WW_{2bc}\t\t\t\t&~\\colonequals~ W_{2bc} + 2 W_{3bc}(\\hat J^{(1)})_3^{\\ 3}~,\\\\\n\\tilde{ \\WW}^{2bc}\t\t&~\\colonequals~ \\tilde W^{2bc} + 2 \\tilde W^{1bc} (\\hat J^{(1)})_1^{\\ 1}~,\\\\\n(\\JJ^{2})_{b_1}^{\\ b_2}\t&~\\colonequals~ (J^{2})_{b_1}^{\\ b_2}~,\\\\\n(\\JJ^{3})_{c_1}^{\\ c_2}\t&~\\colonequals~ (J^{3})_{c_1}^{\\ c_2}~.\n\\end{align}\nOn the other hand, the higher dimensional generators are quite complicated,\n{\\small\n\\begin{align}\\begin{split}\n\\hat{ \\JJ}_{\\text{sum}}&~\\colonequals~ (\\hat J^{1})_1^{\\ 2} + (\\hat J^{1})_2^{\\ 3} \\\\\n& - \\left(-2(2+k)\\partial(\\hat J^{1})_2^{\\ 2} - 4(2+k)\\partial(\\hat J^{1})_3^{\\ 3} +2(\\hat J^{1})_1^{\\ 1}(\\hat J^{1})_2^{\\ 2} + 2 (\\hat J^{1})_1^{\\ 1}(\\hat J^{1})_3^{\\ 3} + 2(\\hat J^{1})_2^{\\ 2}(\\hat J^{1})_3^{\\ 3} \\right)~,\n\\end{split}\\\\\n\\begin{split}\n\\WW_{1bc}&~\\colonequals~ W_{1bc} - \\left(2 W_{2bc}(\\hat J^{1})_1^{\\ 1} -2 W_{3bc}(\\hat J^{1})_2^{\\ 3} \\right)\\\\\n&+ \\left( -4\\left((\\hat J^{1})_1^{\\ 1} + (\\hat J^{1})_2^{\\ 2}\\right) W_{3bc} (\\hat J^{1})_3^{\\ 3} -\\frac{1}{3}(-20-12k)W_{3bc}\\partial(\\hat J^{1})_3^{\\ 3} - \\frac{8}{3} \\partial W_{3bc} (\\hat J^{1})_3^{\\ 3}\\right)~,\n\\end{split}\\\\\n\\begin{split}\n\\tilde{ \\WW}^{3bc}&~\\colonequals~ \\tilde W^{3bc} - \\left( 2\\tilde W^{2bc}(\\hat J^{1})_3^{\\ 3} + 2 \\tilde W^{1bc}(\\hat J^{1})_2^{\\ 3} \\right) +4(\\hat J^{1})_3^{\\ 3} \\tilde W^{1bc} (\\hat J^{1})_2^{\\ 2}\\\\\n& - \\tilde W^{1bc}\\partial\\left(-\\frac{4}{3}(\\hat J^{1})_1^{\\ 1} +(8+4k)(\\hat J^{1})_3^{\\ 3} \\right) - \\partial \\tilde W^{1bc}\\left( -\\frac{4}{3}(\\hat J^{1})_1^{\\ 1} -\\frac{4}{3}(\\hat J^{1})_3^{\\ 3}\\right)~,\n\\end{split}\\\\\n\\begin{split}\n(\\hat{ \\JJ}^{1})_1^{\\ 3}&~\\colonequals~ (\\hat J^{1})_1^{\\ 3} - \\left( 2(k+2)\\partial(\\hat J^{1})_1^{\\ 2} -2 (\\hat J^{1})_2^{\\ 3}\\left((\\hat J^{1})_2^{\\ 2}+(\\hat J^{1})_3^{\\ 3} \\right) -2(\\hat J^{1})_1^{\\ 2}\\left( (\\hat J^{1})_1^{\\ 1} + (\\hat J^{1})_2^{\\ 2} \\right) \\right)\\\\\n&+ 4(4+4k+k^2)\\partial^2(\\hat J^{1})_1^{\\ 1} - 4(2+k)(\\hat J^{1})_1^{\\ 1}\\partial (\\hat J^{1})_1^{\\ 1}+ 4(2+k)(\\hat J^{1})_1^{\\ 1}\\partial (\\hat J^{1})_2^{\\ 2}\\\\\n& -4 \\left( (\\hat J^{1})_1^{\\ 1} + (\\hat J^{1})_3^{\\ 3} \\right)\\left((\\hat J^{1})_2^{\\ 2}+(\\hat J^{1})_3^{\\ 3} \\right)\\left((\\hat J^{1})_1^{\\ 1}+(\\hat J^{1})_2^{\\ 2}\\right)~.\n\\end{split}\n\\end{align}}%\n\nOur next task should be to check for redundancies by computing null relations. This analysis is substantially complicated by the presence of dimension zero fields in the cohomology. This means that we don't have an algorithm for finding such redundancies that must terminate in principle. Instead, we use the nulls of $T_3$ to predict some of the nulls in the cylinder theory.\n\n\\paragraph{Dimension zero nulls}\nStarting with the $\\mathbf{(8,1,1)}$ nulls and specializing the indices to $(a_1,a_2)=(3,1)$ we obtain the null relation\n\\begin{equation}\n0=\\frac{1}{3}W_{3bc}\\tilde W^{1bc} + (J^{1})_{3}^{\\ a} (J^{1})_{a}^{\\ 1} - 3 \\partial (J^{1})_{3}^{\\ 1} = \\frac{1}{4} +\\frac{1}{3} {\\WW}_{3bc} \\tilde {\\WW}^{1bc} + d(\\ldots)~.\n\\end{equation}\nSimilarly, starting with the $\\mathbf{(8,8,1)}$ nulls and specializing the indices to $(a_1,a_2)=(3,1)$ we obtain the null relation\n\\begin{align}\\begin{split}\n0&~=~\\left(W_{3b_1c}\\tilde W^{1b_2c} - \\frac{1}{3}\\delta_{b_1}^{b_2}W_{3bc}\\tilde W^{1bc} \\right) + (J^{(1)})_{3}^{\\ 1} (J^{(2)})_{b_1}^{\\ b_2}~,\\\\\n &~=~ \\left({\\WW}_{3b_1c} \\tilde {\\WW}^{1b_2c} - \\frac{1}{3} \\delta_{b_1}^{b_2} {\\WW}_{3bc} \\tilde {\\WW}^{1bc} \\right) + d(\\ldots)~.\n\\end{split}\\end{align}\nSimilar nulls can be found by interchanging the second and third puncture. In summary, we have the relations\n\\begin{equation}\n\\label{WWtinverse}\n{\\WW}_{3b_1c} \\tilde {\\WW}^{1b_2c} = -\\frac{1}{4} \\delta_{b_1}^{b_2}\\;, \\qquad {\\WW}_{3bc_1} \\tilde {\\WW}^{1bc_2} = - \\frac{1}{4} \\delta_{c_1}^{c_2}~.\n\\end{equation}\nThis shows that, up to a rescaling, $\\WW_{3bc}(z)$ and $\\tilde\\WW^{1bc}(z)$ can be thought of as inverses of one another.\n\nNext, we look at the $\\mathbf{(\\bar{6},3,3)}$ nulls and specialize $a_1=a_2=1$, which gives us\n\\begin{align}\\begin{split}\n0&~=~2(J^{1})_{\\alpha_1}^{\\ 1} W_{\\alpha_2bc}\\epsilon^{\\alpha_1\\alpha_21} + \\tilde W^{1b_1c_1}\\tilde W^{1b_2c_2} \\epsilon_{bb_1b_2}\\epsilon_{cc_1c_2}~,\\\\ \n &~=~\\WW_{3bc} + \\tilde{\\WW}^{1b_1c_1}\\tilde{\\WW}^{1b_2c_2} \\epsilon_{bb_1b_2}\\epsilon_{cc_1c_2} + d(\\dots)~.\n\\end{split}\\end{align}\nSimilarly from the nulls in the $\\mathbf{(6,\\bar{3},\\bar{3})}$ we find\n\\begin{align}\\begin{split}\n0&~=~(J^{1})_{3}^{\\ \\ \\alpha_1} \\tilde W^{\\alpha_2bc}\\epsilon_{\\alpha_1\\alpha_23} +\\frac{1}{2} W_{3b_1c_1} W_{3b_2c_2} \\epsilon^{bb_1b_2}\\epsilon^{cc_1c_2}~,\\\\\n &~=~-\\frac{1}{2} \\tilde{\\WW}^{1bc} +\\frac{1}{2} \\WW_{3b_1c_1} \\WW_{3b_2c_2} \\epsilon^{bb_1b_2}\\epsilon^{cc_1c_2} + d(\\ldots)~.\n\\end{split}\\end{align}\nCombining these with the previous relations, we find that\n\\begin{equation}\n\\frac{1}{3!}\\tilde {\\WW}^{1bc}\\tilde {\\WW}^{1b_1c_1}\\tilde {\\WW}^{1b_2c_2} \\epsilon_{bb_1b_2}\\epsilon_{cc_1c_2} = -\\frac{1}{3!}\\tilde {\\WW}^{1bc}{\\WW}_{3bc} = - \\frac{1}{3!}{\\WW}_{3bc} \\tilde {\\WW}^{1bc} = \\frac{1}{8}~,\n\\end{equation}\nand \n\\begin{equation}\n\\frac{1}{3!}{\\WW}_{3bc}{\\WW}_{3b_1c_1} {\\WW}_{3b_2c_2} \\epsilon^{bb_1b_2}\\epsilon^{cc_1c_2} = \\frac{1}{3!}{\\WW}_{3bc}\\tilde {\\WW}^{1bc} = -\\frac{1}{8}~.\n\\end{equation}\nThese are conditions on the determinants of $\\WW_{3bc}$ and $\\tilde \\WW^{1bc}$ thought of as three-by-three matrices. Note that we used the relation $\\tilde {\\WW}^{1bc}{\\WW}_{3bc} = {\\WW}_{3bc} \\tilde {\\WW}^{1bc}$, which is true in cohomology:\n\\begin{equation}\n\\tilde {\\WW}^{1bc}{\\WW}_{3bc} = {\\WW}_{3bc} \\tilde {\\WW}^{1bc} -d(9 \\partial b_3^{\\ 1})~.\n\\end{equation}\nIf we now introduce rescaled operators $g_{bc} \\colonequals -2 {\\WW}_{3bc}$ and $\\tilde g^{bc} \\colonequals 2 \\tilde {\\WW}^{1bc}$, then $g$ and $\\tilde g$ have unit determinant and are inverses of one another. Because of the determinant condition, this also means that we can rewrite $\\tilde g$ in terms of positive powers of $g$, so only one needs to be considered as an honest generator of the chiral algebra.\n\n\\paragraph{Dimension one nulls}\nWe can continue the same analysis at dimension one. The second relation in the $\\mathbf{(3,3,3)}$ representation gives us\n\\begin{equation}\n(\\JJ^{2})_{b}^{\\ \\beta}\\WW_{3 \\beta c_1} = (\\JJ^{3})_{c_1}^{\\ \\gamma}\\WW_{3 b\\gamma}~.\n\\end{equation}\nBy taking the normal ordered product of both sides with $\\tilde {\\WW}^{1 b c_2}$ and re-ordering (ignoring BRST exact terms), we can make a sequence of replacements using the dimension zero relations of the previous paragraph and end up with the following derivation,\n{\\small\n\\begin{align}\n&&\\tilde {\\WW}^{1 b c_2}({\\JJ}^{(2)})_{b}^{\\ \\beta}{\\WW}_{3 \\beta c_1} &~=~ \\tilde {\\WW}^{1 b c_2}({\\JJ}^{(3)})_{c_1}^{\\ \\gamma}{\\WW}_{3 b\\gamma}\\nn\\\\\n&\\Longrightarrow&({\\JJ}^{(2)})_{b}^{\\ \\beta}{\\WW}_{3 \\beta c_1}\\tilde {\\WW}^{1 b c_2} +\\tfrac{8}{3}{\\WW}_{3 \\beta c_1}\\partial\\tilde {\\WW}^{1 \\beta c_2} &~=~ ({\\JJ}^{(3)})_{c_1}^{\\ \\gamma}{\\WW}_{3 b\\gamma}\\tilde {\\WW}^{1 b c_2} - \\tfrac{1}{3} {\\WW}_{3 \\beta c_1}\\partial\\tilde {\\WW}^{1 \\beta c_2} +\\delta_{c_1}^{c_2} {\\WW}_{3 \\beta \\gamma}\\partial\\tilde {\\WW}^{1 \\beta \\gamma}\\nn\\\\\n&\\Longrightarrow&({\\JJ}^{(2)})_{b}^{\\ \\beta}{\\WW}_{3 \\beta c_1}\\tilde {\\WW}^{1 b c_2} &~=~ ({\\JJ}^{(3})_{c_1}^{\\ \\gamma}{\\WW}_{3 b\\gamma}\\tilde {\\WW}^{1 b c_2} - 3 \\left( {\\WW}_{3 \\beta c_1}\\partial\\tilde {\\WW}^{1 \\beta c_2} -\\tfrac{1}{3}\\delta_{c_1}^{c_2} {\\WW}_{3 \\beta \\gamma}\\partial\\tilde {\\WW}^{1 \\beta \\gamma} \\right)\\nn\\\\\n&\\Longrightarrow&({\\JJ}^{(2)})_{b}^{\\ \\beta}{\\WW}_{3 \\beta c_1}\\tilde {\\WW}^{1 b c_2} &~=~ -\\tfrac{1}{4} ({\\JJ}^{(3)})_{c_1}^{\\ c_2} - 3 \\left( {\\WW}_{3 \\beta c_1}\\partial\\tilde {\\WW}^{1 \\beta c_2} -\\tfrac{1}{3}\\delta_{c_1}^{c_2} {\\WW}_{3 \\beta \\gamma}\\partial\\tilde {\\WW}^{1 \\beta \\gamma} \\right)\\nn\\\\\n&\\Longrightarrow&({\\JJ}^{(2)})_{b}^{\\ \\beta}{ g}_{\\beta c_1}\\tilde { g}^{ b c_2} &~=~ ({\\JJ}^{(3)})_{c_1}^{\\ c_2} - 3 \\left( {g}_{\\beta c_1}\\partial\\tilde { g}^{ \\beta c_2} -\\tfrac{1}{3}\\delta_{c_1}^{c_2} { g}_{\\beta \\gamma}\\partial\\tilde { g}^{\\beta \\gamma} \\right)~.\\label{eq:JintermsofJ}\n\\end{align}}%\nAt last, we see that the current $\\JJ^{(3)}$ is not an independent generator. \n\nOther dimension one nulls can be obtained from the first equality in the $\\mathbf{(3,3,3)}$. Here we find\n\\begin{equation}\n(J^{1})_{3}^{\\ \\alpha}W_{\\alpha bc} = (J^{2})_{b}^{\\ \\beta}W_{3 \\beta c} ~\\Longrightarrow~ \\frac{1}{2}\\WW_{2bc}+\\frac{2}{3}\\partial \\WW_{3bc} = (\\JJ^{2})_{b}^{\\ \\beta}\\WW_{3 \\beta c}~,\n\\end{equation}\nwhich implies that the generator $\\WW_{2bc}$ is not independent. Similarly, from the $\\mathbf{(\\bar3,\\bar3,\\bar3)}$ relations one finds\n\\begin{equation}\n(J^{1})_{\\alpha}^{\\ 1}\\tilde W^{\\alpha bc} = (J^{2})_{\\beta}^{\\ b}\\tilde W^{1 \\beta c} ~\\Longrightarrow~ \\frac{1}{2}\\tilde{\\WW}^{2bc}-\\frac{2}{3}\\partial \\tilde{\\WW}^{1bc} = (\\JJ^{2})_{\\beta}^{\\ b}\\tilde {\\WW}^{1 \\beta c},\n\\end{equation}\nwhich implies that $\\tilde{\\WW}^{2bc}$ is not an independent generator. \n\nBased on the analysis of the index in \\ref{subapp:cylinder_cap_index}, we expect that all higher dimensional generators can be similarly related via null relations to composites of $\\JJ^{2}$ and $g_{bc} = -2\\WW_{3 b c}$. It would be interesting if this could be proven as a consequence of only the null states that are guaranteed to exist based on nulls of the unreduced theory, although such a simplification is not a necessary condition for the existence of the desired nulls.\n\\subsection{Reduction of \\texorpdfstring{$T_3$}{T3} to free hypermultiplets}\n\\label{subapp:e6_to_free}\n\nIn this appendix we provide some details about the reduction of the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra to free symplectic bosons. This corresponds to the subregular embedding $\\mf{su}(2)\\hookrightarrow\\mf{su}(3)$, which is given by\n\\begin{equation}\n\\Lambda(t_0) = \\frac12(T_1^{\\phantom{1}1} - T_3^{\\phantom{1}3})~,\\qquad\n\\Lambda(t_-) = T_3^{\\phantom{1}1}~,\\qquad\n\\Lambda(t_+) = T_1^{\\phantom{1}3}~.\n\\end{equation}\nThe grading on the Lie algebra by the Cartan element $\\Lambda(t_0)$ is half-integral. In order to arrive at first-class constraints, we introduce a different Cartan element $\\delta$ that gives an integral grading. More specifically, we have $\\delta = \\frac13 (T_1^{\\ 1} + T_2^{\\ 2} - 2T_3^{\\ 3})$. With respect to the $\\delta$-grading there are two negatively graded currents and we consequently impose the constraints $\\left(J^{1}\\right)_3^{\\ 1} = 1$ and $\\left(J^{1}\\right)_3^{\\ 2} = 0$. These are implemented via a BRST procedure with differential given by\n\\begin{equation}\nd(z) = \\left(\\left((J^{1})_3^{\\ 1} -1 \\right) c_1^{\\ 3} + (J^{1})_3^{\\ 2} c_2^{\\ 3}\\right)(z)~,\n\\end{equation}\nwhere the ghost pairs $b_3^{\\ 1}, c_1^{\\ 3} $ and $b_3^{\\ 2},c_2^{\\ 3}$ have the usual singular OPEs.\n\nImplementing the first step of the qDS procedure, one obtains the (redundant) generators of the chiral algebra at the level of vector spaces. Applying the tic-tac-toe procedure to produce genuine chiral algebra generators, we obtain the set of generators that were listed in Table \\ref{tab:T3_reduced_generators}. The explicit forms of these generators are given as follows,\n{\\allowdisplaybreaks\n\\begin{align}\n\\label{eq:reduced_generators_def}\n\\JJ_{\\mf{u}(1)} &~\\colonequals~ (\\hat J^{1})_1^{\\ 1}-2 (\\hat J^{1})_2^{\\ 2}+(\\hat J^{1})_3^{\\ 3}\\nn\\\\\n(\\hat {\\JJ}^{1})_1^{\\ 2} &~\\colonequals~ (\\hat J^{1})_1^{\\ 2}\\nn\\\\\n(\\hat {\\JJ}^{1})_1^{\\ 3} &~\\colonequals~ (\\hat J^{1})_1^{\\ 3} -\\left( -(k+1)\\partial (\\hat J^{1})_3^{\\ 3} +(\\hat J^{1})_1^{\\ 1} (\\hat J^{1})_3^{\\ 3} -(\\hat J^{1})_2^{\\ 1}(\\hat J^{1})_1^{\\ 2}\\right)\\nn\\\\\n(\\hat {\\JJ}^{1})_2^{\\ 3} &~\\colonequals~ (\\hat J^{1})_2^{\\ 3} -\\left( (k+2)\\partial (\\hat J^{1})_2^{\\ 1} +(\\hat J^{1})_3^{\\ 3} (\\hat J^{1})_2^{\\ 1} -(\\hat J^{1})_2^{\\ 2}(\\hat J^{1})_2^{\\ 1}\\right)\\nn\\\\\n{\\WW}_{1bc} &~\\colonequals~ W_{1bc} - W_{3bc} (\\hat J^{1})_1^{\\ 1}\\nn\\\\\n{\\WW}_{2bc} &~\\colonequals~ W_{2bc} - W_{3bc}(\\hat J^{1})_2^{\\ 1}\\\\\n{\\WW}_{3bc} &~\\colonequals~ W_{3bc}\\nn\\\\\n\\tilde {\\WW}^{1bc} &~\\colonequals~ \\tilde W^{1bc}\\nn\\\\\n\\tilde {\\WW}^{2bc} &~\\colonequals~ \\tilde W^{2bc}\\nn\\\\\n\\tilde {\\WW}^{3bc} &~\\colonequals~ \\tilde W^{3bc} - \\left(- \\tilde W^{1bc} (\\hat J^{1})_1^{\\ 1} - \\tilde W^{2bc} (\\hat J^{1})_2^{\\ 1} \\right)\\nn\\\\\n(\\JJ^{2})_{b_1}^{\\ b_2}&~\\colonequals~( J^{2})_{b_1}^{\\ b_2}\\nn\\\\\n(\\JJ^{3})_{c_1}^{\\ c_2}&~\\colonequals~( J^{3})_{c_1}^{\\ c_2}~\\nn.\n\\end{align}\n}\n\nThe generators $\\WW_{3bc}$ and $\\tilde\\WW^{1bc}$ have the correct charges and mutual OPE to be identified as the expected symplectic bosons. It follows that the reduction argument will be complete if we can show that at the specific value of the level of interest $k=-3$, all the other generators listed in Eqn. \\eqref{eq:reduced_generators_def} participate in a null state condition that allows them to be equated with composites of $\\WW_{3bc}$ and $\\tilde\\WW^{1bc}$. \n\nIndeed, we do find such relations to account for all additional generators. At level $h=1$, we find\n\\begin{align}\n\\JJ_{\\mf{u}(1)} \t\t\t&~=~ - {\\WW}_{3bc} \\tilde {\\WW}^{1bc}~,\\label{eq:nullJU1}\\\\ \n(\\JJ^{2})_{b_1}^{\\ b_2} &~=~ -\\left({\\WW}_{3b_1c} \\tilde {\\WW}^{1b_2c} - \\frac13 \\delta_{b_1}^{b_2} {\\WW}_{3bc} \\tilde {\\WW}^{1bc} \\right)~,\\label{eq:nullJSU3_2}\\\\\n(\\JJ^{3})_{c_1}^{\\ c_2} &~=~ -\\left({\\WW}_{3bc_1} \\tilde {\\WW}^{1bc_2} - \\frac13 \\delta_{c_1}^{c_2} {\\WW}_{3bc} \\tilde {\\WW}^{1bc} \\right)~,\\label{eq:nullJSU3_3}\\\\\n\\tilde{\\WW}_{2bc} \t\t&~=~ \\frac{1}{2} \\epsilon_{b b_1b_2} \\epsilon_{c c_1c_2} \\tilde {\\WW}^{1b_1c_1} \\tilde {\\WW}^{1b_2c_2}~,\\label{eq:nullWlower}\\\\\n\\tilde{\\WW}^{2bc} \t\t&~=~ -\\frac{1}{2} \\epsilon^{b b_1b_2} \\epsilon^{c c_1c_2} {\\WW}_{3b_1c_1} {\\WW}_{3b_2c_2}~.\\label{eq:nullWupper}\n\\end{align}\nAt dimension $h=3\/2$, one can find the null relations\n\\small\n\\begin{align}\n(\\hat{\\JJ}^{1})_1^{\\ 2} &~=~ \\frac16\\WW_{3b_1c_1}\\WW_{3b_2c_2}\\WW_{3b_3c_3}\\epsilon^{b_1b_2b_3}\\epsilon^{c_1c_2c_3}~,\\\\\n(\\hat{\\JJ}^{1})_2^{\\ 3} &~=~ -\\frac16\\tilde{\\WW}^{1b_1c_1}\\tilde{\\WW}^{1b_2c_2}\\tilde{\\WW}^{1b_3c_3}\\epsilon_{b_1b_2b_3}\\epsilon_{c_1c_2c_3}~,\\\\\n\\WW_{1bc}\t\t\t\t&~=~ 2\\partial\\WW_{3bc} + \\frac{5}{12}\\WW_{3b_1c_1}\\WW_{3b_2c_2}\\tilde{\\WW}^{1b_3c_3}\\epsilon^{\\beta b_1b_2}\\epsilon^{\\gamma c_1c_2}\\epsilon_{\\beta b b_3}\\epsilon_{\\gamma c c_3} -\\frac13 \\WW_{3(b(c}\\WW_{3b_1)c_1)}\\tilde{\\WW}^{1b_1c_1}~,\\\\\n\\WW^{3bc}\t\t\t\t&~=~- \\partial\\tilde{\\WW}^{1bc} + \\frac13 \\tilde{\\WW}^{1b_1c_1}\\tilde{\\WW}^{1b_2c_2}{\\WW}_{3b_3c_3}\\epsilon_{\\beta b_1b_2}\\epsilon_{\\gamma c_1c_2}\\epsilon^{\\beta b b_3}\\epsilon^{\\gamma c c_3} -\\frac{2}{3}\\tilde{\\WW}^{1(b(c}\\tilde{\\WW}^{1b_1)c_1)}{\\WW}_{3b_1c_1}~.\n\\end{align}\n\\normalsize\nFinally, at dimension $h=2$, we find\n\\small\n\\begin{multline}\n(\\hat{\\JJ}^{1})_1^{\\ 3} = \\frac{14}{9}\\WW_{3bc}\\partial\\tilde{\\WW}^{1bc}\n-\\frac{8}{9}\\partial\\WW_{3bc}\\tilde{\\WW}^{1bc}\n+\\frac{2}{9}\\WW_{3(b_1(c_1}\\WW_{3b_2)c_2)}\\tilde{\\WW}^{1(b_1(c_1}\\tilde{\\WW}^{1b_2)c_2)}\\\\\n-\\frac{7}{36}\\WW_{3b_1c_1}\\WW_{3b_2c_2}\\tilde{\\WW}^{1b_3c_3}\\tilde{\\WW}^{1b_4c_4}\\epsilon^{b_1b_2b}\\epsilon^{c_1c_2c}\\epsilon_{b_3b_4b}\\epsilon_{c_3c_4c}~.\n\\end{multline}\n\\normalsize\n\nIt is interesting to see these null relations as a consequence of the nulls in the original chiral algebra. To that effect, let us re-derive the dimension one nulls in this manner. Starting with the $\\mathbf{(8,1,1)}$ null states in Table \\ref{tab:relation_reductions} and specializing the indices to $(a_1,a_2)=(3,1)$, we find the null relation\n\\begin{equation}\n0=\\tfrac13 W_{3bc}\\tilde W^{1bc} + (J^{(1)})_{3}^{\\ a} (J^{(1)})_{a}^{\\ 1} + 3 \\partial (J^{(1)})_{3}^{\\ 1} = \\tfrac13 {\\WW}_{3bc} \\tilde {\\WW}^{1bc} +\\tfrac13 \\JJ_{\\mf{u}(1)} + d(\\ldots)~,\n\\end{equation}\nthus reproducing Eqn. \\eqref{eq:nullJU1}. Alternatively, starting with the null states in the $\\mathbf{(8,8,1)}$ and specializing the indices to $(a_1,a_2)=(3,1)$, we obtain the null relation\n\\begin{multline}\n0=\\left(W_{3b_1c}\\tilde W^{1b_2c} - \\frac13 \\delta_{b_1}^{b_2}W_{3bc}\\tilde W^{1bc} \\right) -\\frac13 \\beta_1 (J^{(1)})_{3}^{\\ 1} (J^{(2)})_{b_1}^{\\ b_2} \\\\\n = \\left({\\WW}_{3b_1c} \\tilde {\\WW}^{1b_2c} - \\frac13 \\delta_{b_1}^{b_2} {\\WW}_{3bc} \\tilde {\\WW}^{1bc} \\right)+ \\left(\\JJ^{(2)}\\right)_{b_1}^{\\ b_2} + d(\\ldots)~,\n\\end{multline}\nwhich precisely matches the null relation of Eqn. \\eqref{eq:nullJSU3_2}. Similarly, one can reproduce \\eqref{eq:nullJSU3_3}. It is straightforward to check that the null relations in Eqns. \\eqref{eq:nullWlower}-\\eqref{eq:nullWupper} can be obtained from the relations in the $\\mathbf{(\\bar{6},3,3)}$ and $\\mathbf{(6,\\bar{3},\\bar{3})}$ and specializing the indices appropriately.\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nA large and interesting class of interacting quantum field theories are the \\emph{theories of class $\\SS$} \\cite{Gaiotto:2009we,Gaiotto:2009hg}. These are superconformal field theories (SCFTs) with half-maximal (\\ie, $\\NN=2$) supersymmetry in four dimensions. The most striking feature of this class of theories is that they assemble into vast duality webs that are neatly describable in the language of two-dimensional conformal geometry. This structure follows from the defining property of theories of class $\\SS$: they can be realized as the low energy limits of (partially twisted) compactifications of six-dimensional CFTs with $(2,0)$ supersymmetry on punctured Riemann surfaces.