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\documentclass[10pt, oneside]{amsart}
\input ./combined_macros_selecta.sty
\addbibresource{Twist.bib}
\usepackage{pdflscape}
\usetikzlibrary{shapes.geometric, arrows, positioning, matrix}
\tikzstyle{s16} = [rectangle, rounded corners, text centered, draw=black,fill=black!90,text=white]
\tikzstyle{s16chiral} = [s16, dashed]
\tikzstyle{s8} = [rectangle, rounded corners, text centered, draw=black,fill=gray!30]
\tikzstyle{s4} = [rectangle, rounded corners, text centered, draw=black]
\tikzstyle{s2chiral} = [rectangle, dashed, rounded corners, text centered, draw=black,fill=green!30]
\tikzstyle{dimension} = [circle, text centered, text width=0.7cm, minimum height=0.7cm, draw=black]
\tikzstyle{arrow} = [thick,->,>=stealth]
\title[A Taxonomy of Twists of SUSY Yang--Mills Theory]{A Taxonomy of Twists of Supersymmetric Yang--Mills Theory}
\author{Chris Elliott\and Pavel Safronov \and Brian Williams}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
We give a complete classification of twists of supersymmetric Yang-Mills theories in dimensions~$2\leq n \leq 10$.
We formulate supersymmetric Yang-Mills theory classically using the BV formalism, and then we construct an action of the supersymmetry algebra using the language of $L_\infty$ algebras. For each orbit in the space of square-zero supercharges in the supersymmetry algebra, under the action of the spin group and the group of R-symmetries, we give a description of the corresponding twisted theory. These twists can be described in terms of mixed holomorphic-topological versions of Chern-Simons and BF theory.
\textbf{Keywords:} Supersymmetry, gauge theory, $L_\infty$ algebra, BV formalism,
\textbf{AMS Subject Classification:} 81Q60, 70S15
\end{abstract}
%\pagestyle{intro}
\setcounter{tocdepth}{2}
\tableofcontents
\section*{Introduction}
\label{sect:intro}
In this paper we calculate supersymmetric twists of super Yang--Mills theories in dimension 2 through 10. Our main tools are the classical Batalin--Vilkovisky formalism, which eliminates the need for auxiliary fields to close the on-shell supersymmetry action, and a consistent use of dimensional reduction which allows us to deduce lower-dimensional statements from higher-dimensional statements.
\subsection*{Classical Field Theories}
Let us begin with an informal discussion of classical field theories. A classical field theory is usually defined in terms of the data of the space of fields $\cF$ equipped with an action functional. To incorporate gauge symmetries, one may either work with $\cF$ as a stack or, as in the BRST formalism, with $\cF$ as a $Q$-manifold, i.e. a graded manifold equipped with a square-zero vector field of cohomological degree $1$ (the BRST differential). In the Batalin--Vilkovisky \cite{BatalinVilkovisky} approach one considers instead the space of BV fields $\cE$, which is equipped with a $(-1)$-shifted symplectic structure; this may be modeled by a $QP$-manifold \cite{Schwarz}. Moreover, we assume that the $Q$-structure is Hamiltonian, i.e. that it is given by a Poisson bracket $\{S, -\}$ with respect to the BV action functional. Here $\cE$ is interpreted as modelling the derived critical locus of the action functional on $\cF$.
In this paper we follow the approach developed in the works of Costello and Gwilliam \cite{CostelloBook,Book1}. As the space of BV fields $\cE$ is an infinite-dimensional manifold, it is difficult to work with it directly (for instance, to make sense of a $(-1)$-shifted symplectic structure). Instead, we zoom in on the neighborhood of a point where $Q$ vanishes (i.e. we consider a given classical solution). We may then consider $\cE$ as the space of sections of a graded vector bundle $E\rightarrow M$ over the spacetime manifold $M$. This allows us to work with finite-dimensional objects throughout. Namely, a $(-1)$-shifted symplectic structure on $\cE$ boils down to a $(-1)$-shifted symplectic pairing $E\cong E^![-1]$, where $E^!=E^*\otimes \Dens_M$. We refer to Definition \ref{def:classicalfieldtheory} for the precise definition of a classical field theory in the BV formalism that we use. Moreover, with this definition we may talk about weak equivalences of classical field theories (a notion inaccessible with $QP$-manifolds) which are simply maps of classical field theories inducing a quasi-isomorphism on $\cE$. We call these \emph{perturbative equivalences} (see Definition \ref{def:perturbativeequivalence}) to emphasize that we are working in a formal neighborhood of a given classical solution. For simplicity, throughout the paper we ignore issues of unitarity: in other words, we always consider complexified bundles of fields.
\subsection*{Classical Supersymmetric Field Theories}
Now consider a classical field theory where the spacetime manifold is $M=\RR^n$, and where the theory is translation-invariant. Given the data of a spinorial representation $\Sigma$ equipped with a symmetric pairing $\Gamma\colon\Sym^2(\Sigma)\rightarrow V=\CC^n$, we may construct a super Lie algebra of supertranslations $\fA=\Pi\Sigma\oplus V$, where $\Pi$ indicates that $\Sigma$ is placed in odd $\ZZ/2\ZZ$-degree, with the only nonvanishing Lie bracket given by $\Gamma$. A supersymmetric classical field theory is then a translation-invariant classical field theory on $\RR^n$ where the translation action on the fields is extended to an action of the super Lie algebra $\fA$. In addition, we may consider an $R$-symmetry group $G_R$ that acts on $\Sigma$ preserving $\Gamma$ and the $\so(n)$-action and also compatibly on the classical field theory.
In most literature on supersymmetry one simply tries to build an action of $\fA$ on the space of ordinary fields $\cF$. However, one often runs into a problem that the supersymmetry action is only \emph{on-shell}: the map from $\fA$ to vector fields $\mathrm{Vect}(\cF)$ preserves Lie brackets only on the critical locus of the action functional. The usual solution is to enlarge the space of fields by adding auxiliary fields with no kinetic terms on which there is an honest (\emph{off-shell}) action of $\fA$. However, this choice may be not canonical. For instance, in 10d $\cN=(1, 0)$ super Yang--Mills one needs to break the Lorentz group $\SO(10)$ to $\Spin(7)\times\SO(2)$ to have an off-shell action of a subalgebra of $\fA$ where the odd part is 9-dimensional (instead of 16-dimensional) \cite{BaulieuBerkovitsBossardMartin}.
We instead take another approach pioneered by Baulieu, Bellon, Ouvry and Wallet \cite{BaulieuBV}. Namely, one may canonically extend the supersymmetry action from the space of ordinary fields $\cF$ to the space of BV fields $\cE$. The property of the action being on-shell now means that the map $\fA\rightarrow \mathrm{Vect}(\cE)$ preserves Lie brackets, but only up to homotopy. One may then try to incorporate these homotopies: to extend the Lie action to an $L_\infty$ action. This contrasts with the auxiliary field approach of the previous paragraph, where one instead builds a resolution of the space of BV fields on which the supersymmetry Lie algebra acts strictly.
In this paper we consider supersymmetric Yang--Mills theories in dimensions 2 through 10. In dimensions 3 through 10 these may be obtained by dimensional reduction of the following theories: 10d $\cN=(1, 0)$ super Yang--Mills, 6d $\cN=(1, 0)$ super Yang--Mills, 4d $\cN=1$ super Yang--Mills and 3d $\cN=1$ super Yang--Mills. These theories depend on a choice of a Lie algebra $\fg$ equipped with a symmetric nondegenerate bilinear pairing. In addition, in dimensions 6, 4 and 3 we may add matter multiplets: in dimension 6 these depend on a choice of a symplectic $\fg$-representation (a hypermultiplet), in dimension 4 these depend on a choice of a $\fg$-representation (a chiral multiplet) and in dimension 3 these depend on a choice of an orthogonal $\fg$-representation. We do not consider superpotential, mass, or Fayet--Iliopoulos terms in this paper. Moreover, as we are working perturbatively, we ignore all topological terms ($\theta$-terms).
The on-shell supersymmetry of pure super Yang--Mills theories in these dimensions can be proven by using a well-known relationship between composition algebras (e.g. division algebras) and supersymmetry (see Section \ref{sect:compositionalgebras}) which goes back to the works \cite{Evans,KugoTownsend}. For instance, we may construct the 10d $\cN=(1, 0)$ supersymmetry from the algebra of octonions $\mathbb{O}$, 6d $\cN=(1, 0)$ supersymmetry from the algebra of quaternions $\mathbb{H}$, 4d $\cN=1$ supersymmetry from the complex numbers $\CC$ and 3d $\cN=1$ supersymmetry from the real numbers $\RR$. Our treatment follows the work of Baez and Huerta \cite{BaezHuerta} and we show how to extend the on-shell $\fA$-action to an $L_\infty$-action using these ideas. As a new result, we also construct an $L_\infty$-action on matter multiplets where the language of composition algebras turns out to be indispensable (see Section \ref{sect:mattermultipletSUSY}). Namely, for any real associative composition algebra $A_\RR$ we simply need a complex $\fg$-representation $P$ equipped with an $A_\RR$-module structure and a symmetric bilinear pairing. We have the following three cases:
\begin{itemize}
\item (\textbf{6d $\cN=(1, 0)$ supersymmetry}) For $A_\RR=\mathbb{H}$ $P$ is forced to take the form $U\otimes W_+$, where $U$ is a symplectic $\fg$-representation and $W_+$ is a 2d complex symplectic vector space (so that $\mathbb{H}\otimes_\RR\CC\cong \eend(W_+)$).
\item (\textbf{4d $\cN=1$ supersymmetry}) For $A_\RR=\mathbb{C}$ $P$ is forced to take the form $R\oplus R^*$, where $R$ is a $\fg$-representation.
\item (\textbf{3d $\cN=1$ supersymmetry}) For $A_\RR=\mathbb{R}$ we simply have an orthogonal $\fg$-representation $P$.
\end{itemize}
In addition to the dimensional reduction of these super Yang--Mills theories, there are also certain special super Yang--Mills theories with chiral supersymmetry in dimension 2: namely, 2d $\cN=(1, 0)$, $\cN=(2, 0)$ and $\cN=(4, 0)$ with matter as well as pure $\cN=(\cN_+, 0)$ theories for any $\cN_+$. We treat these separately (see Section \ref{sect:2dchiral}), but again the language of composition algebras turns out to be convenient.
\subsection*{Supersymmetric Twists}
The notion of supersymmetric twisting for a supersymmetric field theory was introduced by Witten \cite{WittenTQFT}, and further developed mathematically by Costello \cite{CostelloSUSY}.
The definitions we use in this paper will follow our previous work \cite{ElliottSafronov}, so let us briefly recall the important terminology.
Suppose $Q$ is a square-zero supercharge, i.e. an odd element $Q\in\fA$ such that $[Q, Q]=0$. Then it gives rise to a square-zero odd symplectic vector field on the space of BV fields $\cE$. In particular, we may modify the differential on $\cE$ by the replacement $\mathrm{d}\mapsto \mathrm{d}+Q$. Working up to perturbative equivalence, this turns out to drastically simplify the theory as we will shortly see.
The original classical field theory carried a $\ZZ\times\ZZ/2\ZZ$-grading, where $\ZZ$ is the cohomological (in the physics literature: ghost number) grading and $\ZZ/2\ZZ$ is the fermionic grading. We see that $\mathrm{d}$ has bidegree $(1, 0)$ while $Q$ has bidegree $(0, 1)$. So, in general the twisted theory is only $\ZZ/2\ZZ$-graded (with respect to the total grading). To improve that, we may additionally consider a homomorphism $\alpha\colon \U(1)\rightarrow G_R$ into the R-symmetry group under which $Q$ has weight $1$ and such that the $\alpha$-grading modulo 2 coincides with the fermionic grading. Then the $\alpha$-grading gives rise to a $\ZZ$-grading on the twisted theory.
Finally, let us observe that the original classical field theory carried an action of $\Spin(n)$ by rotations of $\RR^n$. But since $Q$ is not preserved under $\Spin(n)$, this action does not survive in the twisted theory. To improve that, we may consider a group $G$ with a \emph{twisting homomorphism} $G\rightarrow \Spin(n)\times G_R$ under which $Q$ is a scalar. Given such a twisting homomorphism, the twisted theory carries a $G$-action.
To summarize, supersymmetric twisting consists of the following three steps:
\begin{enumerate}
\item Choose a square-zero supercharge $Q\in\Sigma$ and modify the differential of the theory as $\mathrm{d}\mapsto\mathrm{d}+Q$.
\item Choose a group $G$ together with a twisting homomorphism $G\rightarrow \Spin(n)\times G_R$ under which $Q$ is scalar. To remove redundancy, we will assume $G\rightarrow \Spin(n)$ is an embedding.
\item Choose a homomorphism $\alpha\colon \U(1)\rightarrow G_R$ under which $Q$ has weight $1$ and such that the $\alpha$-grading modulo 2 is the fermionic grading. This step may not be possible in general.
\end{enumerate}
A classification of possible square-zero supercharges $Q$ was previously done in \cite{ElliottSafronov} and in this paper we use that classification to calculate the twist of super Yang--Mills theories on $\RR^n$ in all dimensions.
\subsection*{Supersymmetric Twists and Supergravity}
In this paper we only consider the case of global supersymmetry for super Yang--Mills theories on $\RR^n$.
In certain cases one may consider coupling of super Yang--Mills to supergravity in which case there is an interpretation of the twisting procedure as performing perturbation theory in a nontrivial supergravity background.
Let us briefly explain this perspective.
A classical solution of supergravity consists, in particular, of the following data: a spacetime manifold $M$, a $\Spin(n)$-bundle $P_\Spin\rightarrow M$ equipped with a connection (spin connection), a $G_R$-bundle $P_R\rightarrow M$ equipped with a connection and a ghost for supertranslations $\eta\in \Gamma(M, (P_\Spin\times P_R)\times^{\Spin(n)\times G_R}\Sigma)$.
The ghost $\eta$ is bosonic: it lives in bidegree $(-1,1)$ for the $\ZZ \times \ZZ/2\ZZ$-grading, so it makes sense to give it a non-zero value.
If we couple super Yang--Mills to supergravity, then the super Yang--Mills fields become sections of the associated bundles to $P_\Spin\times P_R$.
