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\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*{sigma-model -- exposition of higher gauge theories as sigma-models} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{phyiscs}{}\paragraph*{{Phyiscs}}\label{phyiscs} [[!include physicscontents]] \hypertarget{chernsimons_theory}{}\paragraph*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory} [[!include infinity-Chern-Simons theory - contents]] \begin{quote}% This is a sub-entry of [[sigma-model]]. See there for background and context. \end{quote} \vspace{.5em} \hrule \vspace{.5em} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{exposition_of_higher_gauge_theories_as_models}{Exposition of higher gauge theories as $\sigma$-models}\dotfill \pageref*{exposition_of_higher_gauge_theories_as_models} \linebreak \noindent\hyperlink{HigherGeometricTargetSpaces}{Higher geometric target spaces}\dotfill \pageref*{HigherGeometricTargetSpaces} \linebreak \noindent\hyperlink{SigmaCS}{Chern-Simons theory as a $\sigma$-model}\dotfill \pageref*{SigmaCS} \linebreak \noindent\hyperlink{AKSZSigmaModel}{AKSZ theory as a higher Chern-Simons $\sigma$-model}\dotfill \pageref*{AKSZSigmaModel} \linebreak \noindent\hyperlink{summary}{Summary}\dotfill \pageref*{summary} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{SigmaDW}{Dijkgraaf-Witten theory as a $\sigma$-model}\dotfill \pageref*{SigmaDW} \linebreak \noindent\hyperlink{the_yetter_model_as_a_model}{The Yetter model as a $\sigma$-model}\dotfill \pageref*{the_yetter_model_as_a_model} \linebreak \noindent\hyperlink{dijkgraafwitten_theory}{$\infty$-Dijkgraaf-Witten theory}\dotfill \pageref*{dijkgraafwitten_theory} \linebreak \noindent\hyperlink{chernsimons_theory_2}{$\infty$-Chern-Simons theory}\dotfill \pageref*{chernsimons_theory_2} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{exposition_of_higher_gauge_theories_as_models}{}\subsection*{{Exposition of higher gauge theories as $\sigma$-models}}\label{exposition_of_higher_gauge_theories_as_models} We discuss how [[gauge theories]] and their higher analogs are naturally regarded as $\sigma$-models. \hypertarget{HigherGeometricTargetSpaces}{}\subsubsection*{{Higher geometric target spaces}}\label{HigherGeometricTargetSpaces} The [[sigma-model -- exposition of classical sigma-models|classical sigma-models]] all have [[target space]]s that are [[smooth manifold]]s. However, we saw that from [[dimension]] $d \geq 2$ on, the [[background gauge field]]s on these target spaces are naturally no longer just [[principal bundle]]s [[connection on a bundle|with connection]]: instead they are smooth [[principal 2-bundle]]s, then smooth principal 3-bundles, etc. and eventually generally [[principal ∞-bundle]]s with [[connection on an ∞-bundle|∞-connections]]. But the total [[space]] of such higher smooth bundles is no longer in general a [[smooth manifold]]: instead, the total space is a \emph{[[Lie groupoid]]} for $d = 2$, then a \emph{[[Lie 2-groupoid]]} for $d = 3$ and eventually generally a [[smooth ∞-groupoid]]. This means that -- unless we would artifically treat the total space of a [[background gauge field]] bundle on different grounds than its base space -- the general theory of $\sigma$-models should naturally include target spaces that are not just smooth manifolds. At least from $d = 2$ on, for instance, target spaces should be allowed to be [[Lie groupoid]]s. This has a fairly long tradition: the \emph{[[proper groupoid|proper]] [[étale groupoid|étale]]} [[Lie groupoid]]s are precisely \emph{[[orbifold]]s}, spaces that are locally isomorphic to sufficiently nice quotients of a [[Cartesian space]] by a [[group]] [[action]]. Orbifolds have received a lot of attention in the study of [[string]] [[sigma-model]]s. The [[orientifold]] background gauge fields mentioned \href{http://ncatlab.org/nlab/show/sigma-model+--+exposition+of+classical+sigma-models#RelativisticString}{before} involve in general a $\mathbb{Z}_2$-orbifold target space, for instance. But once we pass to the higher geometry of [[Lie groupoid]]s at all, there is no good reason to restrict ourselves to those that are orbifolds. For instance, for any [[Lie group]] $G$ there is its \emph{[[delooping]]} Lie groupoid, the [[action groupoid]] of the trivial action of $G$ on the point, which we shall write \begin{displaymath} \mathbf{B}G := *//G \,; \end{displaymath} and this can perfectly serve as a [[target space]] object for $\sigma$-models too. Here the boldface notation is to indicate that this Lie groupoid is a smooth refinement of the [[classifying space]] $B G \in Top$ of the Lie group. In fact, where $B G$ gives [[isomorphism class]] of smooth $G$-principal bundles, $\mathbf{B}G$ also remembers the [[isomorphism]]s themselves -- and hence in particular the [[automorphism]]s -- of these bundles. It is the \emph{[[moduli stack]]} of smooth $G$-principal bundles: for $\Sigma$ a [[smooth manifold]] we have that the \emph{groupoid} of morphisms of [[smooth infinity-groupoid|smooth groupoid]]s $\Sigma \to \mathbf{B}G$ (the \emph{correct} morphisms, sometimes called \emph{[[Morita morphism]]} to distinguish them from any incorrect notion) is that of smooth $G$-principal bundles and smooth homomorphisms between these \begin{displaymath} SmoothGrpd(\Sigma, \mathbf{B}G) \simeq G Bund(\Sigma) \in Grpd \,, \end{displaymath} whereas the [[geometric realization]] $B G \simeq \vert \mathbf{B}G\vert$ only sees the equivalence classes: \begin{displaymath} [\Sigma, B G] \simeq \pi_0 G Bund(\Sigma) \in Set \,. \end{displaymath} This indicates that ([[nonabelian cohomology|nonabelian]]) [[gauge theory]] on $\Sigma$ should have a formulation as a $\sigma$-model with target ``[[space]]'' $X$ the Lie groupoid $X = \mathbf{B}G$: a $\sigma$-model field $\Sigma \to X = \mathbf{B}G$ is a $G$-bundle, and an isomorphism of field configurations is a [[gauge transformation]] of $G$-bundles. But a field configuration in $G$-gauge theory on $\Sigma$ is not just a $G$-principal bundle, but is a $G$-bundle \emph{with [[connection on a bundle|connection]]}. There is no Lie groupoid that that would similarly represent such connections as a target space object. But there is a \emph{[[smooth infinity-groupoid|smooth groupoid]]} that does: $\mathbf{B}G_{conn}$, the [[groupoid of Lie algebra valued 1-forms]]. Here by a \emph{smooth groupoid} we mean a [[groupoid]] that comes with a rule for which of its collections of objects or morphisms are \emph{smoothly parameterized families}. Technically this is a \emph{[[(2,1)-sheaf]]} or \emph{[[stack]]} on the [[site]] [[CartSp]] of [[Cartesian space]]s and [[smooth function]]s between them. Among all smooth groupoids, Lie groupoids -- and generally [[diffeological space|diffeological groupoid]]s -- are singled out as being the \emph{[[concrete sheaf|concrete]]} objects. While it is useful to know if a given smooth groupoid is [[concrete sheaf|concrete]] or even [[Lie groupoid|Lie]], it is in any case a fact that all of [[higher geometry|higher]] [[differential geometry]] exists for general smooth $\infty$-groupoids just as well. Therefore, if we can allow Lie groupoids as targets for $\sigma$-models, we can allow general smooth groupoids as well. The non-concrete smooth groupoid $\mathbf{B}G_{conn}$ that we just mentioned is defined by the following rule: for $U \in$ [[CartSp]], a smoothly $U$-parameterized family of objects is by definition a $\mathfrak{g}$-[[Lie algebra valued 1-form|valued differential 1-form]] $A \in \Omega^1(U, \mathfrak{g})$ on $U$, where $\mathfrak{g}$ is the [[Lie algebra]] of $G$. A smoothly $U$-parameterized family of morphisms $g : A_1 \to A_2$ is a smooth [[gauge transformation]] $g \in C^\infty(U, G) : A_2 = g A g^{-1} + g d g^{-1}$ between two such form data. (This is ``non-concrete'' because the smooth $U$-parameterized families $U \to \mathbf{B}G_{conn}$ are not $U$-families of points $* \to \mathbf{B}G_{conn}$.) One then finds that the [[mapping space]] groupoid for this target $X = \mathbf{B}G_{conn}$ is the groupoid \begin{displaymath} SmoothGrpd(\Sigma , \mathbf{B}G_{conn}) \simeq G Bund_{conn}(\Sigma) \,, \end{displaymath} whose objects are smooth $G$-principal bundles with connection on $\Sigma$, and whose morphisms are smooth morphisms of principal bundles with connection. This groupoid is the [[configuration space]] of $G$-[[gauge theory]] on $\Sigma$, for instance of $G$-[[Yang-Mills theory]] or of $G$-[[Chern-Simons theory]]: \begin{displaymath} SmoothGrpd(\Sigma, \mathbf{B}G_{conn}) \simeq Conf_{Yang-Mills}(\Sigma) \simeq Conf_{Chern-Simons}(\Sigma) \,. \end{displaymath} Notice that this configuration space is now itself a groupoid: morphisms are [[gauge transformation]]s. In fact, it is naturally itself a smooth groupoid (when we read the [[hom-object]] here as an [[internal hom]] in [[Smooth∞Grpd|SmoothGrpd]]). In the traditional physics literature these Lie groupoidal [[configuration space]]s of fields are best known in terms of their [[infinitesimal object|infinitesimal]] approximation $Lie(Conf(\Sigma)) \in LieAld$, which are \emph{[[Lie algebroid]]s}, and these in turn are best