\n\nGeneric theories of class $\\SS$ are strongly interacting. (In many cases they possess generalized weak-coupling limits wherein the neighborhood of a certain limit point on their conformal manifold can be described by a collection of \\emph{isolated} strongly coupled SCFTs with weakly gauged flavor symmetries.) It is remarkable, then, that one can say much of anything about these theories in the general case. One classic and successful approach has been to restrict attention to the weakly coupled phases of these theories by, for example, studying the physics of Coulomb branch vacua at the level of the low energy effective Lagrangian and the spectrum of BPS states. Relatedly, one may utilize brane constructions of these theories to extract some features of the Coulomb branch physics \\cite{Witten:1997sc,Benini:2009gi}.\n\nAn alternative -- and perhaps more modern -- tactic is to try to constrain or solve for various aspects of these theories using consistency conditions that follow from duality. This approach was successfully carried out in \\cite{Gaiotto:2012xa} (building on the work of \\cite{Gadde:2009kb, Gadde:2010te, Gadde:2011ik, Gadde:2011uv}) to compute the superconformal index of a very general set of class $\\SS$ fixed points (see also \\cite{Lemos:2012ph,Mekareeya:2012tn} for extensions to even more general cases). Subsequently, the framework for implementing this approach to study the (maximal) Higgs branch was established in \\cite{Moore:2011ee}. The general aspiration in this sort of program is that the consistency conditions imposed by generalized $S$-duality and the (known) behavior of these theories under certain partial Higgsing and weak gauging operations may be sufficient to completely determine certain nice observables. In this sense the approach might be thought of as a sort of ``theory space bootstrap''. One expects that this approach has the greatest probability of success when applied to observables of class $\\SS$ theories that are protected against corrections when changing exactly marginal couplings, thus leading to objects that are labelled by topological data and have no dependence on continuous parameters.%\n\\footnote{Observables with a manageable dependence on the marginal couplings, such as $\\Rb^4_{\\epsilon_1\\epsilon_2}$ and $\\Sb^4$ partition functions, also provide natural settings for this type of argument.}\n\nA new class of protected observables for four-dimensional $\\NN=2$ SCFTs was introduced in \\cite{Beem:2013sza}. There it was shown that certain carefully selected local operators, restricted to be coplanar and treated at the level of cohomology with respect to a particular nilpotent supercharge, form a closed subalgebra of the operator algebra. Moreover their operator product expansions and correlation functions are meromorphic functions of the operator insertion points on the plane. This subalgebra consequently acquires the structure of a two-dimensional chiral algebra. The spectrum and structure constants of this chiral algebra are subject to a non-renormalization theorem that renders them independent of marginal couplings. The existence of this sector can formally be summarized by defining a map that associates to any $\\NN=2$ SCFT in four dimensions the chiral algebra that computes the appropriate protected correlation functions,\n\\begin{equation*}\n\\protect\\raisebox{1pt}{$\\chi$}~:~\\bigg\\{\\bigslant{\\text{$\\NN=2$ SCFTs}}{\\text{Marginal deformations}}\\bigg\\}~\\longrightarrow~\\bigg\\{\\text{Chiral algebras}\\bigg\\}~.\n\\end{equation*}\nChiral algebras with the potential to appear on the right hand side of this map are not generic -- they must possess a number of interesting properties that reflect the physics of their four-dimensional ancestors.\n\nIn this paper we initiate the investigation of chiral algebras that are associated in this manner with four-dimensional theories of class $\\SS$. For lack of imagination, we refer to the chiral algebras appearing in this fashion as \\emph{chiral algebras of class $\\SS$}. For a general strongly interacting SCFT, there is at present no straightforward method for identifying the associated chiral algebra. Success in this task would implicitly fix an infinite amount of protected CFT data (spectral data and three-point couplings) that is generally difficult to determine. However, given the rigid nature of chiral algebras, one may be optimistic that chiral algebras of class $\\SS$ can be understood in some detail by leveraging the constraints of generalized $S$-duality and the wealth of information already available about the protected spectrum of these theories. In the present work, we set up the abstract framework of this bootstrap problem in the language of generalized topological quantum field theory, and put into place as many ingredients as possible to define the problem concretely. We perform some explicit calculations in the case of theories of rank one and rank two, and formulate a number of conjectures for the higher rank case. One of our main results is a general prescription to obtain the chiral algebra of a theory with sub-maximal punctures given that of the related theory with all maximal punctures. We demonstrate that the reduction in the rank of a puncture is accomplished in the chiral algebra by quantum Drinfeld-Sokolov reduction, with the chiral algebra procedure mirroring the corresponding four-dimensional procedure involving a certain Higgsing of flavor symmetries.\n\nUltimately we believe that the bootstrap problem for chiral algebras of class $\\SS$ may prove solvable, and we hope that the existence of this remarkable structure will pique the interest of readers with a passion for vertex operator algebras. Characterizing these algebras should prove to be both mathematically and physically rewarding.\n\nThe organization of this paper is as follows. Section \\ref{sec:review} is a two-part review: first of the protected chiral algebra of $\\NN=2$ SCFTs, and then of $\\NN=2$ SCFTs of class $\\SS$. In Section \\ref{sec:TQFT}, we outline the structure of the chiral algebras of class $\\SS$, using the $A_1$ and $A_2$ cases as examples. We also take some steps to formalize the TQFT structure of the chiral algebras of class $\\SS$ so as to emphasize that the structures outlined here are susceptible to rigorous mathematical analysis. In Section \\ref{sec:reducing}, we describe the generalization of our story to the case of theories with sub-maximal punctures. In the process, we are led to consider the problem of quantum Drinfeld-Sokolov reduction for modules of affine Lie algebras. In Section \\ref{sec:cyl_and_cap}, we offer some comments on unphysical chiral algebras that are expected to exist at a formal level in order to complete the TQFT structure. A number of technical details having to do with rank two theories are included in Appendix \\ref{app:level_by_level}. Details having to do with unphysical cylinder and cap chiral algebras appear in Appendix \\ref{app:cylinders_and_caps}. Finally, in Appendix \\ref{app:spectral_sequences} we review the methods for computing the cohomology of a double complex using spectral sequences. These methods are instrumental to the analysis of Section \\ref{sec:reducing}.\n\\section{Background}\n\\label{sec:review}\n\nWe begin with a review of the two main topics being synthesized in this paper: the protected chiral algebras of $\\NN=2$ SCFTs and superconformal theories of class $\\SS$. Readers who have studied our first paper on protected chiral algebras \\cite{Beem:2013sza} should be fine skipping Section \\ref{subsec:chiral_review}, while those familiar with the class $\\SS$ literature (for example, \\cite{Gaiotto:2009we,Gadde:2009kb,Gaiotto:2012xa,Chacaltana:2010ks}) may safely skip Section \\ref{subsec:class_S_review}.\n\n\\medskip\n\\input{.\/sections\/Section_2\/S2_1}\n\\medskip\n\\input{.\/sections\/Section_2\/S2_2}\n\\bigskip\n\\subsection{Review of protected chiral algebras}\n\\label{subsec:chiral_review}\n\nThe observables we aim to study for class $\\SS$ fixed points are those described by the protected chiral algebras introduced in \\cite{Beem:2013sza} (see also \\cite{Beem:2014kka} for the extension to six dimensions). The purpose of this section is to provide a short overview of how those chiral algebras come about and the properties that were deduced for them in the original papers. We simply state the facts in this section; the interested reader is encouraged to consult the original work for explanations.\n\nThe starting point is the $\\NN=2$ superconformal algebra $\\mathfrak{su}(2,2|2)$. The fermionic generators of the algebra are Poincar\\'e supercharges $\\{\\mathcal{Q}^{\\II}_{\\alpha},\\tilde\\mathcal{Q}_{\\dot\\alpha \\JJ}\\}$ and special conformal supercharges $\\{\\SS^{\\alpha}_{\\II},\\tilde\\SS^{\\dot\\alpha\\JJ}\\}$. From these, one can form two interesting nilpotent supercharges that are mixtures of Poincar\\'e and special conformal supercharges,\n\\begin{equation}\n\\label{eq:square_supercharges}\n\\qq_{\\,1}\\ceq \\mathcal{Q}^1_{-}+\\tilde\\SS^{\\dot{-}2}~,\\qquad\\qq_{\\,2}\\ceq \\tilde\\mathcal{Q}_{\\dot{-}2}+\\SS_1^{-}~.\n\\end{equation}\nThese supercharges have the following interesting property. Let us define the subalgebra of the four-dimensional conformal symmetry algebra that acts on a plane $\\Rb^2\\subset\\Rb^4$ as $\\mf{sl}(2)\\times\\overline{\\mf{sl}(2)}$. Let us further denote the complexification of the $\\mf{su}(2)_R$ $R$-symmetry as $\\mf{sl}(2)_R$. These subalgebras have the following nice relationship to the supercharges $\\qq_{\\,i}$,\n\\begin{equation}\n\\label{eq:q_exact_commutators}\n[\\qq_{\\,i},\\mf{sl}(2)]=0~,\\qquad \\{\\qq_{\\,i},\\cdot\\}=\\mathrm{diag}\\left[\\overline{\\mf{sl}(2)}\\times\\mf{sl}(2)_R\\right]~.\n\\end{equation}\nIt follows from these relations that operators that are $\\qq\\,$-closed must behave as meromorphic operators in the plane. They have meromorphic operator product expansions (modulo $\\qq\\,$-exact terms) and their correlation functions are meromorphic functions of the positions. Restricting from the full $\\NN=2$ SCFT to $\\qq\\,$-cohomology therefore defines a two-dimensional chiral algebra. For a pedagogical discussion of chiral algebras, see \\cite{Bouwknegt:1992wg}.\n\nThe conditions for a local operator to define a nontrivial $\\qq\\,$-cohomology element were worked out in \\cite{Beem:2013sza}. It turns out that such operators are restricted to lie in the \\emph{chiral algebra plane}: $\\{x_3=x_4=0\\}$. When inserted at the origin, an operator belongs to a well-defined cohomology class if\nand only if it obeys the conditions \n\\begin{equation}\n\\label{schurconditions}\n\\hat h \\ceq \\frac{E-(j_1+j_2)}{2}-R = 0~,\\qquad \\ZZ \\ceq j_1-j_2+r = 0~.\n\\end{equation}\nUnitarity of the superconformal representation requires $\\hat h \\geqslant \\frac{ | \\ZZ | }{2}$, so the first condition actually implies the second. We refer to operators obeying $\\hat h = 0$ as \\emph{Schur operators}. All Schur operators are necessarily $\\mf{su}(2)_R$ highest weight states. Indeed, if the $\\mf{su}(2)_R$ raising generator did \\emph{not} annihilate a Schur operator, it would generate an operator with $\\hat h < 0$, which would violate unitarity.\n\nAs $\\overline {\\mf{sl}(2)}$ does not commute with $\\qq$, ordinary translations of Schur operators in the chiral algebra plane fail to be $\\qq\\,$-closed away from the origin. Rather, we translate operators using the twisted translation generator $\\widehat{L}_{-1}\\ceq \\overline{L}_{-1}+\\RR_{-}$, where $\\RR_{-}$ is the lowering operator of $\\mf{su}(2)_R$. As shown in Eqn.~\\eqref{eq:q_exact_commutators}, this is a $\\qq\\,$-exact operation. We find that local operators defining nontrivial $\\qq\\,$-cohomology classes can be written in the form\n\\begin{equation}\n\\label{eq:twisted_translated}\n\\OO(z,\\zb)\\ceq u_{\\II_1}(\\zb) \\cdots u_{\\II_k}(\\zb)\\OO^{\\{\\II_1\\cdots\\II_k\\}}(z,\\zb)~,\\qquad \\text{where}\\qquad u_{\\II}(\\zb)\\ceq\\binom{\\,1\\,}{\\zb}~.\n\\end{equation}\nHere $\\OO^{1\\cdots1}(0)$ is a Schur operator, and we are suppressing Lorentz indices. It is these \\emph{twisted-translated} Schur operators, taken at the level of cohomology, that behave as meromorphic operators in two dimensions,\n\\begin{equation}\n\\label{eq:twisted_translated_to_meromorphic}\n\\OO(z) \\ceq [\\OO(z,\\zb)]_{\\qq_{\\,i}}~.\n\\end{equation}\nWe now turn to a recap of the various types of four-dimensional operators that may satisfy the Schur condition, and thus participate in the protected chiral algebra.\n\n\\renewcommand{\\arraystretch}{1.55}\n\\begin{table}\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline \\hline\nMultiplet \t\t\t & $\\OO_{\\rm Schur}$ \t\t \t\t\t\t\t\t & $h\\ceq\\frac{E+j_1+j_2}{2}$ & $\\phantom{-}r$ \t\t \t& Lagrangian ``letters''\\\\ \n\\hline \n$\\hat\\BB_R$ \t\t & $\\Psi^{11\\dots 1}$ \t\t\t\t \t\t\t\t\t & $R$ \t\t \t\t\t\t & $\\phantom{-}0$ \t\t\t& $Q$, $\\tilde Q$ \\\\ \n\\hline\n$\\bar\\DD_{R(j_1,0)}$ & ${\\mathcal{Q}}^1_+\\Psi^{11\\dots 1}_{+\\dots+}$ \t\t\t\t\t & $R+j_1+1$ \t\t\t\t & $-j_1-\\frac12~$ \t& $Q$, $\\tilde Q$, $\\lambda^1_+$ \\\\\n\\hline\n$\\DD_{R(0,j_2)}$ \t & $\\wt{\\mathcal{Q}}^1_{\\dot +}\\Psi^{11\\dots 1}_{\\dot+\\dots\\dot+}$ & $R+j_2+1$\t\t\t\t & $\\phantom{-}j_2+\\frac12$ & $Q$, $\\tilde Q$, $\\tilde\\lambda^1_{\\dot+}$ \\\\\n\\hline\n$\\hat\\CC_{R(j_1,j_2)}$ & $\\mathcal{Q}^1_{+} \\wt{\\mathcal{Q}}^1_{\\dot+}\\Psi^{1\\dots1}_{+\\dots+\\,\\dot+\\dots\\dot+}$ & $R+j_1+j_2 +2$ & $\\phantom{-}j_2-j_1$ & $D_{+\\dot+}^n Q$, $D_{+\\dot+}^n \\tilde Q$, $D_{+\\dot+}^n \\lambda^1_+$, $D_{+\\dot+}^n \\tilde\\lambda^1_{\\dot+}$\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{schurTable} This table summarizes the manner in which Schur operators fit into short multiplets of the $\\NN=2$ superconformal algebra. We use the naming conventions for supermultiplets of Dolan and Osborn \\cite{Dolan:2002zh}. For each supermultiplet, we denote by $\\Psi$ the superconformal primary. There is then a single conformal primary Schur operator ${\\OO}_{\\rm Schur}$, which in general is obtained by the action of some Poincar\\'e supercharges on $\\Psi$. The holomorphic dimension ($h$) and $U(1)_r$ charge ($r$) of ${\\OO}_{\\rm Schur}$ are determined in terms of the quantum numbers $(R,j_1,j_2)$ that label the shortened multiplet. We also indicate the schematic form that ${\\OO}_{\\rm Schur}$ can take in a Lagrangian theory by enumerating the elementary ``letters'' from which the operator may be built. We denote by $Q$ and $\\tilde Q$ the complex scalar fields of a hypermultiplet, by $\\lambda_{\\alpha}^\\II$ and $\\tilde \\lambda_{\\dot \\alpha}^\\II$ the left- and right-handed fermions of a vector multiplet, and by $D_{\\alpha \\dot \\alpha}$ the gauge-covariant derivatives. Note that while in a Lagrangian theory Schur operators are built from these letters, the converse is false -- not \\emph{all} gauge-invariant words of this kind are Schur operators. Only the special combinations with vanishing anomalous dimensions retain this property at finite coupling.} \n\\end{table}\n\n\\subsubsection{Taxonomy of Schur operators}\n\\label{subsubsec:schur_taxonomy}\n\nA Schur operator is annihilated by two Poincar\\'e supercharges of opposite chiralities ($\\mathcal{Q}_-^1$ and $\\widetilde \\mathcal{Q}_{2 \\dot -}$ in our conventions). A summary of the different classes of Schur operators, organized according to how they fit in shortened multiplets of the superconformal algebra, is given in Table \\ref{schurTable} (reproduced from \\cite{Beem:2013sza}). Let us briefly discuss each row in turn.\n\nThe first row describes half-BPS operators that are a part of the Higgs branch chiral ring. These have $E = 2R$ and $j_1 = j_2 = 0$. In a Lagrangian theory, operators of this type schematically take the form $QQ\\cdots\\tilde Q\\tilde Q$. A special case is when $R = 1$, in which case a conserved current is amongst the super-descendants of the primary. The half-BPS primary is then the ``moment map'' operator $\\mu_A$ which has dimension two and transforms in the adjoint representation of the flavor symmetry. The $\\mf{su}(2)_R$ highest weight state of the moment map is a Schur operator.\n\nThe operators in the second row are more general $\\NN=1$ chiral operators, obeying $E = 2 R+|r|$ and $r= -j_1 -\\frac12$. Together with the Higgs branch chiral ring operators (which can be regarded as the special case with $r=0$), they make up the so-called \\emph{Hall-Littlewood chiral ring}. These are precisely the operators that are counted by the Hall-Littlewood limit of the superconformal index \\cite{Gadde:2011uv}. In a Lagrangian theory, these operators are obtained by constructing gauge-invariant words out of $Q$, $\\tilde Q$, and the gaugino field $\\lambda^1_+$ (the bottom component of the field strength chiral superfield $W_\\alpha$ with $\\alpha = +$). In complete analogy, the third line describes $\\NN=1$ \\emph{anti}-chiral operators obeying $E= 2 R+|r|$, $r= j_2 +\\frac12$, which belong to the Hall-Littlewood anti-chiral ring. The second and third lines are CPT conjugate to each other. It is believed that $\\DD$ and $\\overline\\DD$ type operators are absent in any theory arising from a (generalized) quiver description with no loops (\\ie, an \\emph{acyclic quiver}). These are theories for which the Hall-Littlewood superconformal index matches the ``Hilbert series'' for the Higgs branch \\cite{Gadde:2011uv,Benvenuti:2010pq}. Equivalently, these are the theories for which the maximal Higgs branch is an honest Higgs branch, with no low-energy abelian gauge field degrees of freedom surviving.\n\nThe fourth line describes the most general type of Schur operators, which belong to supermultiplets that obey less familiar semi-shortening conditions. An important operator in this class is the conserved current for $\\mf{su}(2)_R$, which belongs to the $\\hat \\CC_{0(0, 0)}$ supermultiplet which also contains the stress-energy tensor and is therefore universally present in any $\\NN=2$ SCFT. This current has one component with $E= 3$, $R=1$, $j_1 = j_2 = \\frac12$ which is a Schur operator.\n\nFinally, let us point out the conspicuous absence of half-BPS operators that belong to the \\emph{Coulomb} branch chiral ring (these take the form $\\mbox{Tr}\\,\\phi^k$ in a Lagrangian theory, where $\\phi$ is the complex scalar of the $\\NN=2$ vector multiplet). These operators are in many ways more familiar than those appearing above due to their connection with Coulomb branch physics. The protected chiral algebra is thus complementary, rather than overlapping, with a Coulomb branch based analysis of class $\\SS$ physics.