We have the following supergravity analogs of the data $(Q, \phi, \alpha)$ for supersymmetric twisting:
\begin{itemize}
\item The supergravity analog of the choice of a square-zero supercharge $Q$ is the value of the ghost $\eta$.
\item The supergravity analog of the twisting homomorphism is a choice of $G$-bundle $P_G\rightarrow M$ with connection so that $P_\Spin\times P_R$ is induced via the homomorphism $G\rightarrow \Spin(n)\times G_R$.
\item The supergravity analog of $\alpha\colon \U(1)\rightarrow G_R$ is a choice of trivial $\U(1)$-subbundle in $G_R$ on which the connection restricts to zero.
\end{itemize}
%^An approach to twisting supergravity with the goal of formulating a twisted version of the AdS/CFT correspondence has been initiated in \cite{CostelloLiSUGRA}.
\subsection*{Applications to Quantization}
The quantization of gauge theories is notoriously subtle and requires a rich theory of renormalization. One attractive application of the descriptions of the twists of supersymmetric gauge theories that we provide is to study quantization in a setting where the machinery required for renormalization is much more rigid.
To rigorously study the quantization of supersymmetric Yang-Mills theory we can work with the mathematical theory of renormalization developed by Costello in \cite{CostelloBook}. This theory of renormalization can been used to study field theories with and without supersymmetry: for example in \cite{CostelloWittengenus, LiLi, BCOV1, ChanLeungLi, GradyLiLi}. In the context of (non-supersymmetric) Yang-Mills theory for instance, it is shown that this theory recovers asymptotic freedom by an explicit analysis of the local counterterms present in the four-dimensional gauge theory \cite{EWY}.
In principle, the existence of local counterterms can be used to analyze the full untwisted supersymmetric gauge theories in a mathematically rigorous way. In practice, however, our approach to renormalization does not provide any significant advantage over traditional approaches used in QFT. However, a significant simplification happens at the level of the {\em twisted} supersymmetric Yang-Mills theories that we study in this work. To start with, for some examples (but not all), the twisted theory turns out to be a {\em topological field theory}. This occurs whenever the bracket $[Q,-]$ with the twisting supercharge surjects onto the space of translations. The theory of renormalization for topological theories can be handled using configuration spaces \cite{Kontsevich, AxelrodSinger}.
In the general setting of this paper, while not every twist results in a topological field theory, it does result in a theory in which some directions of spacetime behave topologically and the remaining directions behave holomorphically. For a mixed holomorphic-topological translation invariant field theory of this type on $\RR^n \times \CC^d$, this means that at least half of the linearly independent translation invariant vector fields act on the field theory in a BRST exact way.
Inspired by the work of Costello and Li in \cite{BCOV1} and Li in \cite{LiFeynman, LiVertex}, the foundations of renormalization for mixed holomorphic-topological field theories on Euclidean space has been developed in \cite{BWhol}. The key result is that the renormalization for mixed holomorphic-topological theories is extremely well-behaved from an analytic perspective. It is shown in the cited work that, to first order in $\hbar$, the renormalization of mixed holomorphic/topological theory is {\em finite}. Furthermore, in \cite{LiVertex}, it is shown that in real dimension two this holds to all orders in $\hbar$.
These results yield a practical approach to the problem of mathematically characterizing the one-loop quantization of every twist of supersymmetric Yang-Mills theory. Furthermore, in all examples of theories obtained via twisting occuring in dimensions 8 and lower, not much is lost when asking for the one-loop quantization. The twisted gauge theories here are all either equivalent to BF-type theories (see Section \ref{gen_BF_section}) or deformations of such theories by a holomorphic differential operator. Such theories admit prequantizations (that is, they define families of effective field theories compatible under renormalization group flow), which are exact at one loop, meaning all higher order corrections vanish identically.
From this starting point, the first natural problem would be to verify whether these one-loop exact prequantizations define actual quantizations of the classical twisted field theory. That is, for each such theory, to compute the one-loop anomalies to the solution of the quantum master equation. This problem comes in two parts: first, to compute the one-loop anomaly to quantization of the theory on flat space $\RR^n \times \CC^d$, and second -- in the case where we can use a twisting homomorphism to define the twisted theory on certain structured $(n+2d)$-manifolds, to calculate the corresponding one-loop anomaly on curved space (in other words, incorporating the computation of a gravitational anomaly). We plan to return to this question in future work.
\subsection*{The Relationship to Factorization Algebras}
In our previous paper \cite{ElliottSafronov}, we discussed supersymmetric twisting with an emphasis not on the classical fields of a supersymmetric field theory, but instead on their classical or quantum \emph{observables}. The factorization algebra formalism of Costello and Gwilliam \cite{Book1, Book2} provides a model for the local structure of the observables in a general quantum field theory. In brief, for every open subset $U \sub M$ of the spacetime manifold, one associates a (possibly $\ZZ \times \ZZ/2\ZZ$-graded) vector space $\obs(U)$ of local observables on $U$. For any pair $U_1, U_2 \sub V$ of disjoint open subsets of an open set $V$, one associates a morphism
\[m_{U_1, U_2}^V \colon \obs(U_1) \otimes \obs(U_2)\to \obs(V),\]
thought of as an \emph{operator product} for local observables. These products should vary smoothly as one varies the open subsets $U_1, U_2$ and $V$. Starting with a classical field theory on $M$, defined using the BV formalism, one can build a factorization algebra $\obs^{\mr{cl}}$ modelling the classical observables of the field theory. If the classical field theory carries the action of a group $G$, so does the associated factorization algebra. Furthermore, Costello and Gwilliam develop techniques for the quantization of such algebras of classical observables, using the theory of renormalization as discussed in the previous section.
In \cite{ElliottSafronov} we studied the supersymmetric twisting procedure as applied to factorization algebras on $\RR^n$ with an action of a supersymmetry algebra. If $Q$ is a topological supercharge, then the $Q$-twist $\obs^Q$ of a supersymmetric factorization algebra automatically satisfies a strong translation invariance condition: all translations must act homotopically trivially. In good circumstances, we can say even more. An \emph{$\bb E_n$-algebra} is an algebra over the operad of little $n$-disks; in the language above, this can be obtained from a factorization algebra for which homotopy equivalent configurations $U_1, U_2 \sub V$ induce homotopy equivalent products.
\begin{nonum}[{\cite[Theorem 3]{ElliottSafronov}}]
If $Q$ is a topological supercharge, and the operator $\obs^Q(B_r(0)) \to \obs^Q(B_R(0))$ associated to the inclusion of concentric balls is an equivalence, then the factorization algebra $\obs^Q$ has the canonical structure of an $\bb E_n$-algebra.
\end{nonum}
The hypothesis of the theorem is automatically satisfied, for example, for superconformal theories, and should be concretely checkable in examples.
In the present work we classify twists of classical field theories, to which one can associate twisted factorization algebras of classical -- and, if the appropriate anomalies vanish, quantum -- observables in the sense of our previous work. In some (topological) examples, these define $\bb E_n$-algebras. In other examples, where the twist is not fully topological, the twisted local observables will define higher analogues of vertex algebras (as in, for instance, \cite{GwilliamWilliamsKM}).
\subsection*{Summary of Twisted Super Yang--Mills Theories}
In this section we will summarize the main results of the paper presented in Part \ref{classification_part}, where we calculate twists of super Yang--Mills theories in dimensions 2 through 10.
Let us begin by explaining what we mean by ``calculation''. Recall that for a Lie algebra $\fg$ there is a $d$-dimensional topological BF theory defined on a $d$-dimensional spacetime manifold $M$ with the space of BV fields $\Omega^\bullet(M; \fg)[1]\oplus \Omega^\bullet(M; \fg^*)[d-2]$, where $\Omega^\bullet$ denotes the space of differential forms equipped with the de Rham differential $\mathrm{d}$. If $M$ is replaced by a complex manifold $X$, we may also consider its version with the space of fields $\Omega^{\bullet, \bullet}(X; \fg)[1]\oplus \Omega^{\bullet, \bullet}(X; \fg^*)[2\dim(X)-2]$, where $\Omega^{\bullet, \bullet}$ is the space of differential forms equipped with the Dolbeault differential. Finally, we have yet another version, a \emph{holomorphic BF theory}, with the space of BV fields $\Omega^{0, \bullet}(X; \fg)[1]\oplus \Omega^{\dim(X), \bullet}(X; \fg^*)[\dim(X)-2]$, again equipped with the Dolbeault differential. We will denote the space of fields in these three examples as $T^*[-1]\map(M_{\mathrm{dR}}, B\fg)$, $T^*[-1]\map(X_{\mathrm{Dol}}, B\fg)$ and $T^*[-1]\map(X, B\fg)$ respectively (the notation is explained in Section \ref{sect:FMPs}). We may also combine these three examples into what we call a \emph{generalized BF theory} with the spaces of fields $T^*[-1]\map(X\times Y_{\mathrm{Dol}}\times M_{\mathrm{dR}}, B\fg)$ (see Definition \ref{def:generalizedBF} for more details).
Let us also recall that if $\fg$ is equipped with a symmetric bilinear nondegenerate pairing, we also have a 3-dimensional topological Chern--Simons theory. If we forgo $\ZZ$-gradings and work with $\ZZ/2\ZZ$-gradings, we may also consider a topological Chern--Simons theory in any odd dimension (see \cite{AlekseevMnev} for a 1-dimensional version and \cite{BakGustavsson2} for a 5-dimensional version). Just like for the BF theory, we also have two other versions which may be combined into a generalized Chern--Simons theory. Another direction we can generalize in is to replace the Lie algebra $\fg$ by a dg Lie algebra, in which case the BF theory itself becomes a particular example of the Chern--Simons theory.
Our goal will then be to show that a particular twist of super Yang--Mills is equivalent to a given generalized Chern--Simons theory. We summarize our results in two forms. In Tables \ref{table_of_twists_16}, \ref{table_of_twists_8} and \ref{table_of_twists_4} we summarize all the possible twists of dimensional reductions of the 10d $\cN=(1, 0)$, 6d $\cN=(1, 0)$ and 4d $\cN=1$ super Yang--Mills theories respectively. In Table \ref{table_of_twists_2d} we summarize the twists of 2d supersymmetric Yang-Mills theory. Before these tables, we will give a short description of each twisted theory in a more physical language, with references to where in the literature it was previously considered.
\textbf{Dimension 10}
\begin{itemize}
\item \emph{$\mc N=(1,0)$ holomorphic twist.} The unique twisted super Yang--Mills theory in 10 dimensions is equivalent to the 5d holomorphic Chern--Simons theory defined on a Calabi--Yau 5-fold. Note that this theory is only $\ZZ/2\ZZ$-graded. This twist was first studied by Baulieu \cite{Baulieu}. As is well-known \cite{GSanomaly}, the theory has a one-loop anomaly and does not admit a quantization.
\end{itemize}
\textbf{Dimension 9}
\begin{itemize}
\item \emph{$\mc N=1$ minimal twist.} The unique twisted super Yang--Mills theory in 9 dimensions is equivalent to a generalized version of the Chern--Simons theory defined on a product of a Calabi--Yau 4-fold and a real 1-manifold. Note that this theory is only $\ZZ/2\ZZ$-graded. Its classical solutions are $G$-bundles holomorphic along the Calabi--Yau manifold and flat along the 1-manifold.
\end{itemize}
\textbf{Dimension 8}
\begin{itemize}
\item \emph{$\mc N=1$ holomorphic twist.} Super Yang--Mills theory in 8 dimensions admits three classes of twists. The minimal twist, by a holomorphic (or, equivalently, pure) supercharge, is equivalent to a holomorphic version of the BF theory defined on a complex 4-fold.
\item \emph{$\mc N=1$ intermediate twist.} The holomorphic twist admits a deformation to a twist by a rank 1 impure spinor. This theory is equivalent to a generalized version of the Chern--Simons theory defined on a product of a Calabi--Yau 3-fold and an oriented surface. Note that this theory is only $\ZZ/2\ZZ$-graded.
\item \emph{$\mc N=1$ topological twist.} The holomorphic twist also admits a deformation to a topological twist defined on $\spin(7)$-manifolds. This theory is perturbatively trivial, in the sense that the classical BV complex is contractible. It was studied in \cite{AcharyaOLoughlinSpence,BaulieuKannoSinger}. The partition function of this theory counts $\spin(7)$-instantons modulo gauge \cite{Lewis,DonaldsonThomas,ReyesCarrion}. If we denote by $\Omega$ the Cayley 4-form on a $\spin(7)$-manifold $M$, then the classical solutions are given by principal $G_\RR$-bundles on $M$ (where $G_\RR$ is a compact Lie group) together with a connection $A$, such that its curvature $F_A$ satisfies the equation
\begin{equation}
F = \ast (\Omega\wedge F).
\end{equation}
\end{itemize}
\textbf{Dimension 7}
\begin{itemize}
\item \emph{$\mc N=1$ minimal twist.} The twists of super Yang--Mills theory in 7 dimensions arise by dimensionally reducing the twists in 8 dimensions. The minimal twist, by a pure spinor, is equivalent to a generalized version of the BF theory defined on a product of a complex 3-fold and a real 1-manifold.
\item \emph{$\mc N=1$ intermediate twist.} The minimal twist admits a deformation to a twist by a rank 1 impure spinor. This theory is equivalent to a generalized version of the Chern--Simons theory defined on a product of a Calabi--Yau surface and a real oriented 3-manifold. Note that this theory is only $\ZZ/2\ZZ$-graded.
\item \emph{$\mc N=1$ topological twist.} The minimal twist also admits a deformation to a topological twist defined on $G_2$-manifolds. This theory is again perturbatively trivial. It was studied in \cite{AcharyaOLoughlinSpence, BaulieuKannoSinger}. The partition function of this theory counts $G_2$-monopoles modulo gauge \cite{DonaldsonSegal}. If we denote by $\psi$ the calibration 4-form on a $G_2$-manifold $M$, then the classical solutions are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$ and a section $\sigma\in\Gamma(M, \mathrm{ad} P)$ satisfying
\begin{equation}
\d_A\sigma = \ast(F\wedge \psi).
