\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{AKSZ sigma-model} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{chernsimons_theory}{}\paragraph*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory} [[!include infinity-Chern-Simons theory - contents]] \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry} [[!include symplectic geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} What is called the \emph{AKSZ formalism} -- after the initials of its four authors -- Alexandrov, [[Maxim Kontsevich]], [[Albert Schwarz]], Oleg Zaboronsky -- is a technique for constructing [[action functional]]s in [[BV-BRST formalism]] for [[sigma model]] [[quantum field theories]] whose [[target space]] is an [[symplectic Lie n-algebroid]] $(\mathfrak{P}, \omega)$. The [[action functional]] of AKSZ theory is that of [[∞-Chern-Simons theory]] induced from the [[Chern-Simons element]] that correspondonds to the [[invariant polynomial]] $\omega$. Details on this are at \href{http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory+--+examples#ASKZTheory}{∞-Chern-Simons theory -- Examples -- AKSZ theory}. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item to a [[Poisson Lie algebroid]] corresponds the [[Poisson sigma-model]]; \item the a [[Courant algebroid]] corresponds the [[Courant sigma-model]]; in particular to a [[semisimple Lie algebra]] corresponds [[Chern-Simons theory]]. \item [[BF-theory]]+[[topological Yang-Mills theory]], \end{itemize} Also, the [[A-model]] and the [[B-model]] topological 2d [[sigma-models]] are examples. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} A [[sigma-model]] [[quantum field theory]] is, roughly, one \begin{itemize}% \item whose fields are maps $\phi : \Sigma \to X$ to some space $X$; \item whose [[action functional]] is, apart from a [[kinetic action|kinetic term]], the [[transgression]] of some kind of [[cocycle]] on $X$ to the [[mapping space]] $\mathrm{Map}(\Sigma,X)$. \end{itemize} Here the terms ``space'', ``maps'' and ``cocycles'' are to be made precise in a suitable context. One says that $\Sigma$ is the \emph{[[worldvolume]]}, $X$ is the \emph{[[target space]]} and the cocycle is the \emph{[[background gauge field]]} . For instance the ordinary charged [[particle]] (for instance an electron) is described by a $\sigma$-model where $\Sigma = (0,t) \subset \mathbb{R}$ is the abstract \emph{[[worldline]]}, where $X$ is a smooth ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] (for instance our [[spacetime]]) and where the background cocycle is a [[circle bundle]] with [[connection on a bundle|connection]] on $X$ (a degree-2 cocycle in [[ordinary differential cohomology]] of $X$, representing a background \emph{[[electromagnetic field]]} : up to a kinetic term the action functional is the [[holonomy]] of the connection over a given [[curve]] $\phi : \Sigma \to X$. The $\sigma$-models to be considered here are \emph{higher} generalizations of this example, where the background gauge field is a cocycle of higher degree (a [[connection on an infinity-bundle|higher bundle with connection]]) and where the worldvolume is accordingly higher dimensional -- and where $X$ is allowed to be not just a manifold but an approximation to a \emph{higher [[orbifold]] (a [[smooth ∞-groupoid]]).} More precisely, here we take the [[category]] of [[space]]s to be [[dg-geometry|smooth dg-manifolds]]. One may imagine that we can equip this with an [[internal hom]] $\mathrm{Maps}(\Sigma,X)$ given by $\mathbb{Z}$-graded objects. Given [[dg-geometry|dg-manifolds]] $\Sigma$ and $X$ their canonical degree-1 vector fields $v_\Sigma$ and $v_X$ acting on the mapping space from the left and right. In this sense their linear combination $v_\Sigma + k \, v_X$ for some $k \in \mathbb{R}$ equips also $\mathrm{Maps}(\Sigma,X)$ with the structure of a differential graded smooth manifold. Moreover, we take the ``cocycle'' on $X$ to be a graded [[symplectic structure]] $\omega$, and assume that there is a kind of Riemannian structure on $\Sigma$ that allows to form the [[transgression]] \begin{displaymath} \int_\Sigma \mathrm{ev}^* \omega := p_! \mathrm{ev}^* \omega \end{displaymath} by [[integral transform|pull-push]] through the canonical [[correspondence]] \begin{displaymath} \mathrm{Maps}(\Sigma,X) \stackrel{p}{\leftarrow} \mathrm{Maps}(\Sigma,X) \times \Sigma \stackrel{ev}{\to} X \,, \end{displaymath} where on the right we have the [[evaluation map]]. Assuming that one succeeds in making precise sense of all this one expects to find that $\int_\Sigma \mathrm{ev}^* \omega$ is in turn a symplectic structure on the mapping space. This implies that the vector field $v_\Sigma + k\, v_X$ on mapping space has a [[Hamiltonian]] $\mathbf{S} \in C^\infty(\mathrm{Maps}(\Sigma,X))$. The grade-0 components $S_{\mathrm{AKSZ}}$ of $\mathbf{S}$ then constitute a functional on the space of maps of graded manifolds $\Sigma \to X$. This is the \textbf{AKSZ action functional} defining the AKSZ $\sigma$-model with target space $X$ and background field/cocycle $\omega$. In (\hyperlink{AKSZ}{AKSZ}) this procedure is indicated only somewhat vaguely. The focus of attention there is a discussion, from this perspective, of the action functionals of the 2-dimensional $\sigma$-models called the \emph{[[A-model]]} and the \emph{[[B-model]]} . In (\hyperlink{Roytenberg}{Roytenberg}), a more detailed discussion of the general construction is given, including an explicit and general formula for $\mathbf{S}$ and hence for $S_{\mathrm{AKSZ}}$ . For $\{x^a\}$ a coordinate chart on $X$ that formula is the following. \begin{defn} \label{TheAKSZAction}\hypertarget{TheAKSZAction}{} For $(X,\omega)$ a [[symplectic Lie n-algebroid|symplectic dg-manifold]] of grade $n$, $\Sigma$ a smooth compact manifold of dimension $(n+1)$ and $k \in \mathbb{R}$, the \textbf{AKSZ action functional} \begin{displaymath} S_{\mathrm{AKSZ},k} : \mathrm{SmoothGrMfd}(\mathfrak{T}\Sigma, X) \to \mathbb{R} \end{displaymath} (where $\mathfrak{T}\Sigma$ is the shifted tangent bundle) is \begin{displaymath} S_{\mathrm{AKSZ},k} : \phi \mapsto \int_\Sigma \left( \frac{1}{2}\omega_{ab} \phi^a \wedge d_{\mathrm{dR}}\phi^b + k \, \phi^* \pi \right) \,, \end{displaymath} where $\pi$ is the [[Hamiltonian]] for $v_X$ with respect to $\omega$ and where on the right we are interpreting fields as forms on $\Sigma$. \end{defn} This formula hence defines an infinite class of $\sigma$-models depending on the target space structure $(X, \omega)$, and on the relative factor $k \in \mathbb{R}$. In (\hyperlink{AKSZ}{AKSZ}) it was already noticed that ordinary [[Chern-Simons theory]] is a special case of this for $\omega$ of grade 2, as is the [[Poisson sigma-model]] for $\omega$ of grade 1 (and hence, as shown there, also the [[A-model]] and the [[B-model]]). The main example in (\hyperlink{Roytenberg}{Roytenberg}) is spelling out the general case for $\omega$ of grade 2, which is called the \emph{[[Courant sigma-model]]} there. One nice aspect of this construction is that it follows immediately that the full Hamiltonian $\mathbf{S}$ on mapping space satisfies $\{\mathbf{S}, \mathbf{S}\} = 0$. Moreover, using the standard formula for the internal hom of chain complexes one finds that the cohomology of $(\mathrm{Maps}(\Sigma,X), v_\Sigma + k v_X)$ in degree 0 is the space of functions on those fields that satisfy the [[Euler-Lagrange equations]] of $S_{\mathrm{AKSZ}}$. Taken together this implies that $\mathbf{S}$ is a solution of the ``master equation'' of a [[BV-BRST complex]] for the quantum field theory defined by $S_{\mathrm{AKSZ}}$. This is a crucial ingredient for the quantization of the model, and this is what the AKSZ construction is mostly used for in the literature. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[sigma-model]] \item [[schreiber:infinity-Chern-Simons theory]] \item [[higher dimensional Chern-Simons theory]] \begin{itemize}% \item [[1d Chern-Simons theory]] \item [[2d Chern-Simons theory]] \item [[3d Chern-Simons theory]] \item [[4d Chern-Simons theory]] \item [[5d Chern-Simons theory]] \item [[6d Chern-Simons theory]] \item [[7d Chern-Simons theory]] \item [[11d Chern-Simons theory]] \item [[string field theory]] \item [[infinite-dimensional Chern-Simons theory]] \item \textbf{AKSZ $\sigma$-model} \begin{itemize}% \item [[Poisson sigma-model]] \begin{itemize}% \item [[A-model]], [[B-model]] \end{itemize} \item [[Courant sigma-model]] \begin{itemize}% \item [[Chern-Simons theory]] \end{itemize} \end{itemize} \end{itemize} \end{itemize} [[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} The original reference is \begin{itemize}% \item M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, \emph{The geometry of the master equation and topological quantum field theory}, Int. J. Modern Phys. A 12(7):1405--1429, 1997 (\href{http://arxiv.org/abs/hep-th/9502010}{arXiv:hep-th/9502010}) \end{itemize} Dmitry Roytenberg wrote a useful exposition of the central idea of the original work and studied the case of the [[Courant sigma-model]] in \begin{itemize}% \item [[Dmitry Roytenberg]], \emph{AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories} Lett.Math.Phys.79:143-159,2007 (\href{http://arxiv.org/abs/hep-th/0608150}{arXiv:hep-th/0608150}). \end{itemize} Other reviews include \begin{itemize}% \item [[Noriaki Ikeda]], \emph{Deformation of graded (Batalin-Volkvisky) Structures} in Dito, Lu, Maeda, Weinstein (eds.) \emph{Poisson geometry in mathematics and physics} Contemp. Math. 450, AMS (2008) \item [[Noriaki Ikeda]], \emph{Lectures on AKSZ Topological Field Theories for Physicists} (\href{http://arxiv.org/abs/1204.3714}{arXiv:1204.3714}) \end{itemize} A cohomological reduction of the formalism is described in \begin{itemize}% \item F. Bonechi, P. Mn\"e{}v, [[Maxim Zabzine]], \emph{Finite dimensional AKSZ-BV-theories} (\href{http://arxiv.org/abs/0903.0995}{arXiv}) \end{itemize} That the AKSZ action on bounding manifolds $\partial \hat \Sigma$ is the integral of the graded symplectic form over $\hat \Sigma$ is theorem 4.4 in \begin{itemize}% \item A. Kotov, T. Strobl, \emph{Characteristic classes associated to Q-bundles} (\href{http://arxiv.org/abs/0711.4106v1}{arXiv:0711.4106v1}) \end{itemize} The discussion of the AKSZ action functional as the [[nLab:∞-Chern-Simons theory]]-functional induced from a [[symplectic Lie n-algebroid]] in [[∞-Chern-Weil theory]] is due discussed in \begin{itemize}% \item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]}, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (\href{http://arxiv.org/abs/1108.4378}{arXiv:1108.4378}) \end{itemize} In the broader context of smooth [[higher geometry]] this is discussed in section 4.3 of \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} Discussion of [[boundary conditions]] for the AKSZ sigma model includes \begin{itemize}% \item [[Peter Bouwknegt]], [[Branislav Jurco]], \emph{AKSZ construction of topological open $p$-brane action and Nambu brackets}, \href{http://arxiv.org/abs/1110.0134}{arxiv/1110.0134} \item [[Noriaki Ikeda]], Xiaomeng Xu, \emph{Canonical functions and differential graded symplectic pairs in supergeometry and AKSZ sigma models with boundary} (\href{http://arxiv.org/abs/1301.4805}{arXiv:1301.4805}) \end{itemize} The AKSZ model is extended to coisotropic boundary conditions in \begin{itemize}% \item [[Theo Johnson-Freyd]], \emph{Exact triangles, Koszul duality, and coisotropic boundary conditions} (\href{https://arxiv.org/abs/1608.08598}{arxiv/1608.08598}) \end{itemize} An example in [[higher spin gauge theory]] is discussed in \begin{itemize}% \item K.B. Alkalaev, Maxim Grigoriev, E.D. Skvortsov, \emph{Uniformizing higher-spin equations} (\href{http://arxiv.org/abs/1409.6507}{arXiv:1409.6507}) \end{itemize} See also \begin{itemize}% \item Theodore Th. Voronov, \emph{Vector fields on mapping spaces and a converse to the AKSZ construction}, \href{http://arxiv.org/abs/1211.6319}{arxiv/1211.6319} \end{itemize} [[!redirects AKSZ sigma-models]] [[!redirects AKSZ sigma model]] [[!redirects AKSZ functional]] [[!redirects AKSZ model]] [[!redirects AKSZ formalism]] [[!redirects AKSZ theory]] [[!redirects AKSZ]] [[!redirects AKSZ-sigma models]] \end{document}