\n\n\\subsubsection{The $4d\/2d$ dictionary}\n\\label{subsubsec:4d_2d_dict}\n\nThere is a rich dictionary relating properties of a four-dimensional SCFT with properties of its associated chiral algebra. Let us briefly review some of the universal entries in this dictionary that were worked out in \\cite{Beem:2013sza}. Interested readers should consult that reference for more detailed explanations.\n\n\\subsubsection*{Virasoro symmetry}\n\nThe stress tensor in a four-dimensional $\\NN=2$ SCFT lives in the $\\hat \\CC_{0(0, 0)}$ supermultiplet, which contains as a Schur operator a component of the $\\mf{su}(2)_R$ conserved current $\\JJ^{(\\II\\JJ)}_{\\alpha\\dot\\alpha}$. The corresponding twisted-translated operator gives rise in cohomology to a two-dimensional meromorphic operator of dimension two, which acts as a two-dimensional stress tensor, $T(z)\\ceq [\\JJ_{+\\dot+}(z,\\bar z)]_{\\qq}$. As a result, the global $\\mf{sl}(2)$ symmetry that is inherited from four dimensions is always enhanced to a local Virasoro symmetry acting on the chiral algebra. From the current-current OPE, which is governed by superconformal Ward identities, one finds a universal expression for the Virasoro central charge,\n\\begin{equation}\n\\label{eq:cc_relation}\nc_{2d} = -12\\,c_{4d}~,\n\\end{equation}\nwhere $c_{4d}$ is the conformal anomaly coefficient of the four-dimensional theory associated to the square of the Weyl tensor. Note that the chiral algebra is necessarily non-unitary due to the negative sign in Eqn.~\\eqref{eq:cc_relation}.\n\n\\subsubsection*{Affine symmetry}\n\nSimilarly, continuous global symmetries of the four-dimensional SCFT (when present) are enhanced to local affine symmetries at the level of the associated chiral algebra. This comes about because the conserved flavor symmetry current sits in the $\\hat\\BB_1$ supermultiplet, whose bottom component is the moment-map operator discussed above. The $\\mf{su}(2)_R$ highest weight component of the moment map operator then gives rise to an affine current, $J_A(z)\\ceq [\\mu_A(z,\\bar z)]_{\\qq}$. The level of the affine current algebra is related to the four-dimensional flavor central charge by another universal relation,\n\\begin{equation}\n\\label{eq:kk_relation}\nk_{2d} = -\\frac12 k_{4d}~.\n\\end{equation}\n\n\\subsubsection*{Hall-Littlewood ring generators as chiral algebra generators}\n\nIdentifying chiral algebra generators is of crucial importance if one is to find an intrinsic characterization of any particular chiral algebra without reference to its four-dimensional parent. A very useful fact is that generators of the Hall-Littlewood chiral ring (and in particular those of the Higgs branch chiral ring) necessarily give rise to generators of the protected chiral algebra after passing to $\\qq\\,$-cohomology. This follows from $\\mf{su}(2)_R$ and $\\mf{u}(1)_r$ selection rules, which forbid such an operator from appearing in any non-singular OPEs. A special case is the aforementioned affine currents, which arise from Higgs branch moment map operators with $E=2R=2$. With the exception of theories with free hypermultiplets, these are always generators.\n \n\\subsubsection*{Exactly marginal gauging}\n\nGiven an SCFT $\\TT$ with a flavor symmetry $G$ that has flavor central charge $k_{4d}=4h^{\\vee}$, one may form a new family of SCFTs $\\TT_G$ by introducing an $\\NN=2$ vector multiplet in the adjoint representation of $G$ and gauging the symmetry. This specific value of the flavor central charge ensures that the gauge coupling beta function vanishes, so the procedure preserves conformal invariance.\n\nThere exists a corresponding procedure at the level of chiral algebras that produces the chiral algebra $\\protect\\raisebox{1pt}{$\\chi$}[\\TT_G]$ given that of the original theory $\\protect\\raisebox{1pt}{$\\chi$}[\\TT]$. In parallel with the introduction of a $G$-valued vector multiplet, one introduces a dimension $(1,0)$ ghost system $(b_A,c^A)$ with $A=1,\\ldots,\\dim G$. In the tensor product of this ghost system and the chiral algebra $\\protect\\raisebox{1pt}{$\\chi$}[\\TT]$, one may form a canonical nilpotent BRST operator given by\n\\begin{equation}\n\\label{eq:BRST_def}\nQ_{\\rm BRST} \\ceq \\oint\\frac{dz}{2\\pi i}\\,j_{\\rm BRST}(z)~,\\qquad j_{\\rm BRST}(z) \\ceq \\left(c^A\\left[J_A-\\frac12 f_{AB}^{\\phantom{AB}C}\\,c^B\\,b_C\\right]\\right)(z)~,\n\\end{equation}\nwhere the affine currents $J_A(z)$ are those associated with the $G$ symmetry of $\\protect\\raisebox{1pt}{$\\chi$}[\\TT]$, and $f_{AB}^{\\phantom{AB}C}$ are the structure constants for $G$. Nilpotency of this BRST operator depends on the precise value of the affine level $k_{2d}=-2h^{\\vee}$, and so the self-consistency of this procedure is intimately connected with the preservation of conformal invariance in four dimensions. The gauged chiral algebra is then obtained as the cohomology of the BRST operator relative to the $b\\,$-ghost zero modes,\n\\begin{equation}\n\\label{eq:gauged_cohomology}\n\\protect\\raisebox{1pt}{$\\chi$}[\\TT_G] = H^{\\star}_{\\rm BRST} \\left[\\psi\\in\\protect\\raisebox{1pt}{$\\chi$}[\\TT]\\otimes\\protect\\raisebox{1pt}{$\\chi$}_{(b,c)}~|~b^A_{0}\\psi=0\\right]\n\\end{equation}\n\n\\subsubsection*{Superconformal index}\n\nThe superconformal index of a superconformal field theory is the Witten index of the radially quantized theory, refined by a set of fugacities that keep track of the maximal set of charges commuting with each other and with a chosen supercharge. For our purposes, we consider the specialization of the index of an $\\NN=2$ SCFT known as the Schur index \\cite{Gadde:2011ik,Gadde:2011uv}. The trace formula for the Schur index reads\n\\begin{equation}\n\\label{eq:SchurSCI}\n\\II^{\\rm (Schur)}(q; {\\bf x}) = \\mbox{Tr}_{\\HH[\\Sb^3]}(-1)^F q^{E-R}\\prod_i {x_i}^{f_i}~,\n\\end{equation}\nwhere $F$ denotes the fermion number and $\\{f_i\\}$ the Cartan generators of the flavor group. The Schur index counts (with signs) precisely the operators obeying the condition (\\ref{schurconditions}). Moreover, for Schur operators $E-R$ coincides with the left-moving conformal weight $h$ (the eigenvalue of $L_0$),\n\\begin{equation}\n\\label{eq:schu_op_dimension}\nE-R ~=~ \\frac{E+j_1+j_2}{2} ~\\eqc~ h~.\n\\end{equation}\nIt follows that the graded character of the chiral algebra is identical to the Schur index,\n\\begin{equation}\n\\label{charschur}\n\\II_{\\chi}(q; {\\bf x}) ~\\ceq~ \\mbox{Tr}_{\\HH_{\\chi}}\\,(-1)^F q^{L_0} ~=~ \\II^{\\rm Schur}(q; {\\bf x})~,\n\\end{equation}\nwhere $\\HH_{\\chi}$ denotes the state space of the chiral algebra. Note that this object is not interpreted as an index when taken as a partition function of the chiral algebra, because (with the exception of chiral algebras associated to $\\NN=4$ theories in four dimensions) the protected chiral algebra itself is not supersymmetric.\n\\subsection{Review of theories of class \\texorpdfstring{$\\SS$}{S}}\n\\label{subsec:class_S_review}\n\nFour-dimensional superconformal field theories of class $\\SS$ may be realized as the low-energy limit of twisted compactifications of an $\\NN=(2,0)$ superconformal field theory in six dimensions on a Riemann surface, possibly in the presence of half-BPS codimension-two defect operators. The resulting four-dimensional theory is specified by the following data:\\footnote{We restrict our attention in this note to \\emph{regular} theories. A larger class of theories can be obtained by additionally allowing for \\emph{irregular} punctures \\cite{Witten:2007td}. Still more possibilities appear when the UV curve is decorated with outer automorphisms twist lines \\cite{Tachikawa:2010vg,Chacaltana:2012ch}.} \n\\begin{itemize}\n\\item A simply-laced Lie algebra $\\mf{g} = \\{A_n, D_n, E_6, E_7, E_8\\}$. This specifies the choice of six-dimensional $(2,0)$ theory.\n\\item A (punctured) Riemann surface ${\\CC}_{g,s}$ known as the \\emph{UV curve}, where $g$ indicates the genus and $s$ the number of punctures. In the low energy limit, only the complex structure of ${\\CC}_{g,s}$ plays a role. The complex structure moduli of the curve are identified with exactly marginal couplings in the SCFT.\n\\item A choice of embedding $\\Lambda_i: \\mf{su}(2) \\to \\mf{g}$ (up to conjugacy) for each puncture $i=1,\\ldots,s$. These choices reflect the choice of codimension-two defect that is present at each puncture in the six-dimensional construction. The centralizer $\\mf{h}_{\\Lambda_i} \\subset \\mf{g}$ of the embedding is the global symmetry associated to the defect. The theory enjoys a global flavor symmetry algebra given by at least $\\oplus_{i=1}^s\\mf{h}_{\\Lambda_i}$.\\footnote{In some exceptional cases the global symmetry of the theory is enhanced due to the existence of additional symmetry generators that are not naturally associated to an individual puncture.}\n\\end{itemize}\nWhen necessary, we will label the corresponding four-dimensional SCFT as $\\TT[\\mf{g}; \\CC_{g,s}; \\{\\Lambda_i\\}]$. Because we are ultimately only interested in theories modulo their exactly marginal couplings, we will not keep track of a point in the complex structure moduli space of the UV curve.\n\nFor the sake of simplicity, we will restrict our attention to theories where $\\mf{g}$ is in the $A$ series. The generalization to $D$ and $E$ series theories (at least in the abstract discussion) should be possible to carry out without a great deal of additional difficulty. In the $A_{n-1}$ case -- \\ie, $\\mf{g}=\\mf{su}(n)$ -- the data at punctures can be reformulated as a partition of $n$: $[n_1^{\\ell_1}\\,n_2^{\\ell_2}\\,\\dots\\,n_k^{\\ell_k}]$ with $\\sum_i \\ell_i n_i = n$ and $n_i > n_{i+1}$. Such a partition indicates how the fundamental representation $\\mf{f}$ of $\\mf{su}(n)$ decomposes into irreps of $\\Lambda(\\mf{su}(2))$,\n\\begin{equation}\n\\label{eq:fundamental_decomposition_partition}\n\\mf{f} \\rightarrow \\bigoplus_{i=1}^{k} \\ell_i V_{\\frac12(n_i-1)}~,\n\\end{equation}\nwhere $V_j$ denotes the spin $j$ representation of $\\mf{su}(2)$. An equivalent description comes from specifying a nilpotent element $e$ in $\\mf{su}(n)$, \\ie, an element for which $\\left({\\rm ad}_e\\right)^r=0$ for some positive integer $r$. The Jordan normal form of such a nilpotent element is given by\n\\begin{equation}\n\\label{eq:jordan_normal_form}\ne = \\bigoplus_{i=1}^{k} \\overbrace{J_{n_i}\\oplus\\cdots\\oplus J_{n_i}}^{\\ell_i~{\\rm times}}~,\n\\end{equation}\nwhere $J_m$ is the elementary Jordan block of size $m$, \\ie, a sparse $m\\times m$ matrix with only ones along the superdiagonal. Thus every nilpotent element specifies a partition of $n$ and vice versa. The $\\mf{su}(2)$ embedding comes from defining $\\mf{su}(2)$ generators $t_0, t_\\pm$ and demanding that $\\Lambda(t_-)=e$.\n\nThe trivial embedding is identified with the partition $[1^n]$ and leads to a defect with maximal flavor symmetry $\\mf{h} = \\mf{su}(n)$. A puncture labelled by this embedding is called \\emph{full} or \\emph{maximal}. The opposite extreme is the principal embedding, which has partition $[n^1]$. This embedding leads to $\\mf{h} = \\varnothing$, and the puncture is effectively absent. Another important case is the subregular embedding, with partition $[n-1,1]$, which leads to $\\mf{h} = \\mf{u}(1)$ (as long as $n>2$). Punctures labelled by the subregular embedding are called \\emph{minimal} or \\emph{simple}.\n\nThe basic entities of class $\\SS$ are the theories associated to thrice-punctured spheres, or \\emph{trinions}. The designations of these theories are conventionally shortened as \n\\begin{equation}\n\\label{eq:trinion_theory_label}\nT_n^{\\Lambda_1\\Lambda_2\\Lambda_3} \\ceq \\TT[\\mf{su}(n); \\CC_{0, 3}; \\{\\Lambda_1\\,\\Lambda_2\\,\\Lambda_3\\}]~.\n\\end{equation} \nFor the special case of all maximal punctures, the convention is to further define $T_n\\ceq T_n^{[1^n][1^n][1^n]}$. All of the trinion theories are isolated SCFTs -- they have no marginal couplings. For most of these theories, no Lagrangian description is known. An important class of exceptions are the theories with two maximal punctures and one minimal puncture: $T_n^{[1^n][1^n][n-1,1]}$. These are theories of $n^2$ free hypermultiplets, which in this context are naturally thought of as transforming in the bifundamental representation of $\\mf{su}(n)\\times\\mf{su}(n)$. In the case $n=2$, the minimal and maximal punctures are the same and the theory of four free hypermultiplets (equivalently, eight free half-hypermultiplets) is the $T_2$ theory. In this case the global symmetry associated to the punctures is $\\mf{su}(2)\\times\\mf{su}(2)\\times\\mf{su}(2)$ which is a subgroup of the full global symmetry $\\mf{usp}(8)$.\n\nAt the level of two-dimensional topology, an arbitrary surface $\\CC_{g,s}$ can be assembled by taking $2g-2+s$ copies of the three-punctured sphere, or ``pairs of pants'', and gluing legs together pairwise $3g-3+s$ times. Each gluing introduces a complex \\emph{plumbing parameter} and for a given construction of this type the plumbing parameters form a set of coordinates for a patch of the Teichmuller space of Riemann surfaces of genus $g$ with $s$ punctures. A parallel procedure is used to construct the class $\\SS$ theory associated to an arbitrary UV curve using the basic trinion theories. Starting with $2g-2+s$ copies of the trinion theory $T_n$, one glues along maximal punctures by gauging the diagonal subgroup of the $\\mf{su}(n)\\times\\mf{su}(n)$ flavor symmetry associated to the punctures. This introduces an $\\mf{su}(n)$ gauge group in the four-dimensional SCFT, and the marginal gauge coupling is related to the plumbing parameter. If one wants, the remaining maximal punctures can then be reduced to sub-maximal punctures using the Higgsing procedure described below.\\footnote{In terms of the low energy SCFT, the operations of Higgsing at external punctures and gauging of internal ones commute, so one may equally well think of gluing together trinions some of whose punctures are not maximal. Our presentation here is not meant to convey the full depth of what is possible in class $\\SS$.} To a given pants decomposition of a UV curve, one associates a ``weakly coupled'' frame of the corresponding SCFT in which the flavor symmetries of a collection of trinion theories are being weakly gauged. The equivalence of different pants decompositions amounts to $S$-duality. It is only in very special cases that a weakly coupled duality frame of this type will actually be described by a Lagrangian field theory.\n\nBy now, quite a few general facts are known about theories of class $\\SS$. Here we simply review some relevant ones while providing pointers to the original literature. The list is not meant to be comprehensive in any sense.\n\n\\medskip\n\\subsubsection*{Central charges}\n\nThe $a$ and $c$ conformal anomalies have been determined for all of the regular $A$-type theories in \\cite{Chacaltana:2010ks,Chacaltana:2012zy}. The answer takes the following form,\n\\begin{equation}\n\\label{eq:4dcentralcharges}\nc_{4d}=\\frac{2 n_v + n_h}{12}~, \\qquad a=\\frac{5 n_v + n_h}{24}~,\n\\end{equation}\nwhere \n\\begin{align}\\label{eq:nv_nh_defs}\n\\begin{split}\nn_v &~=~ \\sum_{i=1}^s n_v(\\Lambda_i) + (g-1)\\left( \\tfrac{4}{3}h^\\vee \\dim \\mf{g} + \\text{rank } \\mf{g} \\right)~, \\\\\nn_h &~=~ \\sum_{i=1}^s n_h(\\Lambda_i) + (g-1)\\left( \\tfrac{4}{3}h^\\vee \\dim \\mf{g} \\right)~,\n\\end{split}\\end{align}\nand\n\\begin{align}\\label{eq:nv_nh_defs_2}\n\\begin{split}\nn_v(\\Lambda) &~=~ 8\\left(\\rho\\cdot \\rho - \\rho \\cdot \\Lambda(t_0)\\right) + \\tfrac{1}{2}(\\text{rank } \\mf{g} - \\dim \\mf{g}_{0})~,\\\\\nn_h(\\Lambda) &~=~ 8\\left(\\rho\\cdot \\rho - \\rho \\cdot \\Lambda(t_0)\\right) + \\tfrac{1}{2} \\dim \\mf{g}_{\\frac{1}{2}}~.\n\\end{split}\\end{align}\nIn these equations, $\\rho$ is the Weyl vector of $\\mf{su}(n)$ and $h^\\vee$ is the dual coxeter number, which is equal to $n$ for $\\mf{g}=\\mf{su}(n)$. The Freudenthal-de Vries strange formula states that $|\\rho|^2 = \\frac{h^\\vee}{12}\\dim\\mf{g}$, which is useful in evaluating these expressions. Additionally, the embedded Cartan generator $\\Lambda(t_0)$ has been used to define a grading on the Lie-algebra, \n\\begin{equation}\n\\label{eq:lie_algebra_grading}\n\\mf{g} = \\bigoplus_{m\\in\\frac{1}{2}\\Zb} \\mf{g}_m~,\\qquad \\mf{g}_m\\ceq\\left\\{ t \\in \\mf{g}\\ |\\ {\\rm ad}_{\\Lambda(t_0)}t = mt \\right\\}~.\n\\end{equation}\nThis grading will make another appearance in Sec. \\ref{sec:reducing}.\n\nThe $\\mf{su}(n)$ flavor symmetry associated to a full puncture comes with flavor central charge $k_{\\mf{su}(n)}=2n$. This is a specialization of the general formula $k_{\\rm ADE}=2h^{\\vee}$. For a non-maximal puncture, the flavor central charge for a given simple factor $\\mf{h}_{\\rm simp}\\subseteq\\mf{h}$ is given by \\cite{Chacaltana:2012zy},\n\\begin{equation}\n\\label{eq:flavor_central_charge_reduced}\nk_{\\mf{h}_{\\rm simp}}\\delta_{AB}= 2 \\sum_j \\mbox{Tr}_{\\RR_j^{(\\text{adj})}}T_A T_B~,\n\\end{equation}\nwhere $T_A,T_B$ are generators of $\\mf{h}_{\\rm simp}$ satisfying the normalization $\\mbox{Tr}_{\\mf{h}_{\\rm simp}} T_A T_B = h^\\vee_{\\mf{h}_{\\rm simp}}\\delta_{AB}$ and we have introduced the decomposition of the adjoint representation of $\\mf{su}(n)$ into representations of $\\mf{h}_\\Lambda \\otimes\\Lambda(\\mf{su}(2))$,\n\\begin{equation}\n\\label{eq:adjdecomposition}\n\\mathrm{adj}_{\\mf{g}} = \\bigoplus_j \\RR_j^{(\\mathrm{adj})} \\otimes V_j~.\n\\end{equation}\nIn cases where there are global symmetries that extend the symmetries associated to punctures, the central charge can be deduced in terms of the embedding index.\n\n\\medskip\n\\subsubsection*{Higgs branch chiral ring and their relations}\n\nOperators in an $\\NN=2$ SCFT whose conformal dimension is equal to twice their $\\mf{su}(2)_R$ spin ($E=2R$) form a ring called the \\emph{Higgs branch chiral ring}. This ring is generally believed to be the ring of holomorphic functions (in a particular complex structure) on the Higgs branch of the moduli space of vacua of the theory. It is expected to be finitely generated, with the generators generally obeying nontrivial algebraic relations. For theories of class $\\SS$ the most general such relations have not been worked out explicitly to the best of our knowledge. However, certain cases of the relations can be understood.