\end{equation}
\end{itemize}
\textbf{Dimension 6}
\begin{itemize}
\item \emph{$\mc N=(1,0)$ and $\mc N=(1,1)$ holomorphic twist.} The holomorphic twist of the 6d $\cN=(1, 0)$ super Yang--Mills theory with matter valued in a symplectic $G$-representation $U$ is equivalent to the theory whose classical solutions are holomorphic maps from a Calabi--Yau 3-fold $X$ to the Hamiltonian reduction of $U$ (a holomorphic version of the gauged Rozansky--Witten model). In general, this theory is only $\ZZ/2\ZZ$-graded. If $U=T^*R$, the theory is $\ZZ$-graded and is the cotangent theory to the space of holomorphic maps from a complex 3-fold to $R/G$. 6d $\cN=(1, 1)$ super Yang--Mills corresponds to the special case $R=\fg$. This twist is also calculated in a forthcoming work of Dylan Butson \cite{Butson}.
\item \emph{$\mc N=(1,1)$ rank $(2, 2)$ twist.} In the $\mc N=(1,1)$ case there are two intermediate twists. The one by a supercharge of rank $(2, 2)$ is equivalent to a generalized version of the Chern--Simons theory defined on a product of a Calabi--Yau curve and a real oriented 4-manifold. Note that this theory is only $\ZZ/2\ZZ$-graded.
\item \emph{$\mc N=(1,1)$ special rank $(1,1)$ twist.} The other intermediate twist, by a rank $(1, 1)$ supercharge, is equivalent to a generalized form of the BF theory defined on a product of a complex surface and a real surface.
\item \emph{$\mc N=(1,1)$ topological twist.} The special rank $(1, 1)$ twist admits a deformation to a topological twist defined on Calabi--Yau 3-folds. This theory is perturbatively trivial. It was studied in \cite{AcharyaOLoughlinSpence, BaulieuKannoSinger}. The partition function of this theory counts solutions to the Donaldson--Thomas equations \cite{Thomas}. If we denote by $\omega$ the K\"ahler form on a Calabi--Yau 3-fold $M$, then the classical solutions are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$ and a 3-form $u\in\Omega^{0, 3}(M, \mathrm{ad} P\otimes_\RR \CC)$ satisfying
\begin{align}
F_{0, 2} + \ol\dd^*_A u &= 0 \\
F_{1, 1}\wedge \omega^2 + [u, \bar{u}] &= 0.
\end{align}
\end{itemize}
\textbf{Dimension 5}
\begin{itemize}
\item \emph{$\mc N=1$ and $\mc N=2$ minimal twist.} The minimal twist of the 5d $\cN=1$ super Yang--Mills theory with matter valued in a symplectic $G$-representation $U$ is equivalent to a theory defined on a product of a Calabi--Yau surface $X$ and a real 1-manifold $M$. The classical solutions are given by maps from $X\times M$ to the symplectic reduction of $U$ holomorphic along $X$ and locally-constant along $M$. In general, this theory is only $\ZZ/2\ZZ$-graded. It was studied by K\"all\'en and Zabzine \cite{KallenZabzine}. If $U=T^* R$, the theory is $\ZZ$-graded and is the cotangent theory to the space of maps from $X\times M$ to $R/G$ holomorphic along $X$ and locally-constant along $M$. 5d $\cN=2$ super Yang--Mills corresponds to the special case $R=\fg$.
\item \emph{$\mc N=2$ intermediate twist.} In the $\mc N=2$ case the minimal twist admits a deformation to an intermediate twist. The corresponding theory is equivalent to a generalized BF theory defined on a product of a complex curve $C$ and a real 3-manifold $M$. The twist was considred in \cite{ElliottPestun} in the case $M=S^1\times \Sigma$ for a Riemann surface $\Sigma$, where the moduli space can be viewed as a multiplicative version of the Hitchin system.
\item \emph{$\mc N=2$ topological A twist.} The intermediate twist admits a deformation to two topological twists. One, which we refer to as the A-twist, arises by dimensionally reducing the topological twist of 6d $\mc N=(1, 1)$ super Yang--Mills theory. This theory is perturbatively trivial. It was studied by Qiu and Zabzine, \cite{QiuZabzine} (see also \cite{Anderson} for a discussion of the twisting homomorphism). The partition function of this theory counts solutions to the Haydys--Witten equations \cite{Haydys,WittenFivebranes}. Let $M$ be a $K$-contact manifold and denote by $R$ the Reeb vector field. The classical solutions are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$ and a section $B\in\Omega^2(M; \mathrm{ad} P)$ satisfying a self-duality equation $\iota_R \ast B = B$ which together satisfy the following equations (see \cite[equations (4) and (5)]{QiuZabzine}; we refer there for the explanation of the notation):
\begin{align}
\iota_R F - (d_A^* B)^H &= 0 \\
F^+_H - \frac{1}{4}B\times B - \frac{1}{2}\iota_R d_A B &= 0.
\end{align}
\item \emph{$\mc N=2$ topological B twist.} Finally, the other topological twist, associated to a rank 4 supercharge, can be identified with a 5d Chern--Simons theory defined on an oriented 5-manifold. Note that this theory is only $\ZZ/2\ZZ$-graded. This twist was identified in work of Geyer--M\"ulsch and of Bak--Gustavsson \cite{GeyerMuelsch, BakGustavsson1,BakGustavsson2}.
\end{itemize}
\textbf{Dimension 4}
\begin{itemize}
\item \emph{$\mc N=1$ holomorphic twist.} The holomorphic twist of the 4d $\cN=1$ super Yang--Mills theory with matter valued in a $G$-representation $R$ is equivalent to the cotangent theory of the theory of holomorphic maps from a complex surface $X$ to $R/G$. This twist was studied by Johansen \cite{Johansen}.
\item \emph{$\cN=2$ and $\cN=4$ holomorphic twist.} We may also consider holomorphic twists of 4d $\cN=2$ and 4d $\cN=4$ super Yang--Mills theories. The holomorphic twist of 4d $\cN=2$ super Yang--Mills with matter valued in a symplectic $G$-representation $U$ is equivalent to the cotangent theory of the theory of holomorphic maps from a Calabi--Yau surface $X$ to the Hamiltonian reduction of $U$. The d $\cN=4$ super Yang--Mills theory corresponds to the case $U=T^*\fg$ in which case the space of classical solutions is a $(-1)$-shifted cotangent bundle to the moduli stack of $G$-Higgs bundles on a complex surface $X$.
\item \emph{$\mc N=2$ and $\mc N=4$ intermediate twist.} There is a deformation of the $\cN=2$ holomorphic twist which is equivalent to a theory of maps from a product of a Calabi--Yau curve $C$ and an oriented surface $\Sigma$ into the Hamiltonian reduction of $U$ which are holomorphic along $C$ and locally-constant along $\Sigma$. This twist was previously studied by Kapustin \cite{KapustinHolo}.
\item \emph{$\mc N=2$ topological rank (2, 0) twist.} The $\cN=2$ holomorphic twist admits a deformation to a topological twist, the \emph{Donaldson twist}. This theory is perturbatively trivial. This theory was first considered in \cite{WittenTQFT}, and the coupling to matter was studied in \cite{AnselmiFre, AlvarezLabastida, HyunParkPark}. The partition function counts solutions to nonabelian Seiberg--Witten equations \cite{Pidstrigach}. Let $G_\RR$ be a compact Lie group and $U$ a quaternionic-unitary $G_\RR$-representation. In particular, $U$ carries a commuting $\SU(2)$-action given by unit quaternions. Suppose $M$ is a spin 4-manifold and let $P_{\Spin}\rightarrow M$ be the corresponding $\Spin(4)$-principal bundle. The classical solutions in this theory are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$ and a section $u\in\Gamma(M, (P\times P_{\Spin})\times^{G_\RR\times \Spin(4)} U)$ which together satisfy
\begin{align}
\sd \d_A u &= 0 \label{eq:SW1} \\
F^+ + \Phi(u) &= 0, \label{eq:SW2}
\end{align}
where $\Phi$ is the moment map and $\sd\d_A$ is the Dirac operator (we refer to \cite{Pidstrigach,HaydysDirac} for more details).
\item \emph{$\cN=4$ topological rank $(2, 0)$ twist.} The same twist may be considered for the 4d $\cN=4$ super Yang--Mills theory, in which case it has three compatible twisting homomorphisms \cite{Yamron}, i.e. there are three ways of interpreting the differential equations on arbitrary oriented 4-manifolds. First, considering the 4d $\cN=4$ super Yang--Mills theory as a 4d $\cN=2$ super Yang--Mills theory with $U=\fg\otimes\mathbb{H}$, we obtain a theory which counts solutions to the nonabelian Seiberg--Witten equations \eqref{eq:SW1}, \eqref{eq:SW2}. Another twisting homomorphism was studied by Vafa and Witten \cite{VafaWitten}. The corresponding theory counts solutions of the Vafa--Witten equations on an oriented 4-manifold $M$. The classical solutions are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$, a self-dual two-form $B\in\Omega^{2, +}(M, \mathrm{ad} P)$ and a section $C\in\Gamma(M, \mathrm{ad} P)$ which satisfy
\begin{align}
-\d_A C + \d_A^* B &= 0 \\
F^+ - \frac{1}{4}B\times B - \frac{1}{2}[C, B] &= 0.
\end{align}
Finally, the third twisting homomorphism was studied by Marcus \cite{Marcus} and Kapustin and Witten \cite{KapustinWitten}. The classical solutions are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$ and a one-form $\phi\in\Omega^1(M, \mathrm{ad} P)$ which satisfy
\begin{align}
(F-\phi\wedge \phi)^+ &= 0\\
(\d_A \phi)^- &= 0 \\
\d^*_A \phi &= 0.
\end{align}
\item \emph{$\mc N=4$ topological B twist.} For the 4d $\cN=4$ super Yang--Mills theory there is a single topological twist which is not perturbatively trivial. It is equivalent to a topological BF theory defined on a 4-manifold (this theory corresponds to the value $t=\pm i$ of the family considered in \cite{KapustinWitten}).
\item \emph{$\cN=4$ topological rank $(2, 2)$ twist.} The $\cN=4$ topological B twist admits a deformation to a perturbatively trivial theory. The corresponding deformation is parametrized by $s\in\CC^\times$, where $s=1$ is the topological B twist. Choosing a parameter $t\in\CC^\times$ satisfying $s=-t^2$, the theory counts solutions of the Kapustin--Witten equations on an oriented 4-manifold $M$. The classical solutions are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$ and a one-form $\phi\in\Omega^1(M, \mathrm{ad} P)$ which satisfy
\begin{align}
(F-\phi\wedge\phi +t \d_A \phi)^+ &= 0 \\
(F-\phi\wedge\phi -t^{-1}\d_A \phi)^- &= 0 \\
\d^*_A \phi &= 0.
\end{align}
\item \emph{$\cN=4$ topological rank $(2, 1)$ twist.} The $\cN=4$ intermediate twist also admits a deformation to a perturbatively trivial theory defined on K\"ahler surface $M$ given by twisting by a rank $(2, 1)$ supercharge. The corresponding equation is a deformation of the Kapustin--Witten equations using the K\"ahler form.
%The classical solutions of this theory are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$ and a one-form $\phi\in\Omega^1(M, \mathrm{ad} P)$ which satisfy
%\begin{align}
%(F-\phi\wedge\phi +\d_A \phi)^+ + [F-\phi\wedge\phi, \omega] &= 0 \\
%(F-\phi\wedge\phi -\d_A \phi)^- &= 0 \\
%\d^*_A \phi &= 0.
%\end{align}
%Here $\omega$ is the K\"ahler form on $M$ and $[-, -]$ is the natural Lie bracket on differential forms given by identifying $\wedge^2 T^*_M\cong \mathfrak{so}(T_M)$.
An analysis of topological twists of the 4d $\cN=4$ super Yang--Mills theory using similar techniques to this paper, but with the aim of obtaining the full derived stack of solutions to the equations of motion, rather than only the perturbative classical field theory, was carried out in \cite{ElliottYoo1}.
\end{itemize}
\textbf{Dimension 3}
\begin{itemize}
\item \emph{$\cN=2$ minimal twist.} The minimal twist of the 3d $\cN=2$ super Yang--Mills theory with matter valued in a $G$-representation $R$ is defined on a product $C\times L$ of a complex curve $C$ and a real 1-manifold $L$. It is equivalent to the cotangent theory of the theory of maps from $C\times L$ to $R/G$ which are holomorphic along $C$ and locally-constant along $L$.
\item \emph{$\cN=4$ and $\cN=8$ minimal twist.} We may also consider the minimal twist of $\cN=4$ and $\cN=8$ super Yang--Mills theories. The minimal twist of the $\cN=4$ super Yang--Mills theory with matter valued in a symplectic $G$-representation $U$ is equivalent to the cotangent theory of the theory of maps from $C\times L$ to the Hamiltonian reduction of $U$ which are holomorphic along $C$ and locally-constant along $L$. The 3d $\cN=8$ theory corresponds to the case $U=T^*\fg$.
\item \emph{$\mc N=4$ topological A twist.} In 3d $\cN=4$ super Yang--Mills theory we may consider a deformation of the minimal twist which gives rise to a perturbatively trivial topological theory defined on spin 3-manifolds. This twist was studied in \cite{BaulieuGrossman,BlauThompson1,Ohta}. From a mathematical point of view the space of states on a two-sphere is studied in \cite{BFN}. The partition function counts solutions to a 3-dimensional version of the nonabelian Seiberg--Witten equations \eqref{eq:SW1}, \eqref{eq:SW2}. Let $G_\RR$ be a compact Lie group and $U$ a quaternionic-unitary $G_\RR$-representation. Let $M$ be a spin 3-manifold and let $P_{\Spin}\rightarrow M$ be the corresponding $\Spin(3)$-principal bundle. The classical solutions in this theory are given by principal $G_\RR$-bundles $P\rightarrow M$ together with a connection $A$, a section $\sigma\in\Gamma(M, \mathrm{ad} P)$ and a section $u\in\Gamma(M, (P\times P_{\Spin})\times^{G_\RR\times \Spin(3)} U)$ which together satisfy
\begin{align}
\sd \d_A u + [\sigma, u] &= 0 \label{eq:3dSW1} \\
F + \ast\d_A \sigma + \Phi(u) &= 0. \label{eq:3dSW2}
\end{align}
\item \emph{$\cN=8$ topological A twist.} We may regard the 3d $\cN=8$ super Yang--Mills theory as a 3d $\cN=4$ super Yang--Mills theory with matter valued in $U=\fg\otimes\mathbb{H}$. In particular, the partition function in the twisted theory counts solutions to the equations \eqref{eq:3dSW1}, \eqref{eq:3dSW2}. We may also consider a different twisting homomorphism obtained by dimensionally reducing the Vafa--Witten or Kapustin--Witten twisting homomorphism. The classical solutions in this theory are given by principal $G$-bundles $P\rightarrow M$ ($G$ is the complexification of the compact Lie group $G_\RR$) together with a connection $A$ and a section $\sigma\in\Gamma(M, \mathrm{ad} P)$ satisfying a complexified version of the Bogomolny equation:
\[
F + \ast \d_A \sigma = 0.