\n\nFor any puncture there is an associated global symmetry $\\mf{h}$, and the conserved currents for that global symmetry will lie in superconformal representations that include \\emph{moment map} operators $\\mu^A$, $A=1,\\ldots,\\dim\\mf{h}$ that belong to the Higgs branch chiral ring. Of primary interest to us are the relations that involve solely these moment map operators. Let us specialize to the case where all punctures are maximal, so $\\mf{h}_i=\\mf{g}$ for all $i=1,\\ldots,s$. There are then chiral ring relations given by\n\\begin{equation}\n\\label{eq:moment_map_relation}\n\\mbox{Tr}\\mu_1^k=\\mbox{Tr}\\mu_2^k=\\cdots=\\mbox{Tr}\\mu_s^k~,\\qquad k=1,2\\ldots~.\n\\end{equation}\nThere are additional Higgs branch chiral ring generators for a general class $\\SS$ theory of the form\n\\begin{equation}\n\\label{eq:higgs_branch_extra_generators}\nQ_{(k)}^{\\II^{(k)}_1\\cdots\\II^{(k)}_s}~,\\qquad k=1,\\ldots,n-1~,\n\\end{equation}\nof dimension $E_k=2R_k=\\frac12 k(n-k)(2g-2+s)$. The multi-indices $\\II^{(k)}$ index the $k$-fold antisymmetric tensor representation of $\\mf{su}(n)$. There are generally additional chiral ring relations involving these $Q_{(k)}$ operators, some of which mix them with the moment maps \\cite{Gadde:2013fma}. The complete form of these extra relations has not been worked out -- a knowledge of such relations would characterize the Higgs branch of that theory as a complex algebraic variety, and such a characterization is presently lacking for all but a small number of special cases. We will not make explicit use of such additional relations in what follows.\n\n\\medskip\n\\subsubsection*{Higgsing and reduction of punctures: generalities}\n\nTheories with non-maximal punctures can be obtained by starting with a theory with maximal punctures and going to a particular locus on the Higgs branch \\cite{Benini:2009gi,Tachikawa:2013kta,Chacaltana:2012zy,Maruyoshi:2013hja}. The flavor symmetry associated to a puncture is reflected in the existence of the above-mentioned half-BPS moment map operators, $\\mu_A$, that transform in the adjoint representation of the flavor symmetry with corresponding index $A=1,\\ldots,n^2-1$. In reducing the flavor symmetry via Higgsing, one aims to give an expectation value to one of the $\\mu_i$'s, say $\\mu_1$, while keeping $\\vev{\\mu_{i \\neq 1} }=0$. Consistency with Eqn. \\eqref{eq:moment_map_relation} then requires that $\\vev{\\mbox{Tr}\\mu_1^k}=0$ for any $k$, or put differently, $\\vev{\\mu_1}$ is a nilpotent $\\mf{su}(n)$ matrix. Since any nilpotent element can be realized as the image of $t_-\\in\\mf{su}(2)$ with respect to some embedding $\\Lambda:\\mf{su}(2)\\hookrightarrow\\mf{su}(n)$, the relevant loci on the Higgs branch are characterized by such an embedding, where we have\n\\begin{equation}\n\\label{eq:invariant_vev}\n\\vev{\\mu_1} = v\\, \\Lambda(t_-)~.\n\\end{equation}\nThe expectation value breaks the $\\mf{su}(n)$ flavor symmetry associated with the puncture to $\\mf{h}_\\Lambda$, the centralizer of the embedded $\\mf{su}(2)$, as well as the $\\mf{su}(2)_R$ symmetry (and also conformal symmetry). It will be important in the following that a linear combination of the flavor and $\\mf{su}(2)_R$ Cartan generators remains unbroken,\\footnote{We suspect not only this Cartan generator, but the full diagonal subalgebra of $\\mf{su}(2)_R$ and the embedded $\\mf{su}(2)$ is preserved on the sublocus of the Higgs branch in question. It should be possible to prove such a thing using the hyperkahler structure on nilpotent cones described in \\cite{Swann}. We thank D. Gaiotto, A. Neitzke, and Y. Tachikawa for helpful conversations on this point.} namely\n\\begin{equation}\n\\label{eq:new_R_cartan}\n\\tilde R \\ceq R + J_0~,\\qquad J_0 \\ceq \\Lambda (t_0)~.\n\\end{equation}\nIn such a vacuum, the low energy limit of the theory is described by the interacting class $\\SS$ SCFT with the same UV curve as the original theory, but with the first puncture replaced by a puncture of type $\\Lambda$. Additionally there will be decoupled free fields arising from the Nambu-Goldstone fields associated to the symmetry breaking \\cite{Maruyoshi:2013hja,Tachikawa:2013kta}. We identify $\\tilde R$ as the Cartan generator of the $\\mf{su}(2)_{\\tilde R}$ symmetry of the infrared fixed point.\n\nIt will prove useful to introduce notation to describe the breaking of $\\mf{su}(n)$ symmetry in greater detail. The generators of $\\mf{su}(n)$ can be relabeled according to the decomposition of Eqn. \\eqref{eq:adjdecomposition},\n\\begin{equation}\n\\label{eq:generator_expansion}\nT_A ~~\\Longrightarrow~~ T_{j,m;\\WW(\\RR_j)}~,\n\\end{equation}\nwhere $m=-j, -j+1,\\ldots,+j$ is the eigenvalue of the generator with respect to $\\Lambda(t_0)$, and $\\WW(\\RR_j)$ runs over the various weights of the representation $\\RR_j$ of $\\mf{h}_{\\Lambda}$. Expanding $\\mu_1$ around its expectation value, we have\n\\begin{equation}\n\\label{eq:mu_expansion}\n\\mu_1 = v\\, \\Lambda(t_-) + \\sum_j \\sum_{m=-j}^{+j} \\sum_{\\WW(\\RR_j)} (\\tilde\\mu_1)_{j;m,\\WW(\\RR_j)}T_{j;m,\\WW(\\RR_j)}~.\n\\end{equation}\nThe operators $(\\tilde\\mu_1)_{j;m,\\WW(\\RR_j)}$ with $m3$ we are guaranteed to have non-linear chiral algebras.\n\nFor $n>3$ the stress tensor must be an independent generator of the chiral algebra. This is because the stress tensor can only be a composite of other chiral algebra operators with dimension $h\\leqslant1$. For an interacting theory there can be no chiral algebra operators of dimension $h=1\/2$, so the only possibility is that the stress tensor is a Sugawara stress tensor built as a composite of affine currents. This can only happen if the $\\mf{su}(n)^3$ symmetry is enhanced, since as we have seen above the affine currents associated to the $\\mf{su}(n)$ symmetries are at the critical level and therefore do not admit a normalizable Sugawara stress tensor. Such an enhancement of the flavor symmetry only happens for the $n=3$ case, as will be discussed in greater detail below.\n\nLet us now consider the two simplest cases of trinion chiral algebras: $n=2$ and $n=3$. These are both exceptional in some sense compared to our expectations for generic $n$, which will ultimately make them easier to work with in our examples.\n\n\\medskip\n\\subsubsection{The \\texorpdfstring{$\\protect\\raisebox{1pt}{$\\chi$}[T_2]$}{Chi[T2]} chiral algebra}\n\\label{subsubsec:t2_chiral_algebra}\n\nIn the rank one case, the trinion SCFT is a theory of free hypermultiplets. This case is exceptional compared to the general free hypermultiplets discussed in Section \\ref{subsec:lagrangian_building_blocks} because for $\\mf{su}(2)$ the maximal puncture and minimal puncture are the same, so the minimal puncture also carries an $\\mf{su}(2)$ flavor symmetry, and instead of $n^2$ hypermultiplets transforming in the bifundamental of $\\mf{su}(n)\\times\\mf{su}(n)$, one instead describes the free fields as $2^3=8$ half hypermultiplets transforming in the trifundamental representation of $\\mf{su}(2)^3$. Consequently the symplectic bosons describing this theory are organized into a trifundamental field $q_{abc}(z)$ with $a,b,c=1,2$, with OPE given by\n\\begin{equation}\n\\label{eq:trifundamental_OPE}\nq_{abc}(z)q_{a'b'c'}(w)\\sim\\frac{\\epsilon_{aa'}\\epsilon_{bb'}\\epsilon_{cc'}}{z-w}~.\n\\end{equation}\nEach of the three $\\mf{su}(2)$ subalgebras has a corresponding $\\wh{\\mf{su}(2)}_{-2}$ affine current algebra in the chiral algebra. For example, the currents associated to the first puncture are given by\n\\begin{align}\\label{eq:T2_affine_currents}\n\\begin{split}\nJ_1^{+}(z) &~\\ceq~ \\frac{1}{2}\\epsilon^{bb'}\\epsilon^{cc'}(q_{1bc}q_{1b'c'})(z)~,\\\\\nJ_1^{-}(z) &~\\ceq~ \\frac{1}{2}\\epsilon^{bb'}\\epsilon^{cc'}(q_{2bc}q_{2b'c'})(z)~,\\\\\nJ_1^{\\,0}(z)&~\\ceq~ \\frac{1}{4}\\epsilon^{bb'}\\epsilon^{cc'}\\Big[(q_{1bc}q_{2b'c'})(z)+(q_{2bc}q_{1b'c'})(z)\\Big]~.\n\\end{split}\\end{align}\nThe currents associated to the second and third punctures are constructed analogously. The stress tensor is now given by\n\\begin{equation}\n\\label{eq:T2_stress_tensor}\nT(z)\\ceq \\e^{aa'}\\e^{bb'}\\e^{cc'}(q_{abc}\\partial q_{a'b'c'})(z)~,\n\\end{equation}\nwith corresponding Virasoro central charge given by $c_{2d}=-4$.\n\nIn this simple case it is easy to explicitly compare the Schur superconformal index for the $T_2$ theory with the vacuum character of the chiral algebra. The Schur index has appeared explicitly in, \\eg, \\cite{Gadde:2009kb}. It is given by a single plethystic exponential,\n\\begin{equation}\n\\label{eq:T2_index}\n\\II(q;{\\bf a},{\\bf b},{\\bf c})={\\rm PE}\\left[\\frac{q^{\\frac12}}{1-q}\\protect\\raisebox{1pt}{$\\chi$}_\\square(\\bf a)\\protect\\raisebox{1pt}{$\\chi$}_\\square(\\bf b)\\protect\\raisebox{1pt}{$\\chi$}_\\square(\\bf c)\\right]~.\n\\end{equation}\nThis is easily recognized as the vacuum character of the symplectic boson system defined here. The only comment that needs to be made is that there are no null states that have to be removed from the freely generated character of the symplectic boson algebra. In the next example this simplifying characteristic will be absent.\n\nCrossing symmetry, or associativity of gluing, was investigated for this chiral algebra in \\cite{Beem:2013sza}. There it was proposed that the complete chiral algebra obtained when gluing two copies of $\\protect\\raisebox{1pt}{$\\chi$}[T_2]$ is the $\\widehat{\\mf{so}(8)}$ affine current algebra at level $k_{\\mf{so}(8)}=-2$, and this proposal was checked up to level $h=5$. If the chiral algebra of the four-punctured sphere is precisely this current algebra, then the crossing symmetry relation is implied immediately. This is because the $\\mf{so}(8)$ current algebra has an automorphism as a consequence of triality that exchanges the $\\mf{su}(2)$ subalgebras in accordance with Figure \\ref{fig:tqft_topological_invariance}. If one could prove that the solution to the BRST problem for this gluing is the $\\wh{\\mf{so}(8)}$ current algebra, one would therefore have a proof of generalized $S$-duality at the level of the chiral algebra for all rank one theories of class $\\SS$. We hope that such a proof will turn out to be attainable in the future.\n\n\\medskip\n\\subsubsection{The \\texorpdfstring{$\\protect\\raisebox{1pt}{$\\chi$}[T_3]$}{Chi[T3]} chiral algebra}\n\\label{subsubsec:t3_chiral_algebra}\n\nThe $T_3$ theory is the rank-one $\\mf{e}_6$ theory of Minahan and Nemeschanksky \\cite{Minahan:1996fg}. Before describing its chiral algebra, let us list a number of known properties of this theory.\n\\begin{enumerate}\n\\item[$\\bullet$]The $a$ and $c_{4d}$ anomaly coefficients are known to be given by $a=\\frac{41}{24}$ and $c_{4d}=\\frac{13}{6}$.\n\\item[$\\bullet$]The global symmetry is $\\mf{e}_6$, for which the flavor central charge is $k_{\\mf{e}_6}=6$. This is an enhancement of the $\\mf{su}(3)^3$ symmetry associated with the punctures. It can be understood as a consequence of the fact that the extra Higgs branch generators have dimension two in this case, which means that they behave as moment maps for additional symmetry generators.\n\\item[$\\bullet$]The Higgs branch of this theory is the $\\mf{e}_6$ one-instanton moduli space, which is the same thing as the minimal nilpotent orbit of $\\mf{e}_6$. This property follows immediately from the realization of this theory as a single D3 brane probing an $\\mf{e}_6$ singularity in F-theory.\n\\item[$\\bullet$]A corollary of this characterization of the Higgs branch is that the Higgs branch chiral ring is finitely generated by the moment map operators $\\mu_A$ for $A=1,\\ldots,78$, subject to the \\emph{Joseph relations} (see {\\it e.g.} \\cite{Gaiotto:2008nz}),\n$$\\restr{(\\mu\\otimes\\mu)}{\\bOn\\oplus\\mathbf{650}}=0~.$$\n\\item[$\\bullet$]The superconformal index of the $T_3$ theory was computed in \\cite{Gadde:2010te}. This leads to a formula for the Schur limit of the index given by \n\\begin{equation}\\begin{split}\n\\II_{T_3}(q)=1\\ &+\\ q\\ \\chi_{\\mathbf{[0,0,0,0,0,1]}}\\ \\\\&+\\ q^2\\ (\\chi_{\\mathbf{[0,0,0,0,0,2]}}+\\chi_{\\mathbf{[0,0,0,0,0,1]}}+1)\\ \\\\ \n&+\\ q^3\\ (\\chi_{\\mathbf{[0,0,0,0,0,3]}}+\\chi_{\\mathbf{[0,0,0,0,0,2]}}+\\chi_{\\mathbf{[0,0,1,0,0,0]}}+2\\ \\chi_{\\mathbf{[0,0,0,0,0,1]}}+1)\\ \\\\\n&+\\ q^4\\ (\\chi_{\\mathbf{[0,0,0,0,0,4]}} + \\chi_{\\mathbf{[0,0,0,0,0,3]}} + \\chi_{\\mathbf{[0,0,1,0,0,1]}} + 3\\ \\chi_{\\mathbf{[0,0,0,0,0,2]}}\\\\\n&\\phantom{+\\ q^4\\ ( }\\ + \\chi_{\\mathbf{[0,0,1,0,0,0]}}+ \\chi_{\\mathbf{[1,0,0,0,1,0]}} + 3\\ \\chi_{\\mathbf{[0,0,0,0,0,1]}} +2)\\ \\notag\\\\\n& + \\ldots\n\\end{split}\\end{equation}\nwhere we denoted the $\\mathfrak{e}_6$ representations by their Dynkin labels and suppressed the fugacity-dependence.\n\\end{enumerate}\n\nThe only chiral algebra generators that are guaranteed to be present on general grounds are the seventy-eight affine currents that descend from the four-dimensional moment map operators. The level of the affine current algebra generated by these operators will be $k=-3$. Note that this is \\emph{not} the critical level for $\\mf{e}_6$. The $\\mf{su}(3)^3$ symmetry associated to the punctures is enhanced, and criticality of the subalgebras does not imply criticality of the enhanced symmetry algebra. For this reason, it is possible to construct a Sugawara stress tensor for the current algebra that is properly normalized, and indeed the correct value of the central charge is given by\n\\begin{equation}\n\\label{eq:sugawara_central_charge_check}\nc_{2d}=-26=\\frac{-3\\dim(\\mf{e}_6)}{-3+h^\\vee_{\\mf{e}_6}}=c_{\\rm Sugawara}~.\n\\end{equation}\nOne then suspects that the chiral algebra does not have an independent stress tensor as a generator, but instead the Sugawara construction yields the true stress tensor. Indeed, this was proven in \\cite{Beem:2013sza} to follow from the saturation of certain unitarity bounds by the central charges of this theory.\n\nThis leads to a natural proposal for the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra that was already put forward in \\cite{Beem:2013sza}. The proposal is that the correct chiral algebra is simply the $\\widehat{\\mf{e}}_6$ affine current algebra at level $k=-3$. The singular OPEs of the seventy-eight affine currents are fixed to the canonical form,\\footnote{Our conventions are that the roots of $\\mf{e}_6$ have squared length equal to two.}\n\\begin{equation}\n\\label{eq:e6_OPE}\nJ_A(z)J_B(0) ~\\sim~ \\frac{-3\\,\\delta_{AB}}{z^2} + \\frac{f_{AB}^{\\phantom{AB}C}\\,J_C(0)}{z}~.\n\\end{equation}\nIt is natural to consider the subalgebra $\\mf{su}(3)^3\\subset \\mf{e}_6$ associated to the three punctures on the UV curve and to decompose the currents accordingly. The adjoint representation of $\\mf{e}_6$ decomposes as\n\\begin{equation}\n\\label{eq:e6_adjoint_decomposition}\n\\mathbf{78}~\\longrightarrow~(\\mathbf{8,1,1})+(\\mathbf{1,8,1})+(\\mathbf{1,1,8})+(\\mathbf{3,3,3})+(\\mathbf{\\bar{3},\\bar 3,\\bar 3})~.\n\\end{equation}\nThe affine currents are therefore rearranged into three sets of $\\mf{su}(3)$ affine currents along with one tri-fundamental and one tri-antifundamental set of dimension one currents,\n\\begin{equation}\n\\label{eq:e6_decomposed_current_list}\nJ_A(z)~\\longrightarrow~\\left\\{(J^{1})_{a}^{\\,a'}(z)~,~(J^{2})_{b}^{\\,b'}(z)~,~(J^{3})_{c}^{\\,c'}(z)~,~W_{abc}(z)~,~{\\wt W}^{abc}(z)\\right\\}~.\n\\end{equation}\nThe singular OPEs for this basis of generators are listed in Appendix \\ref{app:level_by_level}. It is perhaps interesting to note that given this list of generators and the requirement that the $\\mf{su}(3)$ current algebras are all at the critical level, the only solution to crossing symmetry for the chiral algebra that includes no additional generators is the $\\wh{\\mf{e}}_6$ current algebra with $k=-3$. So the chiral algebra is completely inflexible once the generators and their symmetry properties are specified.\n\nA nice check of the whole story is that the Joseph relations are reproduced automatically by the chiral algebra. For the non-singlet relation, this follows in a simple way from the presence of a set of null states in the chiral algebra.\n\\begin{equation}\n\\label{eq: e6 nullprediction in 650}\n\\restr{}{}\\!\\restr{}{}P^{AB}_{\\bf 650}(J_AJ_B)(z)\\restr{}{}\\!\\restr{}{}^2=0 \\quad\\Longleftrightarrow\\quad \\restr{(\\mu\\otimes\\mu)}{\\bf 650}=0~,\n\\end{equation}\nwhere $P^{AB}_{\\bf 650}$ is a projector onto the ${\\bf 650}$ representation. These states are only null at this particular value of the level, so we see a close relationship between the flavor central charge and the geometry of the Higgs branch. Similarly, the singlet relation follows from the identification of the Sugawara stress tensor with the true stress tensor of the chiral algebra,\n\\begin{equation}\nT(z)=\\frac{1}{-3+h^\\vee}(J_AJ_A)(z) \\quad\\Longleftrightarrow\\quad \\restr{(\\mu\\otimes\\mu)}{\\bf 1}=0~.\n\\end{equation}\nSo in this relation we see that the geometry of the Higgs branch is further tied in with the value of the $c$-type central charge in four dimensions.\n\nNote that these successes at the level of reproducing the Higgs branch chiral ring relations follow entirely from the existence of an $\\wh{\\mf{e}}_6$ current algebra at level $k=-3$ in the chiral algebra. However what is not necessarily implied is the absence of additional chiral algebra generators transforming as some module of the affine Lie algebra. We can test the claim that there are no additional generators by comparing the partition function of the current algebra to the Schur limit of the superconformal index for $T_3$ (\\cf\\ \\cite{Gadde:2010te}).\\footnote{Because the current algebra is entirely bosonic, the $\\Zb_2$ graded vacuum character is the same as the ungraded vacuum character. Indeed, it is a prediction of our identification of the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra that there are no cancellations in the Schur index between operators that individually contribute.} This comparison is made somewhat difficult by the fact that affine Lie algebras at negative integer dimension have complicated sets of null states in their vacuum module, and these must be subtracted to produce the correct index. The upshot is that up to level four, the vacuum character does indeed match the superconformal index. In order for this match to work, it is crucial that the $\\wh{\\mf{e}}_6$ current algebra has certain null states at the special value $k=-3$. In Table \\ref{Tab:T3_index}, we show the operator content up to level four of a generic $\\wh{\\mf{e}}_6$ current algebra along with the subtractions that occur at this particular value of the level. It is only after making these subtractions that the vacuum character matches the Schur index. Thus we conclude that if there are any additional generators of the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra, they must have dimension greater than or equal to five.\n\n\\begin{table}\\small\n\\centering\n\\begin{tabular}{cl}\n\\hline\n\\hline\ndimension & $\\mf{e}_6$ representations with multiplicities $m_{\\rm generic}\\blue{\/m_{k=-3}}$\\\\\n\\hline\n$0$ & $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,0]}$\\\\\n$1$ & $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,1]}$\\\\\n$2$ & $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,2]}$,~ $1\\blue{\/0}\\times\\mathbf{[1,0,0,0,1,0]}$,~ $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,1]}$,~ $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,0]}$\\\\\n$3$ & $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,3]}$,~ $1\\blue{\/0}\\times\\mathbf{[1,0,0,0,1,1]}$,~ $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,2]}$,~ $2\\blue{\/1}\\times\\mathbf{[0,0,1,0,0,0]}$,\\\\\n\t& $2\\blue{\/0}\\times\\mathbf{[1,0,0,0,1,0]}$,~ $3\\blue{\/2}\\times\\mathbf{[0,0,0,0,0,1]}$,~ $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,0]}$\\\\\n$4$ & $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,4]}$,~ $1\\blue{\/0}\\times\\mathbf{[1,0,0,0,1,2]}$,~ $1\\blue{\/0}\\times\\mathbf{[2,0,0,0,2,0]}$,~ $1\\phantom{\/0}\\times\\mathbf{[0,0,0,0,0,3]}$,\\\\ \n\t& $2\\blue{\/1}\\times\\mathbf{[0,0,1,0,0,1]}$,~ $1\\blue{\/0}\\times\\mathbf{[0,1,0,1,0,0]}$,~ $3\\blue{\/0}\\times\\mathbf{[1,0,0,0,1,1]}$,~ $2\\blue{\/0}\\times\\mathbf{[1,1,0,0,0,0]}$,\\\\ \n\t& $2\\blue{\/0}\\times\\mathbf{[0,0,0,1,1,0]}$,~ $5\\blue{\/3}\\times\\mathbf{[0,0,0,0,0,2]}$,~ $3\\blue{\/1}\\times\\mathbf{[0,0,1,0,0,0]}$,~ $6\\blue{\/1}\\times\\mathbf{[1,0,0,0,1,0]}$,\\\\\n\t& $6\\blue{\/3}\\times\\mathbf{[0,0,0,0,0,1]}$,~ $3\\blue{\/2}\\times\\mathbf{[0,0,0,0,0,0]}$\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{Tab:T3_index}The operator content of the $\\mf{e}_6$ current algebra up to dimension four. The first multiplicity is valid for generic values of the level, \\ie, any value of $k$ where null states are completely absent. The second multiplicity is valid for $k=-3$, and if no second multiplicity is given then the original multiplicity is also the correct one for $k=-3$. These latter multiplicities precisely reproduce the coefficients appearing in the Schur superconformal index for the $T_3$ theory.}\n\\end{table}\n\nA more refined test of our identification of the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra comes from the requirement of compatibility with Argyres-Seiberg duality \\cite{Argyres:2007cn}. The meaning of Argyres-Seiberg duality at the level of the chiral algebra is as follows. Introduce a pair of symplectic bosons transforming in the fundamental representation of an $\\mf{su}(2)$ flavor symmetry,\n\\begin{equation}\n\\label{eq:AS_sb_OPE}\nq_\\alpha(z) \\tilde q^\\beta (0) \\sim \\frac{\\delta_\\alpha^{\\phantom{a}\\beta}}{z}~,\\qquad \\alpha,\\beta=1,2~.\n\\end{equation}\nIn this symplectic boson algebra one can construct an $\\mf{su}(2)$ current algebra at level $k=-1$. Now take the $\\mf{e}_6$ current algebra and consider an $\\mf{su}(2)\\times\\mf{su}(6)\\subset\\mf{e}_6$ maximal subalgebra. The $\\mf{su}(2)$ current algebra coming from this subalgebra has level $k=-3$. Thus the combined level of the symplectic-boson-plus-$\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ system is $k_{tot}=-4$, and consequently this current algebra can be gauged in the manner described in Section \\ref{subsec:chiral_review} by introducing a $(b,c)$ ghost system in the adjoint of $\\mf{su}(2)$ and passing to the cohomology of the appropriate BRST operator. The resulting chiral algebra should be \\emph{identical} to the chiral algebra obtained by taking two copies of the $n=3$ free hypermultiplet chiral algebra of Section \\ref{subsec:lagrangian_building_blocks} and gauging a diagonal $\\mf{su}(3)$ current algebra. This comparison is detailed in Appendix \\ref{app:level_by_level}. \n\nAlthough we have not been able to completely prove the equivalence of these two chiral algebras (the BRST problem for this type of gauging is not easy to solve), we do find the following. On each side of the duality, we are able to determine the generators of dimensions $h=1$ and $h=3\/2$ which amount to a $\\widehat{\\mf{u}(6)}_{-6}$ current algebra in addition to a pair of dimension $h=\\frac32$ generators transforming in the tri-fundamental and tri-antifundamental representations of $\\mf{u}(6)$, with singular OPEs given by\n\\begin{equation}\nb_{i_1i_2i_3}(z)\\tilde b^{j_1j_2j_3}(0) \\sim \\frac{36\\,\\delta_{[i_1}^{[j_1} \\delta_{\\phantom{[}\\!i_2}^{\\phantom{[}\\!j_2} \\delta_{i_3]}^{j_3]}}{z^3} - \\frac{36\\, \\delta_{[i_1}^{[j_1} \\delta_{\\phantom{[}\\!i_2}^{\\phantom{[}\\!j_2}\\hat J_{i_3]}^{j_3]}(0)}{z^2}+\\frac{18\\, \t\\delta_{[i_1}^{[j_1} \t \\hat J_{\\phantom{[}\\!i_2}^{\\phantom{[}\\!j_2}\\hat J_{i_3]}^{j_3]}(0) - 18\\,\\delta_{[i_1}^{[j_1} \\delta_{\\phantom{[}\\!i_2}^{\\phantom{[}\\!j_2} \\partial \\hat J_{i_3]}^{j_3]}(0)}{z}~.\n\\end{equation}\nThus these operators in addition to the $\\mf{u}(6)$ currents form a closed $\\WW$-algebra which is common to both sides of the duality. We expect that these $\\WW$-algebras are in fact the entire chiral algebras in question. However, it should be noted that the existence of this $\\WW$-algebra actually follows from what we have established about the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra without any additional assumptions. That is to say, the possible addition of generators of dimension greater than four could not disrupt the presence of this $\\WW$-algebra. In this sense, common appearance of this algebra can be taken as a check of Argyres-Seiberg duality that goes well beyond the check of \\cite{Gaiotto:2008nz} at the level of the Higgs branch chiral ring. It not only implies a match of a much larger set of operators than just those appearing in the chiral ring, but it also amounts to a match of the three-point functions for those operators, which include the Higgs branch chiral ring operators.\n\nFinally, let us mention one last consistency check on the identification of $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ to which we will return in Section \\ref{subsec:examples}. When one of the three maximal punctures of the $T_3$ theory is reduced to a minimal puncture by Higgsing, the resulting theory is simply that of nine free hypermultiplets transforming in the bifundamental representation of the remaining $\\mf{su}(3)\\times\\mf{su}(3)$ flavor symmetry (along with a $\\mf{u}(1)$ baryon number symmetry associated to the minimal puncture). Therefore if we have correctly identified the $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ chiral algebra, then it should have the property that when the corresponding reduction procedure is carried out, the result is the symplectic boson chiral algebra of Section \\ref{subsec:lagrangian_building_blocks}. The proposal we have given will indeed pass this check, but we postpone the discussion until after we present the reduction procedure in Section \\ref{sec:reducing}.\n\n\\medskip\n\\subsubsection{A proposal for \\texorpdfstring{$\\protect\\raisebox{1pt}{$\\chi$}[T_n]$}{Chi[Tn]}}\n\\label{subsubsec:tn_chiral_algebra}\n\nWe have seen above that for ranks one and two, the trinion chiral algebras are finitely generated (in the chiral algebra sense) by currents that descend from four-dimensional generators of the Higgs branch chiral ring. We know from the results of \\cite{Beem:2013sza} that this cannot be a characterization that holds true for the chiral algebra of an \\emph{arbitrary} $\\NN=2$ SCFT. Moreover, in an interacting theory where the $\\mf{su}(n)^3$ symmetry is not enhanced to a larger global symmetry algebra, the chiral algebra stress tensor cannot be the Sugawara stress tensor of the dimension one currents. This follows from the fact that the $\\mf{su}(n)$ current algebras are at the critical level, so the Sugawara construction fails to produce an appropriate stress tensor. Therefore there must be at least an additional generator corresponding to the stress tensor. The results of \\cite{Lemos:2014lua} further suggest that there should be additional generators in one-to-one correspondence with the generators of the $\\WW_n$ algebra -- \\ie, generators of dimensions $3,\\ldots,n-1$. Aside from that, however, there is room to hope that there will be no additional generators for the trinion chiral algebras. One piece of partial evidence in favor of this suggestion is the absence of additional HL chiral ring generators on top of those generating the Higgs branch chiral ring. This follows from the fact that the $T_n$ theories have genus zero UV curves. Taking this as sufficient reason to formulate a conjecture, we propose the following:\\footnote{This conjecture is different from the one that appeared in the earliest version of this paper. It has been changed to reflect the results of \\cite{Lemos:2014lua}, where the original version of the conjecture was ruled out and replaced by the modified version that appears here.}\n\n\\begin{conj}[$T_{n\\geqslant3}$ chiral algebras]\n\\label{conj:Tn}\nThe protected chiral algebra of the $T_n$ SCFT for any $n\\geqslant3$ is a $\\WW$-algebra with the following generators\n\\begin{enumerate}\n\\item[$\\bullet$] Three sets of $\\mf{su}(n)^3$ affine currents at the critical level $k=-n$. \n\\item[$\\bullet$] One current of dimension $\\frac12\\ell(n-\\ell)$ transforming in the $(\\wedge^{\\ell},\\wedge^{\\ell},\\wedge^{\\ell})$ representation of $\\mf{su}(n)^3$ for each $\\ell=1,\\ldots,n-1$.\n\\item[$\\bullet$] Operators $W_i$, $i=1,\\ldots,n-1$ of dimension $i+1$ that are $\\mf{su}(n)^3$ singlets. The dimension two operator is identified as a stress tensor $W_1(z)\\equiv T(z)$ with Virasoro central charge equal to $c_{2d}=-2n^3+3n^2+n-2$. In special cases some of these operators may be redundant.\n\\end{enumerate}\n\\end{conj}\n\n\\noindent At any $n\\geqslant4$, the very existence of such a $\\WW$-algebra is quite nontrivial, since for a randomly chosen set of generators one doesn't expect to be able to solve the associated Jacobi identities. In fact if the singular OPEs of such a $\\WW$-algebra can be chosen so that the algebra is associative, it seems likely that the requirements of associativity will \\emph{completely fix} the structure constants, rendering the chiral algebra unique. It is worth observing that precisely such uniqueness occurs in the case of the $T_3$ chiral algebra. The characterization given by the conjecture above for $n=3$ doesn't explicitly imply $\\mf{e}_6$ symmetry enhancement, but the unique chiral algebra satisfying the requirements listed is precisely the $\\mf{e}_6$ current algebra at the appropriate level. A similar uniqueness result is currently under investigation for the $T_4$ chiral algebra \\cite{Lemos:2014lua}.\n\nBefore moving on, let us extrapolate a bit from Conjecture \\ref{conj:Tn} to make a further conjecture that, while not extremely well-supported, is consistent with everything we know at this time.\n\n\\begin{conj}[Genus zero chiral algebras]\n\\label{conj:genus_zero}\nThe protected chiral algebra of any class $\\SS$ SCFT of type $A_n$ whose UV curve has genus zero is a $\\WW$-algebra with singlet generators $W_i$, $i=1,\\ldots, n$ of dimension $i+1$ and additional currents associated to Higgs branch chiral ring generators of the four-dimensional theory. In special cases some of the $W_i$ may be related to composites -- in particular when the central charge is equal to its Sugawara value with respect to the affine currents, then the stress tensor $W_1(z)$ is a composite.\n\\end{conj}\n\n\\noindent The modest evidence in favor of this proposal is that genus zero theories have honest Higgs branches with no residual $U(1)$ gauge fields in the IR, so they don't have any of the additional $\\NN=1$ chiral ring generators discussed in Section \\ref{subsec:chiral_review}. Additionally the examples of \\cite{Beem:2013sza} for which there were chiral algebra generators unrelated to four-dimensional chiral ring generators was a genus one and two theories. It would be interesting to explore this conjecture further, even in the Lagrangian case.\n\\section{Reduced punctures}\n\\label{sec:reducing}\n\nThe $T_n$ building blocks outlined in Sec.\\;\\ref{subsec:building_blocks} only allow us to construct class $\\SS$ chiral algebras associated to undecorated UV curves, while the inclusion of the free hypermultiplet chiral algebras of Sec.\\;\\ref{subsec:lagrangian_building_blocks} allow for decoration by minimal punctures only. The purpose of this section is to develop the tools necessary to describe theories that correspond to UV curves with general non-trivial embeddings decorating some of their punctures.\n\nFrom the TQFT perspective, the most natural way to introduce the necessary additional ingredients is to find a chiral algebra associated to the decorated cap of Fig. \\ref{fig:dec_cap}. This turns out not to be the most obvious approach from a physical perspective since the cap doesn't correspond to any four-dimensional SCFT.%\n\\footnote{It does however correspond to a true compactification of the six-dimensional $(2,0)$ theory \\cite{Gaiotto:2011xs}. We will return to the notion of such a decorated cap in Sec.\\;\\ref{subsec:decorated_cap}.}\nRather, it is more natural to develop a procedure for reducing a maximal puncture to a non-maximal that mimics the Higgsing procedure reviewed in Sec.\\;\\ref{subsec:class_S_review}. Naively, the four-dimensional Higgsing prescription need not lead to a simple recipe for producing the chiral algebra of the Higgsed theory in terms of that of the original theory. This is because the Higgsing spontaneously breaks the superconformal symmetry that is used to argue for the very existence of a chiral algebra, with the theory only recovering superconformal invariance in the low energy limit. Consequently one could imagine that the Higgsing procedure irrecoverably requires that we abandon the chiral algebraic language until reaching the far infrared.\n\nNevertheless, it turns out that the chiral algebra does admit its own Higgsing procedure that has the desired result. Such a procedure cannot literally amount to Higgsing in the chiral algebra, because quantum mechanically in two dimensions there are no continuous moduli spaces of vacua. The best that we can do is to try to impose a quantum-mechanical \\emph{constraint} on the chiral algebra. A natural expectation for the constraint is that it should fix to a non-zero value the chiral algebra operator that corresponds to the Higgs branch chiral ring operator that gets an expectation value. This means imposing the constraint\n\\begin{equation}\n\\label{eq:positive_current_constraint}\nJ_{\\alpha_-}(z)=A~,\n\\end{equation}\nwhere $T_{\\alpha_-}=\\Lambda(t_-)$. Here $A$ is a dimensionful constant that will be irrelevant to the final answer as long as it is nonzero. We might also expect that we should constrain some of the remaining currents to vanish. A motivation for such additional constraints is that when expanded around the new vacuum on the Higgs branch, many of the moment map operators become field operators for the Nambu-Goldstone bosons of spontaneously broken flavor symmetry, and we want to ignore those and focus on the chiral algebra associated to just the interacting part of the reduced theory.\n\nThere happens to be a natural conjecture for the full set of constraints that should be imposed. This conjecture, which was already foreshadowed in \\cite{Beem:2013sza}, is as follows:\n\\begin{conj}\\label{conj:qDS}\nThe chiral algebra associated to a class $\\SS$ theory with a puncture of type $\\Lambda$ is obtained by performing quantum Drinfeld-Sokolov (qDS) reduction with respect to the embedding $\\Lambda$ on the chiral algebra for the theory where the same puncture is maximal.\n\\end{conj}\nQuantum Drinfeld-Sokolov in its most basic form is a procedure by which one obtains a new chiral algebra by imposing constraints on an affine Lie algebra $\\hat\\mf{g}$, with the constraints being specified by an embedding $\\Lambda:\\mf{su}(2)\\hookrightarrow\\mf{g}$. In the case of interest to us, the chiral algebra on which we will impose these constraints is generally larger than just an affine Lie algebra. Nevertheless, these constraints can still be consistently imposed in the same manner. This conjecture therefore amounts to a choice of the additional constraints beyond \\eqref{eq:positive_current_constraint} that should be imposed in order to reduce a puncture. It is interesting to note that the right set of constraints will turn out to fix only \\emph{half} of the currents that are expected to become Nambu-Goldstone bosons. We will see that the removal of the remaining Nambu-Goldstone bosons occurs in a more subtle manner.\n\nBefore delving into the details, we should make the observation that this answer is not unexpected in light of the pre-existing connections between non-maximal defects in the $(2,0)$ theory and qDS reduction \\cite{Alday:2010vg,Chacaltana:2012zy}. Though a sharp connection between the AGT story and the protected chiral algebra construction is still lacking, we take this as a positive indication that such a connection is there and remains to be clarified. We now turn to a more precise description of qDS reduction for chiral algebras with affine symmetry. We will first develop the general machinery for performing such a reduction in the cases of interest, whereafter we will perform a number of tests of the claim that this is the correct procedure for reducing the ranks of punctures in class $\\SS$ chiral algebras.\n\n\\bigskip\n\\input{.\/sections\/Section_4\/S4_1}\n\\bigskip\n\\input{.\/sections\/Section_4\/S4_2}\n\\bigskip\n\\input{.\/sections\/Section_4\/S4_3}\n\\bigskip\n\\input{.