\]
The corresponding field theory in the formalism of extended topological field theories is studied in \cite{BZGN}.
\item \emph{$\mc N=4$ and $\mc N=8$ topological B twist.} The minimal twist of the 3d $\cN=4$ super Yang--Mills theory also admits another deformation to a gauged version of the Rozansky--Witten model valued in the Hamiltonian reduction $U\ham G$ \cite{RozanskyWitten,BlauThompson2}.
\end{itemize}
\textbf{Dimension 2}
\begin{itemize}
\item \emph{$\mc N=(2,2), (4,4), (8,8)$ holomorphic twist.} There is a holomorphic twist in dimension 2 which is defined on complex curves $C$. The twist of 2d $\cN=(2, 2)$ super Yang--Mills theory with matter valued in a $G$-representation $R$ is equivalent to the cotangent theory to the theory of holomorphic maps from $C$ to $R/G$. The twist of 2d $\cN=(4, 4)$ super Yang--Mills theory with matter valued in a symplectic $G$-representation $U$ is equivalent to the cotangent theory to the theory of holomorphic maps from $C$ to the Hamiltonian reduction $U\ham G$. Finally, the case of 2d $\cN=(8, 8)$ super Yang--Mills theory corresponds to choosing $U=T^*\fg$.
\item \emph{$\mc N=(2,2), (4,4), (8,8)$ topological A twist.} In each of these cases, the minimal twist can again be deformed to a topological theory in two inequivalent ways. The first is a perturbatively trivial theory, the gauged A-model. We begin with a description of the twist of the 2d $\cN=(2, 2)$ super Yang--Mills theory with gauge group $G_\RR$ (a compact Lie group) and matter valued in a unitary $G_\RR$-representation $R$ equipped with a moment map $\Phi$. The partition function counts solutions to symplectic vortex equations \cite{CGMRS}. Let $\Sigma$ be an oriented surface equipped with an almost complex structure and a square root $S$ of the line bundle of densities $\Dens_\Sigma$. The classical solutions are given by principal $G_\RR$-bundles $P\rightarrow \Sigma$ equipped with a connection $A$ and a section $u\in\Gamma(M, ((P\times^{G_\RR} R)\otimes_\RR S)$ which satisfy
\begin{align}
\ol\dd_A u &= 0 \label{eq:symplecticvortex1} \\
F + \Phi(u) &= 0. \label{eq:symplecticvortex2}
\end{align}
Next, consider the twist of the 2d $\cN=(4, 4)$ super Yang--Mills theory with a complexified gauge group $G$ and matter valued in a complex symplectic $G$-representation $U$ equipped with a moment map $\Phi$. In this case the classical solutions are given by principal $G$-bundles $P\rightarrow \Sigma$ equipped with a connection $A$ and a section $u\in\Gamma(\Sigma, (P\times^G U)\otimes_\RR S)$ which satisfy a complexified version of \eqref{eq:symplecticvortex1}, \eqref{eq:symplecticvortex2}:
\begin{align}
\ol\dd_A u &= 0 \\
F + \Phi(u) &= 0.
\end{align}
Finally, consider the twist of the 2d $\cN=(8, 8)$ super Yang--Mills theory. The classical solutions are given by principal $G$-bundles $P\rightarrow \Sigma$ equipped with a connection $A$ and sections $u_1\in\Gamma(\Sigma, \mathrm{ad} P)$, $u_2\in\Omega^{1, 1}(\Sigma, \mathrm{coad} P)$ which satisfy
\begin{align}
\ol\dd_A u_1 &= 0 \\
\ol\dd_A u_2 &= 0 \\
F + 2(u_1, u_2) &= 0.
\end{align}
\item \emph{$\mc N=(2,2), (4,4), (8,8)$ topological B twist.} The other topological twist gives rise to a gauged B-model. The twist of the 2d $\cN=(2, 2)$ super Yang--Mills theory with complexified gauge group $G$ and matter valued in a $G$-representation $R$ is equivalent to the cotangent theory to the theory of locally-constant maps from a surface $\Sigma$ to $R/G$. The twist of the 2d $\cN=(4, 4)$ super Yang--Mills theory with complexified gauge group $G$ and matter valued in a symplectic $G$-representation $U$ is equivalent to the cotangent theory of the theory of locally-constant maps from $\Sigma$ to the Hamiltonian reduction $U\ham G$. Finally, the case of the 2d $\cN=(8, 8)$ super Yang--Mills theory corresponds to choosing $U=T^*\fg$. The study of the topological twists of 2d $\mc N=(2,2)$ supersymmetric field theories goes back to the works of Eguchi and Yang \cite{EguchiYang} and Witten \cite{Wittenmirror}.
\item \emph{$(\cN, 0)$ holomorphic twist.} Theories with chiral supersymmetry in 2 dimensions (i.e. with 2d $(\cN, 0)$ supersymmetry) only admit a holomorphic twist. The corresponding twisted theory is equivalent to a cotangent theory to the theory of holomorphic maps from a complex curve $C$ to $\fg^{\cN-2} / G$. Twisted 2-dimensional $(2,0)$ $\sigma$-models were first studied by Witten in \cite{Wittenmirror}, and can be used to obtain the chiral algebra of chiral differential operators \cite{WittenCDO}.
\end{itemize}
\begin{table}[htbp]
\centering
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{c|c|c|c|c}
$d$ & $\mc N$ & Twist & Description & Invariant Directions \\
\hline
\multirow{2}*{10} & \multirow{2}*{$(1,0)$} & \multirow{2}*{\hyperref[sect:10dholomorphictwist]{Rank $(1,0)$}} & Holomorphic Chern-Simons Theory & \multirow{2}*{5 (holomorphic)} \\
&&&$\mr{Map}(\CC^5, B\gg)$ &\\ \hline
\multirow{2}*{9} & \multirow{2}*{1} & \multirow{2}*{\hyperref[sect:9dminimaltwist]{Rank 1}} & Generalized Chern-Simons Theory & \multirow{2}*{5 (minimal)} \\
&&&$\mr{Map}(\CC^4 \times \RR_{\mr{dR}}, B\gg)$ &\\ \hline
\multirow{6}*{8} & \multirow{6}*{1} & \multirow{2}*{\hyperref[sect:8dholomorphictwist] {Rank $(1,0)$ pure}} & Holomorphic BF Theory & \multirow{2}*{4 (holomorphic)} \\
&&& $T^*[-1]\mr{Map}(\CC^4, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:8dpartiallytopologicaltwist]{Rank $(1,1)$}} & Generalized Chern-Simons Theory& \multirow{2}*{5} \\
&&& $\mr{Map}(\CC^3 \times \RR^2_{\mr{dR}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:8dtopologicaltwist]{Rank $(1,0)$ impure}} & Perturbatively trivial ($\Spin(7)$ Instanton)& \multirow{2}*{8 (topological)} \\
&&& $\mr{Map}(\CC^4, B\gg)_\mr{dR}$ & \\ \hline
\multirow{6}*{7} & \multirow{6}*{1} & \multirow{2}*{\hyperref[sect:7dminimaltwist] {Rank $1$ pure}} & Generalized BF Theory & \multirow{2}*{4 (minimal)} \\
&&& $T^*[-1]\mr{Map}(\CC^3 \times \RR_{\mr{dR}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:7dpartialtwist] {Rank $2$}} & Generalized Chern-Simons Theory& \multirow{2}*{5} \\
&&& $\mr{Map}(\CC^2 \times \RR^3_{\mr{dR}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:7dtopologicaltwist]{Rank $1$ impure}} & Perturbatively trivial ($G_2$ Monopole)& \multirow{2}*{7 (topological)} \\
&&& $\mr{Map}(\CC^3 \times \RR_{\mr{dR}}, B\gg)_\mr{dR}$ & \\ \hline
\multirow{8}*{6} & \multirow{8}*{$(1,1)$} & \multirow{2}*{\hyperref[sect:6d11holomorphictwist]{Rank $(1,0)$}} & {Holomorphic BF Theory} & \multirow{2}*{3 (holomorphic)} \\
&&& $T^*[-1]\mr{Map}(\CC^3, \gg/\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:6d11partialtwist]{Rank $(1,1)$ special}} & {Generalized BF Theory} & \multirow{2}*{4} \\
&&& $T^*[-1]\mr{Map}(\CC^2 \times \RR^2_{\mr{dR}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:6drank22twist]{Rank $(2,2)$}} & {Generalized Chern-Simons Theory} & \multirow{2}*{5} \\
&&& $\mr{Map}(\CC \times \RR^4_{\mr{dR}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:6d11topologicaltwist]{Rank $(1,1)$ generic}} & {Perturbatively trivial} & \multirow{2}*{6 (topological)} \\
&&& $\mr{Map}(\CC^2 \times \RR^2_{\mr{dR}}, B\gg)_{\mr{dR}}$ & \\ \hline
\multirow{8}*{5} & \multirow{8}*{$2$} & \multirow{2}*{\hyperref[sect:5dminimaltwist]{Rank $1$}} & {Generalized BF Theory} & \multirow{2}*{3 (minimal)} \\
&&& $T^*[-1]\mr{Map}(\CC^2 \times \RR_{\mr{dR}}, \gg/\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:5dpartialtwist]{Rank $2$ special}} & {Generalized BF Theory} & \multirow{2}*{4} \\
&&& $T^*[-1]\mr{Map}(\CC \times \RR^3_{\mr{dR}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:5drank4twist]{Rank $4$}} & {5d Chern-Simons Theory} & \multirow{2}*{5 (topological)} \\
&&& $\map(\RR^5_{\mr{dR}}, B\fg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:5drank2topologicaltwist] {Rank $2$ generic}} & {Perturbatively trivial} & \multirow{2}*{5 (topological)} \\
&&& $\mr{Map}(\CC \times \RR^3_{\mr{dR}}, B\gg)_{\mr{dR}}$ & \\ \hline
\multirow{12}*{4} & \multirow{12}*{$4$} & \multirow{2}*{\hyperref[sect:4d4holomorphictwist] {Rank $(1,0)$}} & {Holomorphic BF Theory} & \multirow{2}*{2 (holomorphic)} \\
&&& $T^*[-1]\mr{Map}(\CC^2_{\mr{Dol}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:4d4partialtwist] {Rank $(1,1)$}} & Generalized BF Theory & \multirow{2}*{3} \\
&&& $T^*[-1]\mr{Map}(\CC_{\mr{Dol}} \times \RR^2_{\mr{dR}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:4dqgltwist] {Rank $(2,2)$ special}} & BF Theory & \multirow{2}*{4 (topological)} \\
&&& $T^*[-1]\mr{Map}(\RR^4_{\mr{dR}}, B\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:4d4partialtwist] {Rank $(2,1)$}} & {Perturbatively trivial} & \multirow{2}*{4 (topological)} \\
&&& $\mr{Map}(\CC_{\mr{Dol}} \times \RR^2_{\mr{dR}}, B\gg)_{\mr{dR}}$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:4d4Atwist] {Rank $(2,0)$}} & {Perturbatively trivial } & \multirow{2}*{4 (topological)} \\
&&& $\mr{Map}(\CC^2_{\mr{Dol}}, B\gg)_{\mr{dR}}$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:4dqgltwist] {Rank $(2,2)$ generic}} & {Perturbatively trivial} & \multirow{2}*{4 (topological)} \\
&&& $\mr{Map}(\RR^4_{\mr{dR}}, B\gg)_{\mr{dR}}$ & \\ \hline
\multirow{8}*{3} & \multirow{8}*{$8$} & \multirow{2}*{\hyperref[sect:3d8minimal_twist] {Rank $1$}} & {Generalized BF Theory} & \multirow{2}*{2 (minimal)} \\
&&& $T^*[-1]\mr{Map}(\CC_{\rm Dol} \times \RR_{\mr{dR}}, \gg/\gg)$
& \\ \cline{3-5}
&& \multirow{2}*{\hyperref[cor:3dN8Btwist] {Rank $2$ (B)}} & {BF Theory } & \multirow{2}*{3 (topological)} \\
&&& $T^*[-1]\mr{Map}(\RR^3_{\mr{dR}}, \gg/\gg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[cor:3dN8Atwist] {Rank $2$ (A)}} & {Perturbatively trivial} & \multirow{2}*{3 (topological)} \\
&&& $\mr{Map}(\RR^3_{\mr{dR}}, \gg/\gg)_{\mr{dR}}$ & \\ \hline
\end{tabular}
\end{adjustbox}
\caption{Twists of Maximally Supersymmetric Pure Yang-Mills Theories with Lie algebra $\fg$ (16 supercharges).}
\label{table_of_twists_16}
\end{table}
\begin{table}[!ht]
\centering
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{c|c|c|c|c}
$d$ & $\mc N$ & Twist & Description & Invariant Directions \\
\hline
\multirow{2}*{6} & \multirow{2}*{$(1,0)$} & \multirow{2}*{\hyperref[sect:6dholomorphictwist]{Rank $(1,0)$}} & {Holomorphic BF Theory coupled to a holomorphic symplectic boson} & \multirow{2}*{3 (holomorphic)} \\
&&& $\Sect(\CC^3, (U\otimes K_{\CC^3}^{1/2}) \ham \fg)$ & \\ \hline
\multirow{2}*{5} & \multirow{2}*{$1$} & \multirow{2}*{\hyperref[sect:5d1minimaltwist] {Rank $1$}} & {Generalized BF Theory coupled to a generalized symplectic boson} & \multirow{2}*{3 (minimal)} \\
&&& $\Sect(\CC^2 \times \RR_{\mr{dR}}, (U\otimes K_{\CC^2}^{1/2}) \ham \fg)$ & \\ \hline
\multirow{6}*{4} & \multirow{6}*{$2$} & \multirow{2}*{\hyperref[sect:4d_2_holomorphictwist] {Rank $(1,0)$}} & {Holomorphic BF Theory} & \multirow{2}*{2 (holomorphic)} \\