\/sections\/Section_4\/S4_4}\n\\bigskip\n\\bigskip\n\\subsection{Quantum Drinfeld-Sokolov for modules}\n\\label{subsubsec:qDSspecseq}\n\nQuantum Drinfeld-Sokolov reduction is a procedure for imposing a set of constraints given below in Eqn. \\eqref{eq:qDSconstraints} at the quantum level for an affine Lie algebra $\\hat\\mf{g}$ at any level. In the following discussion, we will closely follow the analysis of \\cite{deBoer:1993iz} (see also \\cite{deBoer:1992sy} for a similar discussion for finite dimensional algebras). Although traditionally the starting point for this procedure is a pure affine Lie algebra, our interest is in the case of a more general chiral algebra with an affine Lie subalgebra at the critical level. Said differently, we are interested in performing qDS reduction for nontrivial $\\hat{\\mf{g}}_{-h^\\vee}$ modules. We will utilize essentially the same spectral sequence argument as was used in \\cite{deBoer:1993iz}. Some basic facts about spectral sequences are collected in Appendix \\ref{app:spectral_sequences} for the convenience of the reader.\n\nThe general setup with which we are concerned is the following. We begin with a chiral algebra (for simplicity we take it to be finitely generated) with an $\\widehat{\\mf{su}(n)}_{k}$ affine subalgebra. We denote the generating currents of the affine subalgebra as $J_{A}(z)$, while the additional generators of the chiral algebra will be denoted as $\\{\\phi^i(z)\\}$, each of which transforms in some representation $\\mf{R}_i$ of $\\mf{su}(n)$.\n\nWe now choose some embedding $\\Lambda:\\mf{su}(2)\\hookrightarrow\\mf{su}(N)$, for which the images of the $\\mf{su}(2)$ generators $\\{t_0,t_+,t_-\\}$ will be denoted by $\\{\\Lambda(t_0),\\Lambda(t_+),\\Lambda(t_-\\}$. The embedded Cartan then defines a grading on the Lie-algebra,\n\\begin{equation}\n\\label{eq:cartan_grading_def}\n\\mf{g} = \\bigoplus_{m\\in\\frac12 \\Zb}\\mf{g}_m~,\\qquad \\mf{g}_m\\ceq\\left\\{T_A \\in \\mf{g}~\\vert~{\\rm ad}_{\\Lambda(t_0)}T_A = m\\,T_A \\right\\}~.\n\\end{equation}\nWhen the embedded Cartan is chosen such that some of the currents have half-integral grading, then some of the associated constraints are second-class and cannot be enforced by a straightforward BRST procedure. Fortunately, it has been shown that one may circumvent this problem by selecting an alternative Cartan generator $\\delta$ which exhibits integer grading and imposing the corresponding first class constraints \\cite{Feher:1992ed,deBoer:1992sy,deBoer:1993iz}. We will adopt the convention that an index $\\alpha$ ($\\bar\\alpha$) runs over all roots with negative (non-negative) grading with respect to $\\delta$, while Latin indices run over all roots. The first-class constraints to be imposed are then as follows,\n\\begin{equation}\n\\label{eq:qDSconstraints}\nJ_{\\alpha} = A\\,\\delta_{\\alpha\\alpha_-}~,\n\\end{equation}\nwhere $\\Lambda(t_-) = T_{\\alpha_-}$. These constraints are imposed \\`a la BRST by introducing dimension $(1,0)$ ghost pairs $(c^\\alpha,b_\\alpha)$ in one-to-one correspondence with the generators $T_{\\alpha}$. These ghosts have the usual singular OPE\n\\begin{equation}\n\\label{eq:BRST_ghost_OPE}\nc^\\alpha(z) b_\\beta(0)\\sim \\frac{\\delta^\\alpha_{\\phantom{\\alpha}\\beta}}{z}~,\n\\end{equation}\nand allow us to define a BRST current\n\\begin{equation}\n\\label{eq:BRSTcurrent}\nd(z) = \\left(J_{\\alpha}(z) - A\\,\\delta_{\\alpha\\alpha_-}\\right)c^{\\alpha}(z) - \\frac12 f_{\\alpha\\beta}^{\\phantom{\\alpha\\beta}\\gamma}(b_\\gamma(c^\\alpha c^\\beta))(z)~.\n\\end{equation}\nThe reduced chiral algebra is defined to be the BRST-cohomology of the combined ghost\/matter system. Note that this definition is perfectly reasonable for the case where we are reducing not just the affine current algebra, but a module thereof. The presence of the module doesn't modify the system of constraints of the BRST differential, but as we shall see, the operators in the modules will be modified in a nontrivial way in the constrained theory.\n\nThis cohomological problem can be solved via a modest generalization of the approach of \\cite{Feigin:1990pn,deBoer:1993iz}. We first split the BRST current into a sum of two terms,\n\\begin{align}\\begin{split}\nd_0(z) &~=~ \\left(-A\\,\\delta_{\\alpha\\alpha_-}\\right) c^{\\alpha}(z)~,\\\\\nd_1(z) &~=~ J_{\\alpha}(z)c^{\\alpha}(z)-\\frac12 f_{\\alpha\\beta}^{\\phantom{\\alpha\\beta}\\gamma}(b_\\gamma(c^\\alpha c^\\beta))(z)~.\\label{eq:differential0}\n\\end{split}\\end{align}\nWe now introduce a bi-grading for the currents and ghosts so that the differentials $(d_0,d_1)$ have bi-grades $(1,0)$ and $(0,1)$, respectively,\n\\begin{alignat}{3}\n&\\text{deg }(J_A(z)) \t &~=~& (m,-m)~,\t\t\\qquad &&T_A\\in \\mf{g}_m~,\\notag\\\\\n&\\text{deg }(c^\\alpha(z)) &~=~& (-m,1+m)~, \t\\qquad &&T_\\alpha\\in \\mf{g}_m~,\\label{eq:bigrading}\\\\\n&\\text{deg }(b_\\alpha(z)) &~=~& (m,-m-1)~, \t\t\\qquad &&T_\\alpha\\in \\mf{g}_m\\notag~.\n\\end{alignat}\nThis bi-grading can also be extended to the additional generators $\\phi^i$. We decompose each such generator into weight vectors of $\\mf{su}(n)$ according to\n\\begin{equation}\n\\label{eq:phi_decomposition}\n\\phi^i=\\phi^i_It_I^{(\\mf{R}_i)}~,\\qquad I=1,\\ldots,\\dim\\mf{R}_i~,\n\\end{equation}\nwhere the $t_I^{(\\mf{R}_i)}$ form a weight basis for the representation $\\mf{R}_i$ with weights defined according to \n\\begin{equation}\n\\label{eq:general_rep_weight_basis}\nH_\\alpha\\cdot t_I^{(\\mf{R}_i)} = \\lambda^{(\\mf{R}_i)}_{I,\\alpha}\\, t_I^{(\\mf{R}_i)}~,\n\\end{equation}\nwhere $H_\\alpha$ is an element of the Cartan subalgebra of $\\mf{su}(n)$. Given the element $\\delta$ in terms of which our grading is defined, the bi-grading of the extra generators can be defined according to\n\\begin{equation}\n\\label{eq:bigradingextrafields}\n\\text{deg }(\\phi^i_I) = (\\delta\\cdot t_I^{(\\mf{R}_i)},\\,-\\delta\\cdot t_I^{(\\mf{R}_i)})~.\n\\end{equation}\nThe differentials $(d_0,d_1)$ are each differentials in their own right, that is, they satisfy\n\\begin{equation}\n\\label{eq:double_sequence_differentials}\nd_0^2=d_1^2=d_0d_1+d_1d_0=0~.\n\\end{equation}\nTherefore they define a double complex on the Hilbert space of the ghost\/matter chiral algebra, which is the starting point for a spectral sequence computation of the cohomology.\n\nIt turns out that a simplification occurs if instead of trying to compute the cohomology of the double complex straight off, we first introduce ``hatted currents'' \\cite{Feigin:1990pn,deBoer:1993iz},\n\\begin{equation}\n\\label{eq:hattedcurrents}\n\\hat{J}_A(z) = J_A(z) + f_{A\\beta}^{\\phantom{a\\beta}\\gamma}(b^{\\phantom{a}}_\\gamma c_{\\phantom{a}}^{\\,\\beta})(z)~.\n\\end{equation}\nLet us denote by $\\Ab_1$ the subalgebra generated by $b_\\alpha(z)$ and $\\hat{J}_\\alpha(z)$, and by $\\Ab_2$ the subalgebra produced by the remaining generators $c^\\alpha(z)$, $\\hat{J}_{\\bar\\alpha}(z)$, and $\\phi^i(z)$. One then finds that $d(\\Ab_1)\\subseteq\\Ab_1$ and $d(\\Ab_2)\\subseteq\\Ab_2$, with the generators of $\\Ab_1$ additionally obeying\n\\begin{equation}\n\\label{eq:trivial_cohomology}\nd(b_{\\alpha}(z))= \\hat{J}_{\\alpha}(z)-A\\delta_{\\alpha\\alpha_-}~,\\qquad d(\\hat{J}_{\\alpha}(z))=0~.\n\\end{equation}\nIt follows that the BRST cohomology of $\\Ab_1$ is trivial: $H^*(\\Ab_1,d)=\\Cb$. From the K\\\"unneth formula (see Appendix \\ref{app:spectral_sequences}), it follows that the BRST cohomology of the chiral algebra is isomorphic to the cohomology of the smaller algebra $\\Ab_2$,\n\\begin{equation}\n\\label{eq:cohomology_simplification}\nH^*(\\Ab,d) \\cong H^*(\\Ab_2,d)~.\n\\end{equation}\nOur task then simplifies: we need only compute the cohomology of $\\Ab_2$. We will address this smaller problem by means of a spectral sequence for the double complex $(\\Ab_2,d_0,d_1)$.\n\nThe first step in the spectral sequence computation is to compute the cohomology $H^*(\\Ab_2,d_0)$. The only nontrivial part of this computation is the same as in the case without modules. This is because the additional generators $\\phi^i_{I}(z)$ have vanishing singular OPE with the $c$-ghosts, rendering them $d_0$-closed. Moreover, they can never be $d_0$-exact because the $b$-ghosts are absent from $\\Ab_2$. For the currents and ghosts, one first computes\n\\begin{equation}\n\\label{eq:d0_of_Jhat}\nd_0(\\hat{J}_{\\bar{\\alpha}}(z)) = -A f_{\\bar{\\alpha}\\beta}^{\\phantom{\\alpha\\beta}\\gamma} \\delta^{\\phantom{a}}_{\\gamma\\alpha_-}c^{\\,\\beta}(z) = - \\mbox{Tr}\\left({\\rm ad}_{\\Lambda(t_+)}T_{\\bar{\\alpha}}\\cdot T_\\beta\\right)c^{\\,\\beta}(z)~.\n\\end{equation}\nIt follows that $d_0(\\hat{J}_{\\bar\\alpha}(z))=0$ if and only if $T_{\\bar{\\alpha}}\\in \\ker({\\rm ad}_{\\Lambda(t_+)})$. The same equation implies that the $c^\\alpha(z)$ ghosts are $d_0$-exact for any $\\alpha$. Because the $d_0$-cohomology thus computed is supported entirely at ghost number zero, the spectral sequence terminates at the first step. At the level of vector spaces we find\n\\begin{equation}\n\\label{eq:vector_space_cohomology}\nH^*(\\Ab,d) \\cong H^*(\\Ab_2,d_0)~,\n\\end{equation}\nwith $H^*(\\Ab_2,d_0)$ being generated by the $\\phi^i_I(z)$ and by $J_{\\bar\\alpha}(z)$ for $T_{\\bar\\alpha}\\in\\ker({\\rm ad}_{\\Lambda(t_+)})$.\n\nIn order to improve this result to produce the vertex operator algebra structure on this vector space, we can construct representatives of these with the correct OPEs using the tic-tac-toe procedure. Letting $\\psi(z)$ be a generator satisfying $d_0(\\psi(z))=0$, the corresponding chiral algebra generator $\\Psi(z)$ is given by\n\\begin{equation}\n\\label{eq:tic-tac-toed_generator}\n\\Psi(z) = \\sum_l (-1)^l \\psi_l(z)~,\n\\end{equation}\nwhere $\\psi_l(z)$ is fixed by the condition\n\\begin{equation}\n\\label{eq:tic-tac-toe_condition}\n\\psi_0(z) \\ceq \\psi(z)~,\\quad d_1(\\psi_l(z)) = d_0(\\psi_{l+1}(z))~.\n\\end{equation}\nAt the end, this procedure will give a collection of generators of the qDS reduced theory along with their singular OPEs and it would seem that we are finished. However, it is important to realize that this may not be a minimal set of generators, in that some of the generators may be expressible as composites of lower dimension generators due to null states. The existence of null relations of this type is very sensitive to the detailed structure of the original chiral algebra. For example, the level of the current algebra being reduced plays an important role. In practice, we will find for the class $\\SS$ chiral algebras, \\emph{most} of the generators $\\Psi(z)$ produced by the above construction do in fact participate in such null relations.\n\nSome null states of the reduced theory can be deduced from the presence of null states in the starting chiral algebra. This can be an efficient way to generate redundancies amongst the naive generators of the qDS reduced theory like the ones described above. Abstractly, we can understand this phenomenon as follows. Consider a null operator $N^K(z)$ that is present in the original $\\WW$-algebra, and that transforms in some representation $\\mathfrak{R}$ of the symmetry algebra that is being reduced. Given an embedding $\\Lambda,$ the representation $\\mathfrak{R}$ decomposes as in \\eqref{eq:generaldecomposition} under $\\mathfrak{g}_{\\Lambda}\\oplus \\Lambda(\\mathfrak{su}(2)).$ We can thus split the index $K$ accordingly and obtain $\\{N^{k_j,m_j}(z)\\}_{j\\geqslant 0},$ where $k_j$ is an index labeling the representation $\\mathcal{R}_j^{(\\mathfrak{R})}$ and $m_j$ labels the Cartan of the spin $j$ representation $V_j.$ For fixed values of the index $m_j$ we find an operator that will have proper dimension with respect to the new stress tensor \\eqref{eq:qDSstresstensor}. Moreover, since introducing a set of free ghost pairs naturally preserves the null property of the original operator and restricting oneself to the BRST cohomology does not spoil it either, we find that this operator is null in the qDS reduced theory. In practice, for each value of $m_j$ one chooses a representative of the BRST class $N^{k_j,m_j}(z) + d(\\ldots)$ that only involves the generators of the qDS reduced theory.\n\nThere are a couple of features of the qDS reduced theory that can be deduced without studying the full procedure in specific examples. These features provide us with the most general test of the conjecture that qDS reduction is the correct way to reduce the ranks of punctures in the chiral algebra. The first of these features is the Virasoro central charge of the reduced theory, a subject to which we turn presently.\n\\subsection{Virasoro central charge and the reduced stress tensor}\n\\label{subsec:reduced_central_charges}\n\nA useful feature of qDS reduction is that the stress tensor of a qDS reduced chiral algebra takes a canonical form (up to BRST-exact terms) in which it is written as a shift of the stress tensor of the unreduced theory,\n\\begin{equation}\n\\label{eq:qDSstresstensor}\nT = T_{\\star} - \\partial J_0 + \\partial b_\\alpha c^\\alpha - (1+\\lambda_\\alpha) \\partial (b_\\alpha c^\\alpha)~.\n\\end{equation}\nHere $T_{\\star}$ is the stress tensor of the unreduced theory, $J_0$ is the affine current of the $U(1)$ symmetry corresponding to $\\Lambda(t_0)$, and $\\lambda_\\alpha$ is the weight for $T_\\alpha$ with respect to $\\Lambda(t_0)$ as defined by Eqn. \\eqref{eq:general_rep_weight_basis}.%\n\\footnote{Note that in the case of half-integral gradings, the weights $\\lambda_\\alpha$ are defined with respect to $\\Lambda(t_0)$ and \\emph{not} with respect to the alternate Cartan element $\\delta$.}\nThe dimensions of the ghosts measured by this new stress tensor are $h_{b_\\alpha} = 1+\\lambda_\\alpha$ and $h_{c^\\alpha} = -\\lambda_\\alpha$. Meanwhile the dimensions of all remaining fields are simply shifted by their $J_0$ charge.\n\nThe central charge of the reduced theory can be read off from the most singular term in the self-OPE of the reduced stress tensor. The result is given by \\cite{Feher:1992ed}\n\\begin{align}\\label{eq:central_charge_shift}\n\\begin{split}\nc-c_{\\star}\t&~= \\left(\\dim\\mf{g}_0-\\frac12\\dim\\mf{g}_{\\frac12}-12\\left|\\sqrt{k+h^\\vee}\\Lambda(t_0)-\\frac{\\rho}{\\sqrt{k+h^\\vee}}\\right|^2\\right)-\\left(\\frac{k\\dim\\mf{g}}{k+h^\\vee}\\right)~,\\\\\n\t\t\t&~= \\dim\\mf{g}_0-\\frac12\\dim \\mf{g}_{\\frac{1}{2}}-12(k+h^\\vee)\\left|\\Lambda(t_0)\\right|^2+24\\Lambda(t_0)\\cdot\\rho-\\dim\\mf{g}~.\n\\end{split}\n\\end{align}\nHere $\\rho$ is the Weyl vector of $\\mf{su}(n)$, and in passing to the second line, we have used the Freudenthal-de Vries strange formula $|\\rho|^2 = \\frac{h^\\vee}{12}\\dim\\mf{g}$. In the cases of interest the level of the current algebra is always given by $k=-h^\\vee$ and there is a further simplification,\n\\begin{equation}\n\\label{eq:change2dcentralcharge}\nc=c_{\\star}+\\dim\\mf{g}_0-\\frac12\\dim\\mf{g}_{\\frac12}+24\\Lambda(t_0)\\cdot\\rho-\\dim\\mf{g}~.\n\\end{equation}\n\nThis shift of two-dimensional central charge can be compared to our expectations based on the four-dimensional results in Eqns. \\eqref{eq:4dcentralcharges}-\\eqref{eq:nv_nh_defs_2}. The change of the four-dimensional central charge that occurs upon reducing a maximal puncture down to a smaller puncture labelled by the embedding $\\Lambda$ is given by\n\\begin{align}\\begin{split}\n\\label{eq:change4dcentralcharge}\n-12(c_{4d} - c_{4d,\\text{orig.}} ) &~=~ 2(n_v(\\text{max.}) - n_v(\\Lambda)) + (n_h(\\text{max.}) - n_h(\\Lambda))~,\\\\\n\t\t\t\t\t\t\t\t\t&~=~ \\dim \\mf{g}_0 - \\frac{1}{2}\\dim \\mf{g}_{\\frac{1}{2}} +24 \\Lambda(t_0)\\cdot \\rho -\\dim\\mf{g}~.\n\\end{split}\\end{align}\nThus we see precise agreement with the change in two-dimensional central charge induced by qDS reduction and that of the four-dimensional charge induced by Higgsing after accounting for the relation $c_{2d} = -12 c_{4d}$. We take this as a strong indication the the qDS prescription for reducing chiral algebras is indeed the correct one.\n\\subsection{Reduction of the superconformal index}\n\\label{subsec:reduced_index}\n\nWe can now check that the qDS reduction procedure has an effect on the (graded) partition function of the chiral algebra that mimics the prescription for reducing the Schur superconformal index described in Sec.\\,\\ref{subsec:class_S_review}. As was reviewed above, the Schur limit of the superconformal index is equivalent to a graded partition function of the corresponding chiral algebra,\n\\begin{equation}\n\\label{eq:chiral_character}\n\\II_{\\chi}(q; {\\bf x}) ~\\ceq~ \\mbox{Tr}_{\\HH_{\\chi}}\\,(-1)^F q^{L_0} ~=~ \\II^{\\rm Schur}(q; {\\bf x})~.\n\\end{equation}\nComputing this graded partition function is straightforward for the qDS reduced theory owing to the fact that the BRST differential commutes with all of the fugacities ${\\bf x}$ that may appear in the index and has odd fermion number. This means that we can ignore the cohomological aspect of the reduction and simply compute the partition function of the larger Hilbert space obtained by tensoring the unreduced chiral algebra with the appropriate ghosts system.\\footnote{There is a caveat to this argument, which is that if there are null states in the reduced theory that do not originate as null states in the parent theory, then their subtraction will not necessarily be accomplished by this procedure. We operate under the assumption that such spurious null states do not appear. This assumption appears to be confirmed by the coherence between this procedure and that discussed in Sec.\\,\\ref{subsec:class_S_review}.}\n\nThis simpler problem of computing the partition function of the larger Hilbert space parallels the index computation described in Sec.\\,\\ref{subsec:class_S_review}. There are again two steps -- the inclusion of the ghosts, and the specialization of fugacities to reflect the symmetries preserved by the BRST differential. Including the ghosts in the partition function before specializing the fugacities requires us to assign them charges with respect to the UV symmetries. This can be done in a canonical fashion so that upon specializing the fugacities the BRST current will be neutral with respect to IR symmetries and have conformal dimension one. \n\nRecall that the ghost sector involves one pair of ghosts $(b_\\alpha,c^\\alpha)$ for each generator $T_{\\alpha}$ that is negatively graded with respect to $\\delta$. The charge assignments are then the obvious ones -- namely the charges of $b_\\alpha$ are the same as those of $T_\\alpha$ (let us call them $f_\\alpha$), while those of $c^{\\alpha}$ are minus those of $b_{\\alpha}$. With these charge assignments, the graded partition function of the reduced chiral algebra can be obtained as a specialization that mimics that which led to the superconformal index,\n\\begin{equation}\n\\label{eq:chiral_index_specialization}\n\\II_{\\chi_{\\Lambda}}(q;{\\bf x}_{\\Lambda})=\\lim_{{\\bf x} \\to {\\bf x}_\\Lambda}\\II_{\\chi}(q;{\\bf x})\\,\\II_{(b,c)_{\\Lambda}}(q;{\\bf x})~,\\qquad \\II_{(b,c)_{\\Lambda}}\\ceq{\\rm PE}\\left[-\\!