&&& $T^*[-1]\mr{Sect}(\CC^2, (U \otimes K_{\CC^2}^{1/2}) \ham \fg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:4d_2_11] {Rank $(1,1)$}} & {Generalized BF Theory coupled to a generalized symplectic boson} & \multirow{2}*{3} \\
&&& $\Sect(\CC \times \RR^2_{\mr{dR}}, (U\otimes K_\CC^{1/2}) \ham \fg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:4d2Donaldson] {Rank $(2,0)$}} & {Perturbatively trivial } & \multirow{2}*{4 (topological)} \\
&&& $\mr{Sect}(\CC^2, (U\otimes K_{\CC^2}^{1/2}) \ham \fg)_{\mr{dR}}$ & \\ \hline
\multirow{6}*{3} & \multirow{6}*{$4$} & \multirow{2}*{\hyperref[sect:3d_4_minimal_twist] {Rank $1$}} & {Generalized BF Theory coupled to a generalized symplectic boson} & \multirow{2}*{2 (minimal)} \\
&&& $T^*[-1]\mr{Sect}(\CC \times \RR_{\mr{dR}}, (U\otimes K_\CC^{1/2}) \ham \fg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:3d_4_B_twist] {Rank $2$ (B)}} & {BF Theory coupled to a symplectic boson} & \multirow{2}*{3 (topological)} \\
&&& $\map(\RR^3_{\mr{dR}}, U \ham \fg)$ & \\ \cline{3-5}
&& \multirow{2}*{\hyperref[sect:3d_4_A_twist] {Rank $2$ (A)}} & {Perturbatively trivial } & \multirow{2}*{3 (topological)} \\
&&& $\mr{Sect}(\CC \times \RR_{\mr{dR}}, (U\otimes K_\CC^{1/2}) \ham \fg)_{\mr{dR}}$ & \\ \hline
\end{tabular}
\end{adjustbox}
\caption{Twists of Supersymmetric Yang-Mills Theories with gauge Lie algebra $\fg$ with a hypermultiplet valued in a symplectic representation $U$ (8 supercharges).}
\label{table_of_twists_8}
\end{table}
\begin{table}[hbp]
\centering
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{c|>{\centering}m{5ex}|c|>{\centering}m{65ex}|c}
$d$ & $\mc N$ & Twist & Description & Invariant Directions \\
\hline
\multirow{2}*{4} & \multirow{2}*{$1$} & \multirow{2}{*}{\hyperref[sect:4d1holomorphictwist] {Rank $(1,0)$}} & Holomorphic BF Theory coupled to $R$-matter & \multirow{2}*{2 (holomorphic)} \\
&&& $T^*[-1]\mr{Map}(\CC^2, R/\fg)$ & \\ \hline
\multirow{2}*{3} & \multirow{2}*{$2$} & \multirow{2}*{\hyperref[sect:3dminimaltwist] {Rank $1$}} & {Generalized BF Theory coupled to $R$-matter} & \multirow{2}*{2 (minimal)} \\
&&& $T^*[-1]\mr{Map}(\CC \times \RR_{\mr{dR}}, R/\fg)$ & \\ \hline
\end{tabular}
\end{adjustbox}
\caption{Twists of Supersymmetric Yang-Mills Theories with gauge Lie algebra $\fg$ with a chiral multiplet valued in a representation $R$ (4 supercharges).}
\label{table_of_twists_4}
\end{table}
\begin{table}[!ht]
\centering
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{c|c|c|c}
$\mc N$ & Twist & Description & Invariant Directions \\
\hline
\multirow{6}*{$(4,4)$} & \multirow{2}*{\hyperref[sect:2d44minimaltwist] {Rank $(1,0)$}} & {Holomorphic BF theory coupled to a holomorphic symplectic boson} & \multirow{2}*{1 (holomorphic)} \\
&& {$T^*[-1]\mr{Map}(\CC, T[1](U \ham \gg))$} & \\ \cline{2-4}
& \multirow{2}*{\hyperref[sect:2d44Btwist] {Rank $(1,1)$ (B)}} & {Topological BF theory coupled to a holomorphic symplectic boson} & \multirow{2}*{2 (topological)} \\
&& {$T^*[-1]\mr{Map}(\RR^2_{\mr{dR}}, U \ham \gg)$} & \\ \cline{2-4}
& \multirow{2}*{\hyperref[sect:2d44Atwist] {Rank $(1,1)$ (A)}} & {Perturbatively trivial (A-model)} & \multirow{2}*{2 (topological)} \\
&& {$\mr{Map}(\RR^2_{\mathrm{dR}}, (U \ham \gg)_{\mr{dR}})$} & \\ \hline
\multirow{6}*{$(2,2)$} & \multirow{2}*{\hyperref[sect:2d22minimaltwist] {Rank $(1,0)$}} & {Holomorphic BF theory coupled to $R$ matter} & \multirow{2}*{1 (holomorphic)} \\
&& {$T^*[-1]\mr{Map}(\CC, T[1](R/\gg))$} & \\ \cline{2-4}
& \multirow{2}*{\hyperref[sect:2d22Btwist] {Rank $(1,1)$ (B)}} & {Topological BF theory coupled to $R$ matter} & \multirow{2}*{2 (topological)} \\
&& {$T^*[-1]\mr{Map}(\RR^2_{\mr{dR}}, R/\gg)$} & \\ \cline{2-4}
& \multirow{2}*{\hyperref[sect:2d22Atwist] {Rank $(1,1)$ (A)}} & {Perturbatively trivial (A-model)} & \multirow{2}*{2 (topological)} \\
&& {$T^*[-1]\mr{Map}(\CC, (R/\gg)_{\mr{dR}})$} & \\ \hline
\multirow{2}*{$(\mc N_+, 0)$} & \multirow{2}*{\hyperref[sect:2dN0minimaltwist] {Rank $(1,0)$}} & {Holomorphic BF theory coupled to $\mc N_+ - 2$ free fermions} & \multirow{2}*{1 (holomorphic)} \\
&& {$T^*[-1]\mr{Sect}(\CC, (\fg^{\mc N_+ -2} \otimes K_{\CC}^{1/2}) / \gg)$} & \\ \hline
\multirow{2}*{$(4,0)$} & \multirow{2}*{\hyperref[sect:2d40minimaltwist] {Rank $(1,0)$}} & {Holomorphic BF theory coupled to a holomorphic symplectic boson} & \multirow{2}*{1 (holomorphic)} \\
&& {$T^*[-1]\mr{Sect}(\CC, (U \otimes K_\CC^{1/2}) \ham \gg)$} & \\ \hline
\multirow{2}*{$(2,0)$} & \multirow{2}*{\hyperref[sect:2d20minimaltwist] {Rank $(1,0)$}} & {Holomorphic BF theory coupled to $R$ matter} & \multirow{2}*{1 (holomorphic)} \\
&& {$T^*[-1]\mr{Map}(\CC, R/\gg)$} & \\ \hline
\end{tabular}
\end{adjustbox}
\caption{Twists of Supersymmetric Yang-Mills Theories in two dimensions with gauge group $G$. When $\mc N=(0,2)$ and $(2,2)$ the theory includes a chiral multiplet valued in a representation $R$. When $\mc N=(0,4)$ and $(4,4)$ the theory includes a hypermultiplet valued in a symplectic representation $U$. We can promote the supersymmetry to $\mc N=(8,8)$ when $U = T^*\gg$, but no new twists occur.}
\label{table_of_twists_2d}
\end{table}
\clearpage
\begin{figure}[!hbp]
\begin{adjustbox}{max width=\textwidth}
\begin{tikzpicture}
\matrix (mat) [nodes in empty cells, minimum width=3.5ex, minimum height=3.5ex, column sep=1.8ex,row sep=6ex]{
\node (corner) {}; & \node (d10) {10}; & \node (d9) {9}; & \node (d8) {8}; & \node (d7) {7}; & \node (d6) {6}; & \node {$\qquad \quad \ 5$}; && \node (d5) {};& \node {$\qquad \ 4$}; && \node (d4) {}; & &\node (d3) {3}; & \node (d2) {}; & \\
\node {8}; & & & \node[s16] (88) {$(1,0)_A$}; & & & && &&&& &&& \\
\node {7}; & & & & \node[s16] (77) {$1_A$}; & & && &&&& &&& \\
\node {6}; & & & & & \node[s16] (66) {$(1,1)_A$}; & && &&&& &&& \\
\node {5}; & \node[s16] (105) {$(1,0)$}; & \node[s16] (95) {$1$}; & \node[s16] (85) {$(1,1)$}; & \node[s16] (75) {$2$}; & \node[s16] (65){$(2,2)$}; & \node[s16] (55B) {$4$}; & \node[s16] (55A) {$2_A$}; & &&&& &&& \\
\node {4}; & & & \node[s16] (84) {$(1,0)_B$}; & \node[s16] (74) {$1_B$}; & \node[s16] (64) {$(1,1)_B$}; & \node[s16] (54) {$2_B$}; && \node[s16] (44B) {$(2,2)_B$}; & \node[s16] (44K) {$(2,1)$}; & \node[s8] (44A) {$(2,0)$}; & \node[s16] (44g) {$(2,2)_A$}; & &&& \\
\node {3}; & & & & & \node[s8] (63) {$(1,0)$}; & \node[s8] (53) {$1$}; && \node[s8] (43) {$(1,1)$}; &&&& \node[s8] (33B) {$2_B$}; & \node[s8] (33A) {$2_A$}; & & \\
\node {2}; & & & & & & && \node[s4](42) {$(1,0)$}; &&&& \node[s4] (32) {$1$}; && \\
\node (i1) {1}; & & & & & & & & &&& & && & \\};
\draw[thick] (corner.south west) -- (d2.south west);
\draw[thick] (corner.north east) -- (i1.south east);
\draw[dotted] (0.5,-7) -- (0.5,7);
\draw[dotted] (6.4,-7) -- (6.4,7);
\draw[dotted] (-2,-7) -- (-2,7);
\draw[dotted] (-3.8,-7) -- (-3.8,7);
\draw[dotted] (-4.8,-7) -- (-4.8,7);
\draw[dotted] (-6.3,-7) -- (-6.3,7);
\draw[dotted] (-7.2,-7) -- (-7.2,7);
\draw[arrow] (105) -- (95);
\draw[arrow] (95) -- (85);
\draw[arrow] (88) -- (77);
\draw[arrow] (95) -- (84);
\draw[arrow] (85) -- (75);
\draw[arrow] (85) -- (74);
\draw[arrow] (84) -- (74);
\draw[arrow] (77) -- (66);
\draw[arrow] (75) -- (65);
\draw[arrow] (75) -- (64);
\draw[arrow] (74) -- (64);
\draw[arrow] (74) -- (63);
\draw[arrow] (66) -- (55A);
\draw[arrow] (65) -- (55B);
\draw[arrow] (65) -- (54);
\draw[arrow] (64) -- (54);
\draw[arrow] (64) -- (53);
\draw[arrow] (63) -- (53);
\draw[arrow] (55A) -- (44g);
\draw[arrow] (55B) -- (44B);
\draw[arrow] (54) -- (44B);
\draw[arrow] (54) -- (43);
\draw[arrow] (53) -- (43);
\draw[arrow] (53) -- (42);
\draw[arrow] (44B) -- (33B);
\draw[arrow] (44A) -- (33A);
\draw[arrow] (44K) -- (33A);
\draw[arrow] (43) -- (33B);
\draw[arrow] (43) -- (32);
\draw[arrow] (42) -- (32);
\draw[arrow] (44g) -- (33A);
\end{tikzpicture}
\end{adjustbox}
\caption{This figure shows the orbits of square-zero supercharges in each dimension, and how they relate to one another under dimensional reduction. The labels indicate each orbit: the number refers to the rank, and the subscript indicates the situations where the supercharges of a given rank split into multiple orbits. Each column is labelled by a dimension, and each row by the number of invariant directions of the supercharge. Colours indicate the maximal supersymmetry algebra where the given supercharges live, so black indicates supercharges defined in algebras with 16 supercharges, gray those with 8 supercharges, and white those with 4 supercharges. There is an arrow whenever one twist dimensionally reduces to another twist one dimension lower.}
\label{fig:superchargeorbits}
\end{figure}
\subsection*{Outline of the Paper}
The remainder of the paper is divided into two parts. In Part \ref{formalism_part} we set up the formalism that we will use when we study supersymmetric gauge theories and their twists. The first main ingredient is the Batalin-Vilkovisky formalism for classical field theory (Section \ref{BV_section}).
%, as developed by Costello and Gwilliam in \cite{CostelloBook, Book1, Book2}.
The other main ingredient is the systematic study of supersymmetry algebras and supersymmetric action functionals using normed division algebras (Section \ref{sect:susy}).
%, following Baez and Huerta \cite{BaezHuerta}.
We use this formalism to prove in Section \ref{sect:SYM} that super Yang-Mills theories with matter in dimensions 10, 6, 4 and 3 are in fact supersymmetric, meaning that there is a well-defined $L_\infty$ action of the supersymmetry algebra on the classical BV theory in question. We introduce the idea of dimensional reduction (Section \ref{dim_red_section}) for classical field theories to show that supersymmetry action are well-defined in lower dimensions.
In Part \ref{classification_part} of the paper, we produce the classification of supersymmetric Yang-Mills theories in dimensions 2 to 10 systematically. We start with dimension 10 and work down by dimensional reduction. Each subsection is divided by the number of supersymmetries, and the orbits of square-zero supercharges by which we can twist. Twisted theories are characterized up to perturbative equivalence, including the residual Lorentz symmetry acting on each twisted theory.
\subsection*{Acknowledgements}
We would like to thank Kevin Costello, Owen Gwilliam, Justin Hilburn and Philsang Yoo for helpful discussions during the preparation of this paper.
We are also grateful to Dylan Butson for kind discussion of his related forthcoming work, specifically in reference to supersymmetric theories in dimension $6$.
The research of C.E. on this project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (QUASIFT grant agreement 677368).
The research of P.S. was supported by the NCCR SwissMAP grant of the Swiss National Science Foundation.
The research of B.R.W. was supported by Northeastern University and National Science Foundation Award DMS-1645877.