\\!\\!\\!\\sum_{T_{\\alpha}\\in\\mf{g}_{<0}}\\left(\\frac{q\\,{\\bf x}^{f_{\\alpha}}}{1-q}+\\frac{{\\bf x}^{-f_{\\alpha}}}{1-q}\\right)\\right].\n\\end{equation}\nAs in the discussion of the index in Sec.\\,\\ref{subsec:reduced_index}, we can formally perform the specialization ignoring divergences that occur in both the numerator and the denominator as a consequence of constant terms in the plethystic exponent. In doing this, the flavor fugacities are replaced by fugacities for the Cartan generators of $\\mf{h}_\\Lambda$, while the $q$-grading is shifted by the Cartan element of the embedded $\\mf{su}(2)$. This leads to the following formal expression for the contribution of the ghosts,\\footnote{For simplicity, we write the expression here for the case where $\\Lambda(t_0)$ provides an integral grading so there is no auxiliary $\\delta$. The case of half-interal grading can be treated with modest modifications.}\n\\begin{equation}\n\\label{eq:formal_character_specialized_ghosts}\n\\II_{(b,c)_{\\Lambda}}~~ ``=\" ~~{\\rm PE}\\left[ \\frac{-q}{1-q} \\sum_j \\protect\\raisebox{1pt}{$\\chi$}_{\\RR_j^{(\\mathrm{adj})}}^{\\mf{h}_{\\Lambda}}(\\mathbf{a}_{\\Lambda}) \\sum_{i=-j}^{-1}q^i -\\frac{1}{1-q}\\sum_j \\protect\\raisebox{1pt}{$\\chi$}_{\\RR_j^{(\\mathrm{adj})}}^{\\mf{h}_{\\Lambda}}(\\mathbf{a}^{-1}_{\\Lambda})\\sum_{i=1}^{j}q^i \\right]~.\n\\end{equation}\nAfter a small amount of rearrangement and the recognition that the representations $\\RR_j^{(\\mathrm{adj})}$ are pseudoreal, one finds that this exactly reproduces the formal denominator in Eqn.\\,\\eqref{eq:K_factor_fugacity_replacement_2}. Again, when the limit in Eqn.\\,\\eqref{eq:chiral_index_specialization} is taken carefully, the divergences in this formal denominator cancel against equivalent terms in the $K$-factors of the numerator to produce a finite result. It is interesting that in spite of the asymmetry between $b$ and $c$ ghosts in this procedure, they ultimately play the same role from the point of view of four-dimensional physics -- each ghost is responsible for cancelling the effect of a single Nambu-Goldstone boson from the index.\n\nBefore moving on to examples, we recall that in \\cite{Beem:2014kka} it was observed that the $K$-factor for a maximal puncture matches the character of the corresponding affine Lie algebra at the critical level, and it was conjectured that a similar statement would be true for reduced punctures. That is to say, the $K$-factor associated to the reduction of type $\\Lambda$ should be the character of the qDS reduction of type $\\Lambda$ of the critical affine Lie algebra. Given the analysis to this point, this statement becomes almost a triviality. The qDS reduction of the affine current algebra proceeds by introducing the same collection of ghosts as we have used here, and so the effect on the graded partition function is the introduction of the same ghost term given in Eqn.\\,\\eqref{eq:formal_character_specialized_ghosts} and the same specialization of fugacities. Thus, the identification of the $K$-factors given in Eqn.\\,\\eqref{eq:K-factor} with the character of the qDS reduction of the critical affine Lie algebra depends only on our ability to equate the index (\\ie, the partition function graded by $(-1)^F$) with the ungraded vacuum character. This is a simple consequence of the fact that the starting current algebra consists of all bosonic operators and the spectral sequence calculation of Sec.\\,\\ref{subsubsec:qDSspecseq} only found BRST cohomology elements at ghost number zero.\n\\subsection{Simple examples}\n\\label{subsec:examples}\n\nIn light of the analysis in Section \\ref{subsubsec:qDSspecseq}, the reduction problem admits an algorithmic solution subject to two conditions. (A) the starting chiral algebra should be finitely generated, \\ie, it admits a description as a $\\WW$-algebra. (B) the $L_0$ operator of the reduced theory should have a positive definite spectrum. The latter condition must hold for any reductions where the endpoint corresponds to a physical class $\\SS$ theory, while the former conditions is conjectured to be true for general class $\\SS$ theories but is more certainly true in some simple examples. Given these conditions, the procedure is as follows:\n\\begin{enumerate}\n\\item[$\\bullet$] List the (possibly redundant) generators of the qDS reduced chiral algebra at the level of vector spaces. These are given by the hatted currents $\\hat J_{\\bar \\alpha}$ for which $T_{\\bar{\\alpha}}\\in \\ker({\\rm ad}_{\\Lambda(t_+)})$, along with all of the additional generators $\\{\\phi_i\\}$.\n\\item[$\\bullet$] Apply the tic-tac-toe algorithm to construct genuine generators of the chiral algebra. The OPEs of these reduced chiral algebra generators can be computed directly using the OPEs of the original, unreduced fields.\n\\item[$\\bullet$] Compute the null states at each level up to that of the highest-dimensional generator in order to check for redundancy. Remove any redundant generators. What remains is a description of the reduced chiral algebra as a $\\WW$-algebra.\n\\end{enumerate}\nThis procedure is still morally a correct one when the two conditions listed above fail to be met, but in those cases the algorithm will not necessarily terminate in finite time. In the examples discussed in this subsection, both conditions above will indeed be satisfied, so this algorithm will be sufficient to determine the answer entirely.\n\nWe now consider a pair of simple cases in which the reduction can be performed quite explicitly. Our first example will be the complete closure of a single puncture in the rank one theory of a four-punctured sphere, which as we reviewed above has as its chiral algebra the affine Lie algebra $\\wh{\\mf{so}(8)}_{-2}$. The result of this closure is expected to be the $T_2$ theory (see Figure \\ref{fig:4_to_3_reduction}). The second example will be the partial reduction (corresponding to the semi-regular embedding) of one puncture in the $T_3$ theory to produce a theory of free bifundamental hypermultiplets, which should correspond to free symplectic bosons at the level of the chiral algebra. Details of the second calculation beyond what is included in this summary can be found in Appendix \\ref{subapp:e6_to_free}.\n\n\\bigskip\n\\subsubsection*{Reducing $\\wh{\\mf{so}(8)}_{-2}$ to $\\protect\\raisebox{1pt}{$\\chi$}[T_2]$}\n\nThe starting point for our first reduction is the affine Lie algebra $\\widehat{\\mf{so}(8)}_{-2}$. We first introduce a basis for the affine currents that is appropriate for class $\\SS$ and for the reduction we aim to perform. The adjoint of $\\mf{so}(8)$ decomposes into irreps of the $\\mf{su}(2)^{(1)}\\times\\mf{su}(2)^{(2)}\\times\\mf{su}(2)^{(3)}\\times\\mf{su}(2)^{(4)}$ symmetries associated to punctures according to\n\\begin{equation}\n\\label{eq:SO8currentalgebra}\n\\rep{28}_{\\mf{so}(8)} ~\\to~\n(\\rep3,\\rep1,\\rep1,\\rep1)\\oplus\n(\\rep1,\\rep3,\\rep1,\\rep1)\\oplus\n(\\rep1,\\rep1,\\rep3,\\rep1)\\oplus\n(\\rep1,\\rep1,\\rep1,\\rep3)\\oplus\n(\\rep2,\\rep2,\\rep2,\\rep2)~.\n\\end{equation}\nAccordingly, we assemble the twenty-eight affine currents into these irreps,\n\\begin{equation}\n\\label{eq:so8_relabelled_currents}\nJ_A(z)~\\to~ \\{J^{(1)}_{(a_1b_1)}(z)~,~J^{(2)}_{(a_2b_2)}(z)~,~J^{(3)}_{(a_3b_3)}(z)~,~J^{(4)}_{(a_4b_4)}(z)~,~J_{a_1a_2a_3a_4}(z)\\}~,\n\\end{equation}\nwhere $a_I,b_I$ are fundamental indices of $\\mf{su}(2)^{(I)}$. In this basis, the OPEs of the affine Lie algebra are given by\n{\\small\n\\begin{align}\\label{eq:so8algebra}\n\\begin{split}\n\\makebox[1in][r]{$J^{(I)}_{ab}(z) J^{(J)}_{cd} (w)$}\t&~\\sim~ \\frac{- k (\\e_{ac} \\e_{bd} + \\e_{a d} \\e_{bc}) \\delta^{IJ}}{2(z-w)^2} + \\frac{f^{ef}_{ab ;cd} J^{(I)}_{ef} \\delta^{IJ}}{z-w}~,\\\\\n\\makebox[1in][r]{$J^{(1)}_{ab} (z) J_{c d e f}(w)$} \t&~\\sim~ \\frac{ \\e_{a c} J_{b d e f} + \\e_{b c} J_{a de f}}{2(z-w)}~,\\\\\n\\makebox[1in][r]{$J_{a b c d}(z) J_{efgh}(w)$} \t\t\t&~\\sim~ \\frac{k \\e_{ae} \\e_{bf} \\e_{cg} \\e_{dh}}{(z-w)^2} + \\frac{J^{(1)}_{ae} \\e_{bf} \\e_{cg} \\e_{dh} + \\e_{ae} J^{(2)}_{bf} \\e_{cg} \\e_{dh} + \\e_{ae} \\e_{bf} J^{(3)}_{cg} \\e_{dh}+ \\e_{ae} \\e_{bf} \\e_{cg} J^{(4)}_{dh}}{z-w}~,\n\\end{split}\\end{align}}%\nand similarly for the other $J^{(I)}$. Here the $\\mf{su}(2)$ structure constants are given by $f^{ef}_{ab;cd} = \\frac{1}{2} (\\e_{a c} \\delta_b^e \\delta_d^f + \\e_{bc} \\delta_a^e \\delta_d^f + \\e_{ad} \\delta_b^e \\delta_c^f + \\e_{bd} \\delta_a^e \\delta_c^f)$, and for our case of interest level is fixed to $k=-2$. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[scale=.5]{.\/figures\/4_to_3_reduction.pdf}\n\\caption{Reduction from the $\\mf{so}(8)$ theory to $T_2$.}\n\\label{fig:4_to_3_reduction}\n\\end{figure}\n\nWe will choose the first puncture to close, meaning we will perform qDS reduction on the current algebra generated by $J^{(1)}_{(ab)}$ with respect to the principal embedding,\n\\begin{equation}\n\\label{eq:so8_reduction_embedding}\n\\Lambda(t_+)= -T_{11}~,\\qquad\\Lambda(t_-)= T_{22}~,\\qquad \\Lambda(t_0)= -T_{(12)}~.\n\\end{equation}\nThe grading provided by $\\Lambda(t_0)$ is integral, so we can proceed without introducing any auxiliary grading. The only constraint to be imposed in this case is $J_{22}^{(1)}(z) = 1$. This is accomplished with the help of a single ghost pair $(c^{22},b_{22})$, in terms of which the BRST operator is given by\n\\begin{equation}\n\\label{eq:so8_brst_operator}\nd(z)=c^{22}(J_{22}-1)(z)~.\n\\end{equation}\nThe remaining three sets of $\\mf{su}(2)$ affine currents can be thought of as trivial modules of the reduced currents, while the quadrilinear currents provide a nontrivial module. In the language of the previous subsection we have\\footnote{We should note that there is something slightly unconventional about the reduction procedure here. In this example the entire starting chiral algebra is an affine current algebra, so one could in principle perform qDS reduction in the entirely standard manner. This is \\emph{not} what our prescription tells us to do. Instead, we treat a single $\\mf{su}(2)$ subalgebra as the target of the reduction, and the rest as modules. The two procedures are naively inequivalent, although we have not checked in detail to make sure that the results don't turn out the same.}\n\\begin{equation}\n\\label{eq:so8_extra_generators}\n\\{\\phi^i\\} = \\{J^{(2)}_{(a_2b_2)}~,J^{(3)}_{(a_3b_3)}~,J^{(4)}_{(a_4b_4)}~, J_{a_1a_2a_3a_4}\\}~.\n\\end{equation}\n\nThe reduced generators of step one are simply the hatted current $\\hat{J}^{(1)}_{{11}} = J^{(1)}_{{11}}$ along with the additional generators in \\eqref{eq:so8_extra_generators}. Applying the tic-tac-toe procedure produces true generators of the reduced chiral algebra,\n\\begin{align}\\label{eq:finalgeneratorsSO8}\n\\begin{split}\n\\makebox[11ex][l]{$\\hat{\\JJ}^{(1)}_{{11}}$} \t\t&~\\colonequals~\t\\hat J^{(1)}_{{11}} - \\hat J^{(1)}_{{12}} \\hat J^{(1)}_{{12}} - (k+1) \\partial(\\hat J^{(1)}_{{12}})~,\\\\\n\\makebox[11ex][l]{$(\\JJ_1)_{a_2a_3a_4}$}\t\t\t&~\\colonequals~\tJ_{{1}a_2a_3a_4} - \\hat J^{(1)}_{{12}} J_{{2}a_2a_3a_4}~,\\\\\n\\makebox[11ex][l]{$(\\JJ_2)_{a_2a_3a_4}$}\t\t\t&~\\colonequals~\tJ_{{2}a_2a_3a_4}~,\\\\\n\\makebox[11ex][l]{$\\JJ^{(I=\\{2,3,4\\})}_{a_Ib_I}$}\t&~\\colonequals~\tJ^{(I=\\{2,3,4\\})}_{a_Ib_I}~,\n\\end{split}\\end{align}\nwhere $\\hat J^{(1)}_{{12}} \\colonequals J^{(1)}_{{12}} + b_{{22}}c^{{2 2}}$. \n\nThe stress tensor of the reduced algebra takes the form given in Eqn. \\eqref{eq:qDSstresstensor}, where the original stress tensor was the Sugawara stress tensor of $\\widehat{\\mf{so}(8)}_{-2}$ and the generator of the embedded Cartan is $J_0 = - J^{(1)}_{{12}}$. We can then compute the conformal dimensions of the new generators and we find\n\\begin{align}\\begin{split}\n\\makebox[1.1in][l]{$[\\hat{\\JJ}^{(1)}_{{11}}]=2$~,}\\qquad &[\\JJ^{(I)}_{a_Ib_I}]=1~,\\\\\n\\makebox[1.1in][l]{$[(\\JJ_1)_{a_2a_3a_4}]=3\/2$~,} \\qquad &[(\\JJ_2)_{a_2a_3a_4}]=1\/2~.\n\\end{split}\\end{align}\nThe currents $\\JJ^{(I)}_{a_Ib_I}$ persist as affine currents of $\\mf{su}(2)$ subalgebras, so all of their singular OPEs with other generators are determined by the symmetry properties of the latter. Explicit calculation determines the OPEs that are not fixed by symmetry to take the following form,\n{\\small\n\\begin{align}\\begin{split}\n\\hat{\\JJ}^{(1)}_{{11}}(z)\\hat{\\JJ}^{(1)}_{{11}}(0)\t&~\\sim~ -\\frac{1}{2}\\frac{(2+k)(1+2k)(4+3k)}{z^4} - \\frac{2(2+k)\\hat{\\JJ}^{(1)}_{{11}}(0)}{z^2}-\\frac{(2+k)\\partial \\hat{\\JJ}^{(1)}_{{11}}(0)}{z}\\\\\n\\hat{\\JJ}^{(1)}_{{11}}(z) (\\JJ_1)_{a_2a_3a_4}(0) \t&~\\sim~ -\\frac{1}{2}\\frac{(2+k)(1+2k)(\\JJ_2)_{a_2a_3a_4}(0)}{z^3}-\\frac{1}{4}\\frac{(7+2k)(\\JJ_1)_{a_2a_3a_4}(0)}{z^2} - \\frac{(\\hat{\\JJ}^{(1)}_{{11}}(\\JJ_2)_{a_2a_3a_4})(0)}{z}\\\\\n\\hat{\\JJ}^{(1)}_{{11}}(z) (\\JJ_2)_{a_2a_3a_4}(0) \t&~\\sim~ \\phantom{-}\\frac{1}{4}\\frac{(1+2k)(\\JJ_2)_{a_2a_3a_4}(0)}{z^2} + \\frac{(\\JJ_1)_{a_2a_3a_4}(0)}{z}\\\\\n(\\JJ_1)_{a_2a_3a_4}(z) (\\JJ_2)_{b_2b_3b_4}(0) \t\t&~\\sim~ -\\frac{1}{2}\\frac{(1+2k)\\epsilon_{a_2b_2}\\epsilon_{a_3b_3}\\epsilon_{a_4b_4}}{z^2}+\\frac{-\\frac{1}{2}((\\JJ_2)_{a_2a_3a_4} (\\JJ_2)_{b_2b_3b_4})(0) + \\mathfrak{J}_{a_2a_3a_4;b_2b_3b_4}(0)}{z}\\\\\n(\\JJ_2)_{a_2a_3a_4}(z) (\\JJ_2)_{b_2b_3b_4}(0) \t\t&~\\sim~ \\phantom{-}\\frac{\\epsilon_{a_2b_2}\\epsilon_{a_3b_3}\\epsilon_{a_4b_4}}{z}\\\\\n(\\JJ_1)_{a_2a_3a_4}(z) (\\JJ_1)_{b_2b_3b_4}(0) \t\t&~\\sim~ \\phantom{-}\\frac{3}{4}\\frac{(1+2k)\\epsilon_{a_2b_2}\\epsilon_{a_3b_3}\\epsilon_{a_4b_4}}{z^3}+\\frac{\\frac{1}{4}(3+2k)((\\JJ_2)_{a_2a_3a_4} (\\JJ_2)_{b_2b_3b_4})(0) -\\mathfrak{J}_{a_2a_3a_4;b_2b_3b_4}(0)}{z^2}\\\\\n&\\qquad\\qquad+\\frac{\\frac{1}{4}((\\JJ_2)_{a_2a_3a_4} \\partial(\\JJ_2)_{b_2b_3b_4})(0)+\\frac{1}{2}(1+k)(\\partial(\\JJ_2)_{a_2a_3a_4} (\\JJ_2)_{b_2b_3b_4})(0)}{z}\\\\\n&\\qquad\\qquad\\qquad-\\frac{1}{2}\\frac{ \\partial\\mathfrak{J}_{a_2a_3a_4;b_2b_3b_4}(0) }{z}~,\n\\end{split}\\end{align}\n}\nwhere \n\\begin{equation}\n\\mathfrak{J}_{a_2a_3a_4;b_2b_3b_4}(z)=\\JJ^{(2)}_{a_2b_2}(z)\\epsilon_{a_3b_3}\\epsilon_{a_4b_4} + \\JJ^{(3)}_{a_3b_3}(z)\\epsilon_{a_2b_2}\\epsilon_{a_4b_4}+\\JJ^{(4)}_{a_4b_4}(z)\\epsilon_{a_2b_2}\\epsilon_{a_3b_3}~,\n\\end{equation}\nand we have removed $d$-exact terms.\n\nWe expect the result of this reduction procedure to be the trifundamental symplectic boson algebra $\\protect\\raisebox{1pt}{$\\chi$}[T_2]$, and $(\\JJ_2)_{a_2a_3a_4}(z)$ has the correct dimension and OPE to be identified with the trifundamental generator $q_{a_2a_3a_4}$. In order to complete the argument, we need all of the remaining reduced generators to be expressible as composites of this basic generator. Indeed it turns out to be a straightforward exercise to compute the null states in the reduced algebra at dimensions $h=1,\\frac32,2$ and to verify that null relations allow all the other generators to be written as normal ordered products of (derivatives of) $(\\JJ_2)_{a_2a_3a_4}(z)$. For example, we should expect that the $\\mf{su}(2)$ affine currents should be equivalent to the bilinears currents of Eqn. \\eqref{eq:T2_affine_currents}, and indeed there are null relations (only for $k=-2$) that allow us to declare such an equivalence,\n\\begin{align}\\label{eq:null_relation_free_currents}\n\\begin{split}\n\\tfrac12(\\JJ_2)_{abc}(\\JJ_2)_{a'b'c'}\\e^{bb'}\\e^{cc'} &~=~ \\JJ^{(2)}_{aa'}~,\\\\ \n\\tfrac12(\\JJ_2)_{abc}(\\JJ_2)_{a'b'c'}\\e^{aa'}\\e^{cc'} &~=~ \\JJ^{(3)}_{bb'}~,\\\\\n\\tfrac12(\\JJ_2)_{abc}(\\JJ_2)_{a'b'c'}\\e^{aa'}\\e^{bb'} &~=~ \\JJ^{(4)}_{cc'}~,\n\\end{split}\\end{align}\nAt dimensions $h=3\/2$ and $h=2$ there are additional null states for our special value of the level,\n\\begin{align}\n(\\JJ_1)_{bcd}\t\t\t&~=~ -\\tfrac{3}{2}\\partial (\\JJ_2)_{bcd} + \\tfrac{2}{3}(\\JJ_2)_{(b_1(c_1(d_1}(\\JJ_2)_{b_2)c_2)d_2)}(\\JJ_2)_{b_3c_3d_3} \\epsilon^{b_2b_3}\\epsilon^{c_2c_3}\\epsilon^{d_2d_3}~,\\\\\n\\begin{split}\n\\hat{\\JJ}^{(1)}_{{11}} &~=~ -\\tfrac{3}{4} (\\JJ_2)_{b_1c_1d_1} \\partial (\\JJ_2)_{b_2c_2d_2} \\e^{b_1b_2}\\e^{c_1c_2}\\e^{d_1d_2}\\\\ \n\t\t\t\t\t\t&~\\phantom{=}~ -\\tfrac{1}{6}(\\JJ_2)_{b_1c_1d_1}(\\JJ_2)_{(b_2(c_2(d_2}(\\JJ_2)_{b_3)c_3)d_3)}(\\JJ_2)_{b_4c_4d_4}\\e^{b_1b_2}\\e^{c_1c_2}\\e^{d_1d_2}\\e^{b_3b_4}\\e^{c_3c_4}\\e^{d_3d_4}~.\n\\end{split}\n\\end{align}\nThus all of the additional generators are realized as composites of the basic field $(\\JJ_2)_{abc}(z)$, and we have reproduced the $\\protect\\raisebox{1pt}{$\\chi$}[T_2]$ chiral algebra from qDS reduction of the $\\mf{so}(8)$ affine current algebra at level $k=-2$. We should re-emphasize that the redundancy amongst generators due to null states depends crucially on the precise value of the level. This is another instance of a general lesson that we have learned: the protected chiral algebras of $\\NN=2$ SCFTs realize very special values of their central charges and levels at which nontrivial cancellations tend to take place. We will see more of this phenomenon in the next example.\n\n\\bigskip\n\\subsubsection*{Reducing $(\\,\\wh{\\mf{e}}_6\\,)_{-3}$ to symplectic bosons}\n\nIn this case, our starting point is again an affine Lie algebra, this time $(\\wh{\\mf{e}}_6)_{-3}$. Also we are again led to decompose the adjoint representation of $\\mf{e}_6$ under the maximal $\\mf{su}(3)_1\\times\\mf{su}(3)_2\\times\\mf{su}(3)_3$ subalgebra associated to the punctures on the UV curve as was done in \\eqref{eq:e6_adjoint_decomposition}, leading to a basis of currents given by \\eqref{eq:e6_decomposed_current_list} subject to singular OPEs given by Eqn.~\\eqref{eq:e6_decomposed_current_OPEs}. Our aim is now to perform a partial reduction of the first puncture. Accordingly, we divide the generating currents as usual,\n\\begin{equation}\n\\label{eq:e6_generators_reduction_basis}\n(J^{1})_{a}^{\\phantom{a}a'}~, \\qquad\\qquad \\{\\phi_i\\} = \\{(J^{2})_{b}^{\\phantom{b}b'}~, (J^{3})_{c}^{\\phantom{c}c'}~, W_{abc}~, \\tilde W^{abc} \\}~,\n\\end{equation}\nwhere now $a,b,c$ are fundamental indices of $\\mf{su}(3)_{1,2,3}$, and the adjoint representation is represented by a fundamental and antifundamental index subject to a tracelessness condition.