%\pagestyle{standard}
\part{Supersymmetric Gauge Theory} \label{formalism_part}
\section{The BV-BRST Formalism} \label{BV_section}
In this section we will set up the homological formalism in which we study classical field theory: the BV-BRST formalism. Much of the material in this section is not original. We refer the reader to \cite{CostelloBook, Book2} for more details on this perspective. We will conclude the section by describing a number of fundamental examples of classical field theories that are highly structured: mixed holomorphic-topological theories. We will also discuss the concept of \emph{dimensional reduction} of a classical field theory on $M$ along a fibration $M \to N$. We will use the idea of dimensional reduction to construct many of the supersymmetric field theories which we will consider in the next section.
\subsection{Conventions}
Throughout the paper we will frequently study obects, for instance vector bundles, equipped with a $\ZZ\times\ZZ/2\ZZ$-grading. \emph{Degree} will refer to the first (cohomological) grading and \emph{odd} or \emph{even} to the second (fermionic) grading. We will write $\Pi E$ to denote $E$ placed in odd $\ZZ/2\ZZ$ degree. For an element $x$ we denote by $|x|\in\ZZ/2\ZZ$ the total degree.
Given a vector bundle $E\rightarrow M$ we denote by $\cE$ the topological vector space of smooth sections of $E$ and by $\cE_c$ the topological vector space of smooth compactly supported sections.
We denote by $\cO(\cE)$ (respectively $\cO(\cE_c)$) the completed algebra of symmetric functions on $\cE$ (respectively $\cE_c)$.
We denote by $\oloc(\cE)$ the space of local functionals on $\cE$ (see \cite[Definition 4.5.1.1]{Book2}). An element of $\oloc(\cE)$ will be denoted symbolically by an expression of the form
\[\int_M f (\phi, \phi', \dots),\]
where $f$ is a density on $M$ depending on infinite jets of sections of $E$. Note, however, that the integral here is a formal symbol.
The space of local functionals can be viewed as a subspace
\[
\oloc(\cE) \subset \cO(\cE_c)
\]
where the integral symbol makes sense in earnest when applied to sections which are compactly supported.
We denote by $\oloc^+(\cE)\subset \oloc(\cE)$ the subspace of local functionals which are at least cubic.
Given two vector bundles $E, F$ on $M$ we can also make sense of the space of local functionals from $E$ to $F$.
By definition, this is
\[
{\rm Fun}_{\rm loc}(\cE, \cF) = \prod_{n \geq 0} {\rm PolyDiff}(\cE^{\times n}, \cF)_{S_n}
\]
where ${\rm PolyDiff}(\cE^{\times n}, \cF)$ denotes the space of polydifferential operators, and we take coinvariants for the obvious symmetric group action.
When $\cF = \cE$, we refer to ${\rm Fun}_{\rm loc}(\cE, \cE)$ as the space of local vector fields on $E$.
There is a natural Lie bracket on ${\rm Fun}_{\rm loc}(\cE, \cE)$ and a canonical action of this Lie algebra on local functionals.
\subsection{Formal Moduli Problems and Classical Field Theories}
\label{sect:FMPs}
The classical BV (Batalin-Vilkovisky) formalism \cite{BatalinVilkovisky} is a model for classical field theory from the Lagrangian perspective. In brief, the classical BV formalism produces a local model for the critical locus of an action functional, but considered in the derived sense. That is, given a space $\mc F$ of fields and an action functional with derivative $\d S$, one considers not just the usual locus in $\mc F$ of fields with $\d S(\phi) = 0$, but the derived intersection $\mr{dCrit}(S) = \mc F \cap^h_{T^*\mc F} \Gamma_{\d S}$ of the zero section in $T^*\mc F$ with the graph of $\d S$. The formalism we describe below can be interpreted as an abstract formalism for modelling the tangent complex at a point to a derived critical locus $\mr{dCrit}(S)$ as a formal moduli problem.
Recall that a formal moduli problem is a functor from connective dg Artinian algebras $(R, \mathfrak{m})$ to simplicial sets which satisfies a derived version of Schlessinger's condition. We refer to \cite{DAGX,PridhamFMP,Toen} for more details.
For instance, if $\fg$ is an $L_\infty$ algebra, we have a formal moduli problem $B\fg$ defined by
\[(B\fg)(R, \mathfrak{m})=\mathrm{MC}(\fg\otimes \mathfrak{m}),\]
where $\mathrm{MC}(\fg\otimes \mathfrak{m})$ is the simplicial set of Maurer--Cartan elements. The main result of \cite{DAGX,PridhamFMP} is that the functor $B$ defines an equivalence of $\infty$-categories between $L_\infty$ algebras and formal moduli problems. The inverse functor sends a formal moduli problem $\cM$ to the $L_\infty$-algebra $T_{\cM, \ast}[-1]$, the shifted tangent complex of $\cM$ at the basepoint. This important result will serve as motivation for the main definition in this section (Definition \ref{def:classicalfieldtheory}).
Let $V$ be a $\fg$-representation. Then we may construct an $L_\infty$ algebra
\[L_{V, \fg}=\fg\oplus V[-1]\]
with the only nontrivial brackets coming from the $L_\infty$ brackets on $\fg$ and the action map of $\fg$ on $V$. We introduce the notation
\[V/\fg:= B L_{V, \fg}.\]
\begin{example}
If $\fg$ is an $L_\infty$ algebra, it has an adjoint representation $\fg$. We define the \defterm{$n$-shifted tangent bundle} of $B\fg$ to be
\[T[n] B\fg = \fg[n+1] / \fg.\]
\label{ex:tangentBg}
\end{example}
\begin{example}
Suppose $\fg$ is an $L_\infty$ algebra which is bounded as a complex and has finite-dimensional graded pieces. Then $\fg^*$ is a coadjoint representation of $\fg$. We define the \defterm{$n$-shifted cotangent bundle} of $B\fg$ to be
\[T^*[n] B\fg = \fg^*[n-1] / \fg.\]
\label{ex:cotangentBg}
\end{example}
\begin{definition}
Let $\fg$ be an $L_\infty$ algebra. A \defterm{$\Gm$-action} on a formal moduli problem $B\fg$ is a weight grading $\fg = \bigoplus_m \fg(m)$ compatible with the $L_\infty$ structure.
\end{definition}
\begin{example}
Suppose $\fg$ is an $L_\infty$ algebra and $V$ is a $\fg$-representation. Then $V/\fg$ carries a $\Gm$-action: the underlying $L_\infty$ algebra $\fg\oplus V[-1]$ carries a grading where $\fg$ has weight $0$ and $V[-1]$ has weight $1$. For instance, $T[n] B\fg$ and $T^*[n] B\fg$ carry $\Gm$-actions.
\end{example}
\begin{example}
Suppose $\fg$ is a dg Lie algebra and $U$ a $\fg$-representation equipped with an $n$-shifted symplectic pairing $U\otimes U\rightarrow \CC[n]$. Consider the dg Lie algebra
\[\fh = \fg\oplus U[-1]\oplus \fg^*[n-2]\]
with the brackets $\fg\otimes \fg\rightarrow \fg$ given by the Lie bracket on $\fg$, $\fg\otimes U\rightarrow U$ given by the $\fg$-action on $U$, $\fg\otimes \fg^*\rightarrow \fg^*$ given by the coadjoint action and $\mu\colon U\otimes U\rightarrow \fg^*[d-1]$ defined by $(\mu(v, w), x)_\fg = ([x, v], w)_U$. The dg Lie algebra $\fh$ carries nondegenerate invariant symmetric pairing of cohomological degree $n-2$ given by pairing $\fg$ and $\fg^*$ and pairing $U$ with itself. We denote
\[U\ham \fg := B\fh.\]
This formal moduli problem is equipped with a $\Gm$-action where $\fg$ has weight 0, $U[-1]$ has weight $1$ and $\fg^*[n-2]$ has weight $2$. This may be thought of as an infinitesimal version of the Hamiltonian reduction of $U$ by the $\fg$-action.
\end{example}
Now suppose $L$ is a local $L_\infty$ algebra on a manifold $M$. For every open subset $U\subset M$ we have a formal moduli problem
\[(B L)(U) = B L(U),\]
i.e. $L$ defines a presheaf $B L$ of formal moduli problems on $M$. The following definition was introduced in \cite[Definition 4.1.3.3]{Book2}.
\begin{definition}
A \defterm{local formal moduli problem on $M$} is a presheaf of formal moduli problems on $M$ represented by a local $L_\infty$ algebra.
\end{definition}
\begin{remark}
In \cite{Book2} an extra assumption of ellipticity is required for the local $L_\infty$ algebras considered. It is only relevant for quantization, which we do not consider in this paper, so for simplicity we will not require ellipticity (though in fact all examples we consider will end up being elliptic).
\end{remark}
Given a local formal moduli problem $\cM=B L$, we may consider the space of local functionals which is defined as
\[\oloc(\cM) := \oloc(\cL[1]).\]
The local $L_\infty$ structure on $\cL$ induces a Chevalley--Eilenberg diffrential on $\oloc(\cM)$.
\begin{example}
Let $X,Y$ be complex manifolds, let $M$ be a smooth manifold and let $B\fg$ be a formal moduli problem represented by an $L_\infty$ algebra $\fg$. Then we may define the following local formal moduli problem on $X\times Y\times M$. Let $\Omega^{0, \bullet}_X$ be the graded vector bundle of $(0, n)$-forms on $X$, $\Omega^{\bullet, \bullet}_Y$ be the graded vector bundle of $(p, q)$-forms on $Y$ and $\Omega^\bullet_M$ a graded vector bundle of differential forms on $M$. We may then consider a graded vector bundle
\[L = \Omega^{0,\bullet}_X\otimes \Omega^{\bullet, \bullet}_Y\otimes \Omega^\bullet_M\otimes \fg\]
on $X\times Y\times M$. It carries a differential given by the sum $\ol\dd_X + \ol\dd_Y + \d_M + \d_\fg$. It also carries a local $L_\infty$ structure which uses the $L_\infty$ structure on $\fg$ and the wedge product of differential forms. We then define
\[\map(X\times Y_{\mathrm{Dol}}\times M_{\mathrm{dR}}, B\fg) := B L.\]
\label{ex:mappingspace}
\end{example}
\begin{remark}
A smooth complex algebraic variety $X$ gives rise to derived stacks $X_{\mathrm{Dol}}$ and $X_{\mathrm{dR}}$ defined by Simpson \cite{Simpson,PTVV}. So, given smooth complex algebraic varieties $X,Y,M$ and a derived stack $F$ we may consider the mapping stack
\[\map(X\times Y_{\mathrm{Dol}}\times M_{\mathrm{dR}}, F).\]
Example \ref{ex:mappingspace} is an analogous construction in the world of formal moduli problems.
\end{remark}
\begin{example}
Consider $X, Y, M, \fg$ as in Example \ref{ex:mappingspace} and suppose $E$ is a line bundle on $X\times Y\times M$, equipped with a holomorphic structure along $X\times Y$ and a flat connection along $M$. Moreover, assume $B\fg$ carries a $\Gm$-action. We then have a local $L_\infty$ algebra
\[L = \bigoplus_m \Omega^{0,\bullet}_X\otimes \Omega^{\bullet, \bullet}_Y\otimes \Omega^\bullet_M\otimes \fg(m)\otimes E^{\otimes m}\]
on $X\times Y\times M$. We define
\[\Sect(X\times Y_{\mathrm{Dol}}\times M_{\mathrm{dR}}, B\fg\times_{\Gm} L) := B L.\]
\end{example}
As in Examples \ref{ex:tangentBg} and \ref{ex:cotangentBg}, we may define shifted tangent and cotangent bundles of a local formal moduli problem which give more examples.
\begin{prop}
Consider $X,Y,M, \fg$ as in Example \ref{ex:mappingspace}. The local formal moduli problem $T^*[n]\map(X\times Y_{\mathrm{Dol}}\times M_{\mathrm{dR}}, B\fg)$ is isomorphic to the local formal moduli problem
\[\Sect\left(X\times Y_{\mathrm{Dol}}\times M_{\mathrm{dR}}, T^*[n+\dim(X)+2\dim(Y)+\dim(M)] B\fg\times_{\Gm} (K_X\otimes \Dens_M)\right),\]
where $K_X$ is the canonical bundle of $X$ and $\Dens_M$ is the line bundle of densities on $M$.
\label{prop:mapintocotangent}
\end{prop}
\begin{proof}
The claim follows from the following isomorphisms of graded vector bundles:
\begin{align*}
\left(\Omega^{0, \bullet}_X\right)^!&\cong \Omega^{0, \bullet}_X\otimes K_X[\dim(X)] \\
\left(\Omega^{\bullet, \bullet}_Y\right)^!&\cong \Omega^{\bullet, \bullet}_Y[2\dim(Y)] \\
\left(\Omega^\bullet_M\right)^!&\cong \Omega^\bullet_M\otimes \Dens_M[\dim(M)].
\end{align*}
\end{proof}
\begin{corollary}
Suppose $X,Y,M,\fg$ are as in Example \ref{ex:mappingspace} and, moreover, that $X$ is equipped with a holomorphic volume form and $M$ is oriented. Then
\[T^*[n]\map(X\times Y_{\mathrm{Dol}}\times M_{\mathrm{dR}}, B\fg)\cong \map(X\times Y_{\mathrm{Dol}}\times M_{\mathrm{dR}}, T^*[n+\dim(X)+2\dim(Y)+\dim(M)] B\fg).\]
\label{cor:mapintocotangent}
\end{corollary}
Given a local formal moduli problem, we may talk about shifted symplectic structures \cite{PTVV} on it. In this paper we will only be interested in a strict notion as follows.
\begin{definition}
Let $\cM$ be a local formal moduli problem on $M$ represented by a local $L_\infty$ algebra $L$. A \defterm{strict $n$-shifted symplectic structure} on $\cM$ is a pairing $\omega\colon L\otimes L\rightarrow \Dens_M[n-2]$ satisfying the following conditions:
\begin{enumerate}
\item It is fiberwise nondegenerate.
\item It is graded skew symmetric.
\item The pairing $\cL_c\otimes \cL_c \rightarrow \CC$ defined by
\[\alpha\otimes \beta \mapsto \int_M \omega( \alpha, \beta)\]
is an invariant pairing on the $L_\infty$ algebra $\cL_c$.
\end{enumerate}
\end{definition}
We can now state a concise definition of a classical field theory in the BV formalism.
\begin{definition}
A \defterm{classical BV field theory} (or, simply, classical field theory) is a local formal moduli problem on the spacetime manifold $M$ equipped with a strict $(-1)$-shifted symplectic structure.