\n\nThe partial closing down to a minimal puncture is accomplished by means of the subregular embedding,\n\\begin{equation}\n\\Lambda(t_0) = \\frac12(T_1^{\\phantom{1}1} - T_3^{\\phantom{1}3})~,\\qquad\n\\Lambda(t_-) = T_3^{\\phantom{1}1}~,\\qquad\n\\Lambda(t_+) = T_1^{\\phantom{1}3}~.\n\\end{equation}\nThe grading induced by the embedded Cartan turns out to be half-integral in this case and must therefore be supplanted by the integral $\\delta$ grading. Under this grading the generators $\\Lambda(t_-) = T_3^{\\phantom{1}1}$ and $T_{3}^{\\phantom{1}2}$ are negative and of grade minus one. The relevant constraints are thus $\\left(J^{1}\\right)_3^{\\ 1} = 1$ and $\\left(J^{1}\\right)_3^{\\ 2} = 0.$ The implementation of these constraints via the BRST procedure introduces two ghost pairs $b_3^{\\ 1}, c_1^{\\ 3} $ and $b_3^{\\ 2},c_2^{\\ 3}.$\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[scale=.5]{.\/figures\/3_to_hyper_reduction.pdf}\n\\caption{Reduction from the $\\mf{e}_6$ theory to free hypermultiplets.}\n\\label{fig:3_to_hyper_reduction}\n\\end{figure}\n\nIn the reduction of $\\protect\\raisebox{1pt}{$\\chi$}[T_3],$ one finds that the currents $(\\hat J^{1})_{\\bar \\alpha}$ such that $T_{\\bar{\\alpha}}\\in \\ker(ad(\\Lambda(t_+))),$ are given by $(\\hat J^{1})_1^{\\ 2},(\\hat J^{1})_1^{\\ 3},(\\hat J^{1})_2^{\\ 3},$ and the current generating the reduced $\\mf{u}(1)$ symmetry\n\\begin{equation}\n\\JJ_{\\mf{u}(1)} = (\\hat J^{1})_1^{\\ 1}-2(\\hat J^{1})_2^{\\ 2}+(\\hat J^{1})_3^{\\ 3}\\;.\n\\end{equation}\nTogether with the additional generators in \\eqref{eq:e6_generators_reduction_basis}, these constitute the generators of the cohomology at the level of vector spaces. The tic-tac-toe procedure produces honest chiral algebra generators, which we denote by the calligraphic version of the same letter as the vector space generator. The quantum numbers of these redundant generators are summarized in Table \\ref{tab:T3_reduced_generators}. Their precise expressions can be found in Appendix \\ref{subapp:e6_to_free}.\n\\begin{table}[ht!]\n\\begin{center}\n\\begin{tabular}{ c|c|c|c|c }\n\\hline\\hline\n~~Generator~~ & Dimension & $U(1)$ & $SU(3)_2$ & ~~$SU(3)_3$~~ \\\\\n\\hline\n$\\JJ_{\\mf{u}(1)} $ \t\t\t\t\t& $1$ \t\t&$\\phantom{-}0$ \t\t \t& $\\bOn$ \t\t& $\\bOn$\t \t\\\\\n$(\\hat {\\JJ}^{1})_{1\\phantom{\\frac{1}{2}}}^{\\phantom{1}2}$ \t& $\\frac32$ & $\\phantom{-}3$\t\t \t& $\\bOn$ \t\t& $\\bOn$\t \t\\\\\n$(\\hat {\\JJ}^{1})_{1\\phantom{\\frac{1}{2}}}^{\\ph1 3} $ \t& $2$ \t\t& $\\phantom{-}0$ \t \t& $\\bOn$ \t\t& $\\bOn$\t\t\\\\\n$(\\hat {\\JJ}^{1})_{2\\phantom{\\frac{1}{2}}}^{\\ph1 3} $ \t& $\\frac32$ & $-3$ \t\t\t \t& $\\bOn$ \t\t& $\\bOn$\t\t\\\\\n${\\WW}_{1bc}$ \t\t\t\t\t\t\t\t& $\\frac32$ & $\\phantom{-}1$ \t& $\\bTh$ \t\t& $\\bTh$\t\t\\\\\n${\\WW}_{2bc}$ \t\t\t\t\t\t\t\t& $1$ \t\t& $-2$\t \t & $\\bTh$ \t\t& $\\bTh$\t\t\\\\\n${\\WW}_{3bc}$ \t\t\t\t\t\t\t\t& $\\frac12$ & $\\phantom{-}1$\t & $\\bTh$ \t\t& $\\bTh$\t\t\\\\\n$\\tilde{\\WW}^{1bc}$ \t\t\t\t\t\t& $\\frac12$ & $-1$ \t \t & $\\bar{\\bTh}$ \t& $\\bar{\\bTh}$\t\\\\\n$\\tilde{\\WW}^{2bc}$ \t\t\t\t\t\t& $1$ \t\t& $\\phantom{-}2$\t & $\\bar{\\bTh}$ \t& $\\bar{\\bTh}$\t\\\\\n$\\tilde{\\WW}^{3bc}$ \t\t\t\t\t\t& $\\frac32$ & $-1$\t \t & $\\bar{\\bTh}$ \t& $\\bar{\\bTh}$\t\\\\\n$(\\JJ^{2})_{b}^{\\phantom{b}b'}$ \t\t\t\t& $1$\t\t& $\\phantom{-}0$\t\t \t& $\\mathbf{8}$ \t& $\\bOn$\t\t\\\\\n$(\\JJ^{3})_{c}^{\\phantom{c}c'}$ \t\t\t\t& $1$\t\t& $\\phantom{-}0$\t\t \t& $\\bOn$ \t\t& $\\mathbf{8}$\t\\\\\n\\end{tabular}\n\\end{center}\n\\caption{The quantum numbers of redundant generators of the reduced $T_3$ chiral algebra.\\label{tab:T3_reduced_generators}}\n\\end{table}\n\nAgain, we see that there are dimension one half generators $(\\WW_3)_{bc}=W_{3bc}$ and $(\\tilde\\WW^1)^{bc}=\\tilde W^{1bc}$ that one naturally expects should be identified as the symplectic bosons of the reduced theory. Indeed, up to $d$-exact terms, the OPE for these generators is exactly what we expect from the desired symplectic bosons,\n\\begin{equation}\n\\label{eq:reduced_symplectic_boson_OPE}\n(\\WW_3)_{bc}(z) (\\tilde{\\WW}^1)^{b'c'}(0)\\sim \\frac{\\delta_{b}^{\\phantom{b}b'}\\delta_{c}^{\\phantom{c}c'}}{z}~.\n\\end{equation}\nThese generators thus have the correct dimension, charges and OPE to be identified with the expected hypermultiplet generators. Again, by studying the null relations of the reduced chiral algebra at levels $h=1,\\frac32,2$ one finds that precisely when the level $k=-3$, all of the higher dimensional generators in Table \\ref{tab:T3_reduced_generators} are related to composites of $(\\WW_3)_{bc}$ and $(\\tilde {\\WW}^1)^{bc}$ (see Appendix \\ref{subapp:e6_to_free}). In particular, one can verify that the $\\mf{u}(1)\\oplus \\mf{su}(3)_2 \\oplus \\mf{su}(3)_3$ currents are equal to their usual free field expression modulo null states.\n\\section{Cylinders and Caps}\n\\label{sec:cyl_and_cap}\n\nThe procedure we have introduced for reducing punctures is sufficiently general that there is no obstacle to formally defining chiral algebras associated to unphysical curves such as the cylinder and (decorated) cap. These are unphysical curves from the point of view of class $\\SS$ SCFTs, although they have a physical interpretation in terms of theories perturbed by irrelevant operators that correspond to assigning a finite area to the UV curve \\cite{Gaiotto:2011xs}. It would be interesting to interpret the chiral algebras associated with these curves in terms of those constructions, although naively extrapolating away from conformal fixed points seems impossible. (There are other unphysical curves, such as a thrice-punctured sphere with two minimal punctures and one maximal puncture, and the chiral algebras for these can also be defined. We focus on cylinders and caps in this section as they are particularly natural objects in the TQFT.) \n\nThe chiral algebra associated to a cylinder is a particularly natural object to consider from the TQFT perspective because it corresponds to the identity morphism (when taken with one ingoing and one outgoing leg). When taken with two ingoing or two outgoing legs, it is the chiral algebra avatar of the evaluation and coevaluation maps, respectively, of an ordinary two-dimensional TQFT. Similarly, the chiral algebra of the undecorated cap is the chiral algebra version of the trace map.\n\nOn the whole, we have not been able to systematically solve the BRST problem for these theories in the general case. This is because, as we shall see, the chiral algebras involve dimension zero (or negative dimension) operators, which prevent us from applying the simple algorithm set forth in Sec. \\ref{sec:reducing}. Nevertheless, we are able to develop a compelling picture of the mechanics of the cylinder chiral algebra. It would be interesting from a purely vertex operator algebra point of view to construct these algebras rigorously.\n\n\\bigskip\n\\input{.\/sections\/Section_5\/S5_1}\n\\bigskip\n\\input{.\/sections\/Section_5\/S5_2}\n\\bigskip\\bigskip\n\\subsection{The (decorated) cap chiral algebra}\n\\label{subsec:decorated_cap}\n\nThe chiral algebra associated to a decorated cap can be defined by partially reducing one puncture of the cylinder chiral algebra. The resulting chiral algebra should have the interesting property that if you glue it to another class $\\SS$ chiral algebra using the standard gauging BRST construction, it effectively performs the appropriate qDS reduction on the original chiral algebra.\n\nIn trying to characterize this chiral algebra, one immediately encounters the problem that it includes operators of negative dimension. Namely, consider the first steps of the general reduction procedure as applied to the cylinder chiral algebra. The (potentially redundant) generators for the decorated cap labeled by an embedding $\\Lambda$ include the usual currents $\\hat J_{\\bar\\alpha}$ for $T_{\\bar\\alpha} \\in \\ker( ad_{{\\Lambda}(t_+)})$, the dimensions of which are shifted by their ${\\Lambda}(t_0)$ weight. However, there are additional generators coming from the dimension zero bifundamental fields $g_{ab}$ of the cylinder theory. In terms of the reduced symmetry associated with the decoration, these fields are reorganized as follows: for each irrep of $\\mf{su}(2)$ in the decomposition \\eqref{eq:generaldecomposition} of the fundamental representation there are $2j+1$ generators transforming in representation $\\mf{f}\\otimes\\RR_j^{(\\mf{f})}$ with dimensions $-j,-j+1,\\ldots,j$. The dimension zero null relation corresponding to the determinant condition in the cylinder theory of the cylinder theory is expected to descend to the cap theory. The superconformal index (see App. \\ref{subapp:cylinder_cap_index}) supports this expectation, and further suggests that there may be no additional redundancies.\n\nThe existence of negative dimension operators makes this a rather exotic chiral algebra, and we will not explore it much further. Nevertheless, let us offer a couple of brief comments. In the description of the cap chiral algebra given in the previous paragraph, it is not immediately clear that an affine current algebra associated to the maximal puncture survives. However, one finds that the necessary dimension one currents can be constructed using the above fields in a manner similar to \\eqref{eqn: Rightcurrent_cylinder}, using only those elements of the left current algebra that survive in the cap chiral algebra. When gluing the cap to another theory $\\TT$, this current algebra will enter in the BRST current \\eqref{eqn: BRSTcurrentcylinder}. As in the case of the cylinder, the Gauss law constraint can be solved by constructing transferred fields, which thanks to nonzero conformal dimension of the various components of $g_{ab}$ end up with their dimensions shifted correctly. It remains to verify that restricting to the BRST cohomology removes the transferred versions of the currents $J^{\\TT}_{A}$ for $T_{A} \\not\\in \\ker( ad_{{\\Lambda}(t_+)})$.\n\\subsection{The cylinder chiral algebra}\n\\label{subsec:cylinder}\n\nThe chiral algebra associated to a cylinder should be obtained by performing a complete qDS reduction on one puncture of the trinion chiral algebra $\\protect\\raisebox{1pt}{$\\chi$}[T_n]$. In the generalized TQFT, the cylinder chiral algebra plays the role of the identity morphism for a single copy of the affine Lie algebra, ${\\rm Id}:\\wh{\\mf{su}(n)}_{-n}\\mapsto\\wh{\\mf{su}(n)}_{-n}$. The essential property associated with an identity morphism is illustrated in Figure \\ref{fig:tqft_identity}.\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[scale=.5]{.\/figures\/identity_morphism.pdf}\n\\caption{Characteristic property of the identity morphism.}\n\\label{fig:tqft_identity}\n\\end{figure}\nAs a statement about chiral algebras, the identity property is quite interesting. It means that the chiral algebra should have the property that when tensored with another class $\\SS$ chiral algebra $\\protect\\raisebox{1pt}{$\\chi$}[\\TT]$ along with the usual $(b,c)$ ghosts, restriction to the appropriate BRST cohomology produces a chiral algebra that is isomorphic to the original class $\\SS$ chiral algebra,\n\\begin{equation}\nH^*_{\\rm BRST}\\left(\\psi\\in\\protect\\raisebox{1pt}{$\\chi$}_{\\rm Id}\\otimes\\protect\\raisebox{1pt}{$\\chi$}[\\TT]\\otimes\\protect\\raisebox{1pt}{$\\chi$}_{bc}\\,\\restr{}{}\\,b_0\\psi=0\\right)\\cong\\protect\\raisebox{1pt}{$\\chi$}[\\TT]~.\n\\end{equation}\n\nAs stated above, the qDS reduction problem in this case is substantially complicated by the fact that amongst the list of naive generators of the reduced chiral algebra, there will always be dimension zero currents. Consequently, a systematic solution of the BRST problem that removes redundancies from the list of generators is difficult even in the case of the $\\protect\\raisebox{1pt}{$\\chi$}[T_2]$ and $\\protect\\raisebox{1pt}{$\\chi$}[T_3]$ theories, for which the starting point of the reduction is known. A somewhat detailed analysis of the $\\mf{su}(3)$ case can be found in Appendix \\ref{app:cylinders_and_caps}.\n\nAlthough we don't have a general first principles solution, the general structure of the reduction and our intuition gained from other examples suggests a simple characterization of the cylinder chiral algebra. We state this here as a conjecture.\n\\begin{conj}[Cylinder chiral algebra]\\label{conj_cylinder}\nThe chiral algebra associated to a cylinder of type $\\mf{su}(n)$ is finitely generated by an $\\widehat{\\mf{su}(n)}_{-n}$ affine current algebra $\\{(\\JJ_{L})_A(z)$, $A=1,\\ldots,n^2-1\\}$, along with dimension zero currents $\\{g_{ab}(z),~a,b=1,\\ldots,n\\}$ that are acted upon on the left by the affine currents. These dimension zero currents further obey a determinant condition $\\det g=1$, \\ie, they form a matrix that belongs to $SL(n,\\Cb)$.\n\\end{conj}\nThis turns out to be a surprisingly interesting chiral algebra. Let us mention a few of its properties.\n\nThe first key property -- one which is not completely obvious from the description -- is that this chiral algebra actually has two commuting $\\widehat{\\mf{su}(n)}_{-n}$ current algebras. The second set of affine currents are defined as follows\n\\begin{equation}\\label{eqn: Rightcurrent_cylinder}\n\\left(\\JJ_{R}\\right)_c^{\\ c^\\prime}(z)\\colonequals \\left(\\JJ_{L}\\right)_{b^\\prime}^{\\ b}\\ g_{b c}\\ g^{b^\\prime c^\\prime} + n\\left(g_{bc}\\ \\partial g^{bc^\\prime} - \\frac{1}{n} \\delta_c^{c^\\prime} g_{bd}\\ \\partial g^{bd} \\right)~,\n\\end{equation}\nwhere we have traded the adjoint index for a fundamental and antifundamental index satisfying a tracelessness condition, and we've also introduced the shorthand\n\\begin{equation}\\label{eqn: determinantconstraint}\ng^{ab}(z) = \\frac{1}{n!}\\epsilon^{a a_2\\ldots a_n}\\epsilon^{b b_2\\ldots b_n} \\left(g_{a_2b_2}\\ldots g_{a_nb_n}\\right)(z)~.\n\\end{equation}\nBecause of the determinant condition, this can be thought of as the inverse of $g_{ab}(z)$. The currents $(\\JJ_R)_A(z)$ act on the dimension zero currents on the right. The $\\JJ_R$ currents and the $\\JJ_L$ currents have nonsingular OPE with one another, so they generate commuting affine symmetries. These are the symmetries associated with the two full punctures of the cylinder.\n\nThe key feature of this chiral algebra should be its behavior as the identity under gluing to other class $\\SS$ chiral algebras. Let us thus consider a chiral algebra associated to a UV curve $\\CC_{g,s\\geq1}$ with at least one maximal puncture. Let us consider a general operator in this theory which will take the form $X_{a_1 a_2\\ldots a_p}^{b_1b_2\\ldots b_q}$, with $p$ fundamental indices and $q$ antifundamental indices (possibly subject to (anti)symmetrizations and tracelessness conditions) of the flavor symmetry associated to the maximal puncture and with its transformation properties under other flavor symmetries suppressed. Then our expectations is that after gluing in the cylinder, there will be a new operator of the same dimension of the same form, but where its transformation under the symmetry of the original maximal puncture has been replaced with a transformation under the symmetry at the unglued end of the cylinder.\n\nWe can see how this might come about. Gluing a cylinder to the maximal puncture means tensoring the original chiral algebra with the chiral algebra of conjecture \\ref{conj_cylinder} in addition to the usual adjoint $(b,c)$ system of dimensions $(1,0)$. We then restrict ourselves to the BRST cohomology (relative to the $b$-ghost zero modes) of the nilpotent operator\n\\begin{equation}\\label{eqn: BRSTcurrentcylinder}\nQ_{\\rm BRST} = \\oint dz\\,c^{A} ((\\JJ_L)_{A} + J^{\\TT}_{A} + \\frac12 J^{\\text{gh}}_{A})~,\n\\end{equation}\nwhere $J^{\\TT}_{A}$ is the current for the symmetry associated to the puncture on $\\CC_{g,s\\geq1}$ that is being glued. Our original operator, which was charged under the $\\mf{su}(n)$ that is being gauged and therefore does not survive the passage to BRST cohomology, has a related \\emph{transferred operator} of the following form\n\\begin{equation}\n\\hat X^{c_1 c_2\\ldots c_p}_{d_1d_2\\ldots d_q} = X_{a_1 a_2\\ldots a_p}^{b_1b_2\\ldots b_q}\\ g^{a_1c_1}\\ g^{a_2c_2}\\ldots g^{a_pc_p}\\ g_{b_1d_1}\\ g_{b_2d_2}\\ldots g_{b_qd_q}~.\n\\end{equation}\nThis operator \\emph{is} gauge invariant, since the gauged symmetry acts on $g_{ab}, g^{ab}$ on the left. In this sense the $g_{ab}$ fields effectively transfer and conjugate the symmetry from one end of the cylinder to the other. Notice that the transferred operators have the same dimension as before, because the $g_{ab}(z)$ have dimension zero. What's more, by virtue of the unit determinant condition on $g_{ab}$, we see that the OPE of the transferred fields ends up being exactly the conjugate of the OPE of the original fields. It therefore seems likely that we recover precisely the same chiral algebra on the other end of the cylinder (up to conjugation of $\\mf{su}(n)$ representations). Of course, for this construction to work we have to assume that the spectrum of physical operators will consist only of the transferred operators. It would be interesting to prove this conjecture.\n\nFinally, one can't help but notice the similarities between this description of the cylinder chiral algebra and the discussions of \\cite{Moore:2011ee} regarding the holomorphic symplectic manifold associated with the cylinder in the Higgs branch TQFT. In that work, the hyperk\\\"ahler manifold $T^* G_{\\Cb}$ was associated to the cylinder. It is interesting to note that the chiral algebra we have described in Conjecture \\ref{conj_cylinder} seems to be precisely what one obtains from studying the half-twisted $(0,2)$ supersymmetric sigma model on $G_{\\Cb}$ \\cite{Witten:2005px,Kapustin:2005pt}. Alternatively, it describes the global sections of the sheaf of chiral differential operators on $G_{\\Cb}$ as defined in \\cite{Malikov:cdr1,Malikov:cdr2,Malikov:gerbes1,Malikov:gerbes2,Malikov:gerbes3}. This connection is exciting, but remains mostly mysterious to the authors at present.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}