\label{def:classicalfieldtheory}
\end{definition}
Given a local formal moduli problem $\cM = BL$ equipped with a strict $n$-shifted symplectic structure, the space of local functionals $\oloc(\cM)$ is equipped with a Poisson bracket (see \cite[Chapter 5.3]{CostelloBook})
\[
\{-,-\}\colon \oloc(\cM) \times \oloc(\cM) \to \oloc(\cM)[-n]
\]
This bracket is a graded version of the so-called Soloviev bracket \cite{Soloviev} defined on the $\infty$-jets, as described in \cite[Section 4]{GetzlerBracket}.
We explain how to define the Poisson bracket in our context. Write $E = L[1]$ for convenience. First note that there is a linear map
\[
\d_{\mr{dR}} \colon \oloc(\cE) \to {\rm Fun}_{\rm loc}(\cE, \cE^!)
\]
defined as follows.
A local functional $F \in \oloc(\cE)$ can be written as an equivalence class of a sum of densities of the form
\[
D_1(-) \cdots D_m(-) \Omega
\]
where $D_i$ is a differential operator $D_i \colon \cE \to C^\infty_M$ and $\Omega$ is a density on $M$.
Without loss of generality, suppose $F$ is of this form.
Then, we can view $F$ as a functional in $\cO(\cE_c)$ by the assignment
\[
\phi \mapsto \int_M D_1(\phi) \cdots D_m(\phi) \Omega
\]
where $\phi$ denotes a compactly supported section.
Define the symmetric multilinear map
\begin{align*}
\d_{\mr{dR}} F \colon \cE_c^{\times (m-1)} & \to \cE^\vee \\
(\phi_1, \ldots, \phi_{n-1}) & \mapsto D_1(\phi_1) \cdots D_{m-1}(\phi_{m-1}) D_{m} (-) + \{{\rm symmetric\;terms}\} .
\end{align*}
Integrating by parts, we see that for any $(m-1)$-tuple $(\phi_1, \ldots, \phi_{m-1}) \in \cE_c^{m-1}$ the linear functional $(\d_{\mr{dR}} F) (\phi_1,\ldots, \phi_{m-1})$ is an element of $\cE^!$.
This implies that $\d_{\mr{dR}} F \in {\rm Fun}_{\rm loc}(\cE, \cE^!)$.
The non-degenerate pairing $\omega$ determines a bundle isomorphism $\omega \colon E \cong E^! [n]$ and hence an isomorphism of local functions
\[
\omega \colon {\rm Fun}_{\rm loc}(\cE, \cE^!) \cong {\rm Fun}_{\rm loc}(\cE, \cE[-n]) .
\]
We recognize the right hand side as the space of local vector fields placed in a shifted cohomological degree.
In total, we see that a local functional $F$ determines a local vector field by applying this isomorphism to $\d_{\mr{dR}}F$:
\[
X_F := \omega \circ \d_{\mr{dR}} (F) \in {\rm Fun}_{\rm loc}(\cE, \cE[-n]) .
\]
This is the Hamiltonian vector field corresponding to $F$. We can now define the Poisson bracket.
\begin{definition}
The Poisson bracket between local functionals $F, G$ is defined by
\[\{F, G\} = X_F (G).\]
\end{definition}
The Poisson bracket enjoys the graded skew symmetry property
\[\{F, G\} = (-1)^{|F||G|+n+1} \{G, F\}\]
as well as the graded Jacobi identity.
The differential on a local $L_\infty$ algebra $L$ is given by a differential operator $Q_{\mathrm{BV}}\colon L\rightarrow L$. The structure of the $L_\infty$ brackets can be encoded into its potential. In the same way, the structure of a local $L_\infty$ algebra $L$ together with an $n$-shifted symplectic structure on $BL$ may be encoded into the \defterm{action functional} $S\in\oloc(BL)$ of cohomological degree $n+1$ such that
\[S = \frac{1}{2}\int_M\omega(e, Q_{BV} e) + I,\]
where $e\in L$ and $I\in\oloc(BL)$ is at least cubic. Moreover, the action functional satisfies the \defterm{classical master equation}
\[\{S, S\} = 0.\]
We refer to \cite[Proposition 5.4.0.2]{Book2} for this construction.
Given a classical field theory represented by a local $L_\infty$ algebra $L$ on $M$, as in Definition \ref{def:classicalfieldtheory}, we call $E=L[1]$ the \defterm{bundle of BV fields}, and we call the complex $(E, Q_{\mr{BV}})$ the \defterm{classical BV complex}. We call the Poisson bracket on $\oloc(\cM)$ the \defterm{BV bracket}. It will sometimes be convenient to think of a classical field theory as a quadruple $(E, \omega, Q_{\mr{BV}}, I)$ consisting of the bundle of BV fields equipped with a $(-1)$-shifted symplectic pairing $\omega$, a classical BV differential $Q_{\mr{BV}}$ and an interaction functional $I$. We characterize such data in the following way.
\begin{definition}
A \defterm{free BV theory} on a manifold $M$ is the data of:
\begin{itemize}
\item a finite rank graded vector bundle $E \to M$ equipped with an even differential operator of cohomological degree $+1$
\[
Q_{\mr{BV}} \colon \cE \to \cE [1]
\]
such that $(1)$: $Q_{\mr{BV}}^2 = 0$ and $(2)$: the pair $(\cE , Q_{\mr{BV}})$ is an elliptic complex;
\item a map of bundles
\[
\omega\colon E \otimes E \to \Dens_M [-1]
\]
that is
\begin{enumerate}
\item[$(1)$] fiberwise nondegenerate,
\item[$(2)$] graded skew symmetric, and
\item[$(3)$] satisfies $\int_M \omega(e_0, Q_{\mr{BV}} e_1) = (-1)^{|e_0|} \int_M \omega(Q_{\mr{BV}} e_0, e_1)$ where $e_i$ are compactly supported sections of $E$ .
\end{enumerate}
\end{itemize}
\end{definition}
The datum of a classical BV field theory as in Definition \ref{def:classicalfieldtheory} is equivalent to the datum of a free BV theory $(E, Q, \omega)$ equipped with an even functional
\[I \in \oloc^+(\cE)\]
of cohomological degree zero satisfying the Maurer-Cartan equation
\[Q_{\mr{BV}} I + \frac{1}{2} \{I,I\} = 0,\]
under the identification of the BV action as
\[S = \frac{1}{2} \int_M \omega(e, Q_{\mr{BV}} e) + I\in \oloc(E).\]
\begin{example} \label{def:cotangent_type}
Let $\cM=BL$ be a local formal moduli problem. Then the $(-1)$-shifted cotangent bundle $T^*[-1]\cM$ carries a natural $(-1)$-shifted symplectic structure. Indeed, $T^*[-1]\cM = B(L\oplus L^![-3])$ and we simply pair $L$ and $L^!$. Classical field theories arising via this construction are called \defterm{cotangent type} theories.
\end{example}
We will also consider $\CC[t]$-families of classical field theories.
\begin{definition} \label{family_of_BV_theories_def}
A \defterm{$\CC[t]$-family of classical field theories} is a graded bundle of locally-free $\CC[t]$-modules $L$ on $M$ equipped with a structure of a $\CC[t]$-linear local $L_\infty$ algebra and a $\CC[t]$-linear $(-1)$-shifted symplectic structure $\omega\colon L\otimes_{\CC[t]} L\rightarrow \CC[t]\otimes \Dens_M[-3]$.
\end{definition}
We will consider $\CC[t]$-families of classical field theories where $L = \CC[t]\otimes L_0$ and the pairing
\[\omega\colon L\otimes_{\CC[t]} L\rightarrow \CC[t]\otimes \Dens_M[-3]\]
comes from a pairing
\[\omega_0\colon L_0\otimes L_0\rightarrow \Dens_M[-3].\]
In this case the local $L_\infty$ structure is encoded in a $t$-dependent action functional $S$.
\begin{remark}
Besides the $\ZZ$-graded classical field theories defined above, we may consider the following variants of the above definition:
\begin{itemize}
\item A $\ZZ\times \ZZ/2\ZZ$-graded local $L_\infty$ algebra is a $\ZZ$-graded local $L_\infty$ algebra $L$ equipped with an extra $\ZZ/2\ZZ$-grading (the \defterm{fermionic grading}) with respect to which all operations are even. An $n$-shifted symplectic structure on a $\ZZ\times\ZZ/2\ZZ$-graded local formal moduli problem $BL$ is a pairing $L\otimes L\rightarrow \Dens_M[n-2]$, which is even with respect to the fermionic grading.
\item A $\ZZ/2\ZZ$-graded classical field theory is defined in the same way as a $\ZZ$-graded classical field theory where we only consider the cohomological grading modulo 2.
\end{itemize}
\end{remark}
\subsection{Perturbative Equivalence of Classical Field Theories}
Next, we formulate the notion of a morphism, and an equivalence, of classical BV theories.
\begin{definition}
A \defterm{morphism} $\Phi\colon (E, \omega, S)\rightsquigarrow (E', \omega', S')$ of classical field theories over the same manifold $M$ is a collection $\Phi =\sum_{n\geq 1}^\infty \Phi_n$ of poly-differential operators $\Phi_n\colon \Sym^n(E)\rightarrow E'$ that intertwine the pairings $\omega, \omega'$ and define an $L_\infty$ map $\cE[-1]\rightarrow \cE'[-1]$. A morphism is a \defterm{perturbative equivalence} if the map $\Phi_1\colon (\cE, Q_{BV})\rightarrow (\cE', Q'_{BV})$ is a quasi-isomorphism. A classical field theory is \defterm{perturbatively trivial} if it is perturbatively equivalent to the zero theory ($E = 0$).
\label{def:perturbativeequivalence}
\end{definition}
The interpretation of this definition is that $\Phi$ is a non-linear map between the bundles of BV fields, and $\Phi_n$ is its $n^{\text{th}}$ Taylor coefficient.
We will now describe two primitive examples of equivalences of classical field theories that will be useful in simplifying twisted theories.
First, we consider the process of eliminating an auxiliary field.
\begin{prop}\label{prop:integrateoutfield}
Fix a volume form $\dvol_M$ on $M$. Suppose $(E, \omega, S)$ is a classical field theory, where $E\cong E_0\oplus (\cO_M\oplus \Dens_M[-1])$ with the symplectic pairing $\omega$ given by a sum of a symplectic pairing $\omega_0$ on $E_0$ and the standard symplectic pairing on the second summand. Denote by $\phi$ a section of $\cO_M$ and by $\phi^*$ a section of $\Dens_M[-1]$. Suppose the BV action is
\[S = S_0 + \frac{1}{2} \int \dvol_M (\phi^2 - 2\phi S_1),\]
where $S_0$ is a local functional independent of $\phi,\phi^*$ and $S_1$ is a $\cO_M$-valued polydifferential operator which is independent of $\phi$.
Then the theory $(E, \omega, S)$ is perturbatively equivalent to the theory $(E_0, \omega_0, S')$ with the BV action $S' = S_0 - S_1^2/2$, where we set $\phi = S_1$ and $\phi^* = 0$.
\end{prop}
\begin{proof}
Concretely, suppose that the linear part of $S_1$ is given by an operator $Q_1$, and that the interacting part of $S_1$ is given by a functional $I_1 = \sum_{n=1}^\infty I_1^{(n)}$.
The desired equivalence $\Phi \colon (E, \omega, S) \to (E_0, \omega_0, S')$ is given by the natural projection $\Phi = \Phi_1 \colon E \to E_0$.
The quasi-inverse $\Psi \colon (E_0, \omega_0, S') \to (E, \omega, S)$ is defined as follows.
First $\Psi_1(e) = (e, Q_1(e), 0) \in E$.
For $n > 1$, define
\begin{align*}
\Psi_n \colon \sym^n(E_0) &\to E \\
e_1 \otimes \cdots \otimes e_n &\mapsto (0, I_1^{(n)}(e_1, \ldots, e_n), 0).
\end{align*}
These $\Psi_n$ manifestly intertwine the pairings $\omega$ and $\omega'$. To see that they intertwine the action functionals, we observe that
\begin{align*}
S(\Psi(e)) &= S(e, S_1(e), 0) \\
&= S_0(e) + \frac 12 \dvol_M \int (S_1(e)^2 - 2 S_1(e)^2) \\
&= S_0(e) - \frac 12 S_1(e)^2 \\
&= S'(e).
\end{align*}
\end{proof}
\begin{remark}
In terms of the classical BV complex, this proposition has the following interpretation. We consider theories where the classical BV complex is of the form
\[\xymatrix{
\cdots & \ul{0} & \ul{1} & \cdots \\
\cdots \ar[r] & E_0^0 \ar[r]^{Q_0} \ar@{.>}[dr] & E_0^1 \ar[r] &\cdots \\
&\cO_M \ar^-{{\rm dvol}}[r] \ar@{.>}[ur] &\dens_M. &
}\]
The bottom map multiplies a function by the volume element.
The dotted arrows are induced from $S_1$.
The proposition implies that we can replace this with a quasi-isomorphic cochain complex consisting of only the first line, provided we make a suitable modification of the classical action functional.
\end{remark}
We may also eliminate pairs of fields as follows.
\begin{prop}
\label{prop:BRSTdoublet}
Let $(E_0, \omega_0, S_0)$ be a classical BV theory and let $F \to M$ be a graded vector bundle.
Consider the theory $(E, \omega, S)$ with underlying graded vector bundle
\[
E = E_0 \oplus \left(F \oplus F^! [-1]\right) \oplus \left(F^! \oplus F[-1] \right)
\]
whose sections we denote by $e_0 + \phi + \phi^* + \psi + \psi^*$ according to the above decomposition.
The shifted symplectic form $\omega$ is given by the sum of $\omega_0$ and the standard degree $+1$ pairings between $F, F^! [-1]$ and $F^!, F[-1]$.
Suppose further that the local functional
\[
S = S_0 + \int \phi \psi^* - \int \phi I_\phi - \int \psi^* I_{\psi^*} - \int \phi^* I_{\phi^*} - \int \psi I_{\psi}
\]
satisfies the classical master equation, where $I_{\phi}, I_{\psi^*}, I_{\phi^*}, I_{\psi}$ are polydifferential operators on fields valued in $F^!$, $F$, $F$, $F^!$ respectively, and which are independent of $\phi$ and $\psi^*$.
Then the classical BV theory $(E, \omega, S)$ is perturbatively equivalent to the BV theory $(E_0, \omega_0, S')$ where $S'$ is given by setting $\phi = I_{\psi^*}, \phi^* = 0$ and $\psi^* = I_{\phi}, \psi = 0$ in the original action functional $S$.
\end{prop}
\begin{proof}
Concretely, we'll write $\sum_{n \ge 1} I^{(n)}_\phi$ and $\sum_{n \ge 1} I^{(n)}_{\psi^*}$ for the Taylor expansions of $I_\phi$ and $I_{\psi^*}$ respectively.
The desired equivalence $\Phi \colon (E, \omega, S) \to (E_0, \omega_0, S')$ is given by the natural projection $\Phi = \Phi_1 \colon E \to E_0$.
The quasi-inverse $\Psi \colon (E_0, \omega_0, S') \to (E, \omega, S)$ is defined as follows.
The linear term is $\Psi_1(e) = (e, I^{(1)}_{\phi}(e),0,0, I^{(1)}_{\psi^*}(e))$, and for $n > 1$ we have
\[\Psi_n(e_1\otimes \cdots \otimes e_n) = (0, I^{(n)}_{\phi}(e_1, \ldots, e_n), 0, I^{(n)}_{\psi^*}(e_1, \ldots, e_n)).\]
The maps $\Psi_n$ manifestly intertwine the pairings on $E_0$ and $E$, since the image of $\Psi_n$ lands in an isotropic summand of the $E_1\oplus E_1^![-1]\oplus E_1^!\oplus E_1[-1]$ part of $E$.
Also, by construction, the $\Psi_n$ intertwine the action functionals, since
\begin{align*}
S(F(e)) &= S_0(e) + \frac{1}{2} \int_M \omega(I_{\psi^*}(e), I_\phi(e) - I_\phi(e)) + \omega(I_\phi (e), I_{\psi^*}(e) - I_{\psi^*}(e)) \\
&= S'(e).
\end{align*}
\end{proof}
\begin{remark}
For the classical BV theory $(E, \omega, S)$ as in the proposition, the linearized BV differential defines the following cochain complex of fields:
\[\xymatrix{
\cdots & \ul{-1} & \ul{0} & \ul{1} & \ul{2} & \cdots \\
\cdots \ar[r] & E_0^{-1}\ar[r] \ar@{.>}[dr] \ar@{.>}[ddr] &E_0^0 \ar@{.>}[ddr] \ar@{.>}[dr] \ar[r] &E_0^1 \ar[r] &E_0^2 \ar[r] &\cdots \\
&&F_\phi \ar^{~}[dr] \ar@{.>}[ur] &F^!_{\phi*} \ar@{.>}[ur] && \\
&&F^!_{\psi^*} \ar^{~}[ur] \ar@{.>}[uur] &F_\psi \ar@{.>}[uur] && \\
}\]
where the subscripts match the notation for the fields in the statement above \footnote{Note that we are writing $F$ as if it is concentrated in a single cohomological degree, but the proposition applies for any graded vector bundle as in the statement of the proposition.}. The top line is the underlying cochain complex of the theory with fields $E_0$. The arrows $F_\phi \to F_\psi$ and $F^!_{\psi^*} \to F^!_{\phi^*}$ are given by the identity. The dotted arrows represent terms in the differential arising from $I_\phi, I_{\psi^*}, I_{\phi^*}, I_{\psi}$. The above proposition implies we can replace this cochain complex of fields with a quasi-isomorphic complex consisting of only the first line, provided we make a suitable modification of the classical action functional.
\end{remark}
\begin{remark}
We will call the pair $(\phi, \psi)$ satisfying the conditions of the previous proposition a \defterm{trivial BRST doublet}.
\end{remark}
%There is a corollary of this proposition obtained by iterating the procedure.
%We will only use a single iteration.
%
%\begin{corollary} \label{cor:quad}
%Let $(E_0, \omega_0, S_0)$ be a classical BV theory and $F \to M$ a graded vector bundle.
%Consider the theory $(E, \omega, S)$ with underlying graded vector bundle
%\[
%E = E_0 \oplus \bigoplus_{i=1}^2 \left(F \oplus F^! [-1]\right) \oplus \bigoplus_{i=1}^2 \left(F^! \oplus F[-1] \right)
%\]
%whose sections we denote by $e_0 + \sum_{i=1}^2 (\phi + \phi_i^*) + \sum_{i=1}^2(\psi_i + \psi_i^*)$ according to the above decomposition.
%The shifted symplectic form $\omega$ is given by the sum of $\omega_0$ plus the standard degree $+1$ pairings between $F, F^! [-1]$ and $F^!, F[-1]$.
%Suppose further that the local functional
%\[
%S = S_0 + \int \left(\phi_2\psi^*_1 + (\phi_2 - \phi_1)\psi_2^*\right) - \sum_{i=1}^2 \int \phi_i I_{\phi,i} - \sum_{i=1}^2 \int \psi_i^* I_{\psi^*,i} - \sum_{i=1}^2 \int \phi^*_i I_{\phi^*,i} - \sum_{i=1}^2 \int \psi_i I_{\psi,i}
%\]
%satisfies the classical master equation, where $I_{\phi,i}, I_{\psi^*,i}, I_{\phi^*,i}, I_{\psi,i}$ are polydifferential operators on fields valued in $F^!$, $F$, $F$, $F^!$ respectively, and which are independent of the fields $\phi_i$ and $\psi^*_i$.
%Then, the classical BV theory $(E, \omega, S)$ is perturbatively equivalent to the BV theory $(E', \omega_0, S')$ where
%\[
%E' = E_0 \oplus \left(F_{\phi_1} \oplus F^!_{\phi_1^*} [-1]\right) \oplus \left(F^!_{\psi_1^*} \oplus F[-1]_{\psi_1} \right)
%\]
%and
%\[
%S' = S_0 + \int \phi_1 \psi_1^* - S''
%\]
%where $S'$ is given by setting $\phi = I_{\psi^*}, \phi^* = 0$ and $\psi^* = I_{\phi}, \psi = 0$ in the functional functional $S$
%\end{corollary}
%\begin{proof}
%%Notice that the collection of fields $(e_0, \phi_1, \phi_1^*, \phi_2, \phi_2^*, \psi_1, \psi_1^*, \psi_2, \psi_2^*)$ have the same shifted Poisson brackets as the collection of fields $(\phi_1, \phi_1^*
%We begin by making the following change of coordinates on the BV fields:
%\begin{align*}
%\Tilde{\phi}_2 & = \phi_2 - \phi_1 \\
%\Tilde{\psi}_2 & = \psi_2 + \psi_1 \\
%%\Tilde{\phi}^*_2 & = \phi^*_2 - \phi^*_1 \\
%%\Tilde{\psi}^*_2 & = \psi^*_2 + \psi^*_1
%\end{align*}
%with $\phi_1, \psi_1$ left fixed.
%This change of coordinates is compatible with the $(-1)$-symplectic pairing.
%Further, in the new coordinates, the action takes the form
%\[
%S = S_0 + \int \Tilde{\phi}_2\Tilde{\psi}^*_2 + \int \phi_1\psi_2^* - \sum_{i=1}^2 \int \phi_i I_{\phi,i} - \sum_{i=1}^2 \int \psi_i^* I_{\psi^*,i}
%\]
%\end{proof}
\subsection{Symmetries in the Classical BV Formalism} \label{symmetry_section}
In this section we define what it means for a (super) Lie algebra to act on a classical field theory (see also \cite[Chapter 11]{Book2} for a related discussion). Let $(E, \omega, S)$ be a classical field theory and $\fg$ a super Lie algebra. We will define $\gg$-equivariant local observables in the classical field theory by introducing $\gg$-valued background fields into our classical field theory and extending the action functional to a functional that involves these background fields, but still satisfies the classical master equation. We begin by defining an appropriate version of the Chevalley--Eilenberg cochain complex.
\begin{definition}
The \defterm{Chevalley--Eilenberg complex} for the Lie algebra $\gg$, with coefficients in $\oloc(\cE)$, is defined as follows. Consider the graded vector space
\[C^\bullet(\fg, \oloc(\cE)) = \bigoplus_n \hom(\wedge^n \fg, \oloc(\cE))[-n]\]
parametrizing multilinear maps $f\colon \fg^{\otimes n}\rightarrow \oloc(\cE)$ that satisfy the antisymmetry property
\[f(x_1, \dots, x_i, x_{i+1}, \dots, x_n) = (-1)^{|x_1||x_2|+1} f(x_1, \dots, x_{i+1}, x_i, \dots, x_n)\]
where $x_j \in \fg$. The Chevalley-Eilenberg differential is given, following the sign conventions of \cite{SafronovCoisoInt}, by the formula
\[(\d_{\mr{CE}} f)(x_1, \dots, x_n) = \sum_{i < j}(-1)^{|x_i| \sum_{p=1}^{i-1} |x_p| + |x_j| \sum_{p=1,p\neq i}^{j-1} |x_p| +i+j+|f|} f([x_i, x_j], x_1, \dots, \widehat{x}_i, \dots, \widehat{x}_j, \dots, x_n).\]
The complex is additionally equipped with a degree $+1$ BV bracket via the formula
\[\{f, g\}(x_1, \dots, x_{k+l}) = \sum_{\sigma\in S_{k, l}} \mathrm{sgn}(\sigma) (-1)^{\epsilon+\epsilon_1} \{f(x_{\sigma(1)}, \dots, x_{\sigma(k)}), g(x_{\sigma(k+1)}, \dots, x_{\sigma(k+l)})\},\]
where $S_{k, l}$ is the set of $(k, l)$-shuffles, $\epsilon$ is the usual Koszul sign and
\[\epsilon_1 = |g|k + \sum_{i=1}^k |x_{\sigma(i)}|(l+|g|).\]
\end{definition}
The operator $Q_{\mr{BV}}$ on $\oloc(\cE)$ extends $C^\bu(\fg)$-linearly to an operator on $C^\bullet(\fg, \oloc(\cE))$ by the rule
\[
(Q_{\mr{BV}} f)(x_1,\ldots, x_n) = Q_{\mr{BV}} f(x_1,\ldots, x_n)
\]
where $f \colon \fg^{\otimes n} \to \oloc(\cE)$.
The differentials $\d_{\mr{CE}}$ and $Q_{\mr{BV}}$ are compatible in the sense that $(\d_{\mr{CE}} + Q_{\mr{BV}})^2 = 0$ making $C^\bu(\fg, \oloc(\cE))$ into a cochain complex with total differential $\d_{\mr{CE}} + Q_{\mr{BV}}$.
Via the BV bracket, the shift of this cochain complex $C^\bu(\fg, \oloc(\cE))[-1]$ is a dg Lie algebra. This shifted cotangent complex will model equivariant local observables in our classical field theory, but to finish defining the $\gg$ action we must define the equivariant version of the classical interaction. This is defined as follows.
\begin{definition}
\label{infinitesimal_action_def}
Let $(E, \omega, S)$ be a classical field theory. An \defterm{action} of a super Lie algebra $\fg$ on $(E, \omega, S)$ is an element
\[S_\fg = \sum_{k\geq 0} S_{\fg}^{(k)}\in C^\bullet(\fg, \oloc(\cE))\]
of cohomological degree zero, where $S_\fg^{(k)}$ is a multilinear map $\fg^{\otimes k} \to \oloc(\cE)$, that satisfies the following three conditions:
\begin{itemize}
\item[(a)] $S_\fg^{(0)} = S$.
\item[(b)] For each $k \geq 1$ and $x_1, \ldots, x_k \in \fg$ the local functional $S^{(k)}_\fg (x_1,\ldots, x_k)$ is at least quadratic in the fields.
\item[(c)] $S_\fg$ satisfies the Maurer--Cartan equation:
\[\d_{\mr{CE}} S_\fg + \frac{1}{2} \{S_\fg, S_\fg\} = 0.\]
\end{itemize}
\end{definition}
\begin{remark}
We have defined an action of a Lie algebra on a classical field theory in terms of a Noether current $S_\fg$.
Such data gives rise to an $L_\infty$ action of $\fg$ on the space of fields $\cE$ in the following way.
By the Maurer-Cartan equation, the operator $\d_{\mr{CE}} + \{S_\fg, -\}$
defines a differential on the graded vector space $\cO(\fg[1] \oplus \cE)$.
By assumption that the Noether current is at least quadratic in the fields, we see that this differential defines a family of maps
\[
\fg^{\otimes k} \otimes \cE^{\otimes \ell} \to \cE
\]
combining to give $\cE$ the structure of an $L_\infty$-module for $\fg$.
\end{remark}
\begin{remark}
We have seen that a classical BV theory can also be presented in terms of a BV differential $Q_{\mr{BV}}$ and an interaction $I$ satisfying the Maurer-Cartan equation
\[Q_{\mr{BV}} I + \frac{1}{2} \{I,I\} = 0 .\]
One can also formulate actions of a Lie algebra on a classical theory in these terms.
The data of an action of a Lie algebra $\fg$ on a classical field theory $(E, \omega, S)$ is equivalent to the choice of a local interaction functional
\[I_\fg = \sum_{k\geq 0} I_{\fg}^{(k)} \text{ in } C^\bullet(\fg, \oloc(\cE)),\]
satisfying the Maurer-Cartan equation
\[(\d_{\mr{CE}} + Q_{\mr{BV}}) I_\fg + \frac{1}{2} \{I_\fg, I_\fg\} = 0.\]
\end{remark}
We may also define actions of supergroups on classical field theories. The action of a supergroup $G$ is more data than the action of a super Lie algebra $\gg$: it includes the infinitesimal action of the Lie algebra $\gg$, along with an action of $G$ on the fields exponentiating this infinitesimal action. That is, we make the following definition.
\begin{definition}
\label{group_action_def}
Let $(E, \omega, S)$ be a classical field theory, and let $G$ be a supergroup acting on spacetime $M$. An \defterm{action} of $G$ on $(E, \omega, S)$ is given by the following data:
\begin{itemize}
\item An action of $G$ on $\cE$ compatible with the $G$-action on $M$.
\item An action $S_\fg$ of its super Lie algebra $\fg$ with $S^{(k)}_\fg = 0$ for $k\geq 2$
\end{itemize}
These are required to satisfy the following conditions:
\begin{itemize}