known in terms of their [[function algebras on ∞-stacks|function algebras]], called the \emph{[[Chevalley-Eilenberg algebra]]s} $CE(Lie(Conf(\Sigma)))$: this [[dg-algebra]] is in physics called the [[BRST complex]]. Its degree-1 generators, the [[cotangent vector|cotangents]] to the [[morphism]]s of $Conf(\Sigma)$, are called the \emph{ghost fields} of gauge theory. Of course we already saw secretly groupoidal configuration spaces in the \href{ExpositionClassical}{above} list of examples of $\sigma$-models of relativistic [[branes]]. We said that their configuration spaces $C^\infty(\Sigma,X)//Diff(\Sigma)$ were [[quotient]]s; but really they are to be taken as higher categorical quotients, known as \emph{[[homotopy colimit|homotopy quotients]]} or \emph{[[2-colimit|weak quotients]]} : they are the [[action groupoid]]s of $Diff(\Sigma)$ acting on $C^\infty(\Sigma,X)$. We will see in the examples below that there is, of course, no reason to stop after passing from target manifolds to smooth target groupoids. At least as the $\sigma$-model increases in dimension, it is natural to consider smooth target [[2-groupoid]]s, target [[3-groupoid]]s, \ldots{} target [[n-groupoid]]s and eventually [[smooth ∞-groupoids]]. The full context of smooth $\infty$-groupoids is the natural completion of traditional [[differential geometry]] to \emph{[[higher geometry]]} . Given that it does thus make sense to regard general [[smooth ∞-groupoid]]s as target spaces for $\sigma$-models, the questions is if there are useful [[background gauge field]]s on such. This is indeed the case: for instance we have a that says that for $G$ a compact Lie group, there is, for every integral cohomology class $c \in H^{n+1}(B G, \mathbb{Z})$ of the classifying space of $G$ -- a [[characteristic class]] for $G$-principal bundles -- up to equivalence a unique smooth lift $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ to a smooth [[circle n-bundle]] on the smooth $\mathbf{B}G$. Moreover, we have a [[infinity-Chern-Weil theory|theorem]] that for sufficiently highly connected Lie groups or smooth $\infty$-groups $G$, this refines canonically to a [[circle n-bundle with connection]] on the differentially refined smooth moduli space $\mathbf{B}G_{conn}$, given by a morphism: \begin{displaymath} \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} \,. \end{displaymath} This assignment generalizes the classical [[Chern-Weil homomorphism]]: we may speak of the \emph{[[∞-Chern-Weil homomorphism]]} . The first example \hyperlink{SigmaCS}{below} shows that ordinary [[Chern-Simons theory]] is a $\sigma$-model that arises this way. Generally we may this speak of $\sigma$-models with target space a [[smooth ∞-groupoid]] and [[background gauge field]]s given by the [[∞-Chern-Weil homomorphism]] this way as [[schreiber:∞-Chern-Simons theories]]. The second example \hyperlink{SigmaDW}{below} shows that ordinatry [[Dijkgraaf-Witten theory]] is a $\sigma$-model that arises this way when $G$ is a [[discrete group]]. Generally we may thus speak of $\sigma$-models with target space a [[discrete ∞-groupoid]] and [[background gauge field]]s given by the [[∞-Chern-Weil homomorphism]] this way as [[schreiber:∞-Dijkgraaf-Witten theories]]. \hypertarget{SigmaCS}{}\subsubsection*{{Chern-Simons theory as a $\sigma$-model}}\label{SigmaCS} One of the earliest [[topological quantum field theories]] ever considered in detail is \emph{[[Chern-Simons theory]]} . We introduce this from the point of view of $\sigma$-models with higher geometric target spaces as discussed \hyperlink{HigherGeometricTargetSpaces}{above}. An ordinary (as opposed to higher) \emph{[[gauge theory]]} is a [[quantum field theory]] whose field configurations on a manifold $\Sigma$ are [[connection on a bundle|connections]] on $G$-[[principal bundle]]s over $\Sigma$, for $G$ some [[Lie group]]. The word \emph{[[gauge transformation]]} is essentially the physics equivalent of the word \emph{[[isomorphism]]} , referring to isomorphisms in a [[configuration space]] of a field theory and specifically to isomorphisms between such bundles with connection. The [[action functional]] of a gauge theory is to be \emph{[[gauge invariance|gauge invariant]]} meaning that it assigns the same value to configurations that are related by a gauge transformaiton. This means precisely that the exponentiated action is a [[functor]] \begin{displaymath} \exp(i S(-)) : G Bund_{conn}(\Sigma) \to U(1) \end{displaymath} from the [[groupoid]] of gauge field configurations and gauge transformaitons, to the [[circle group]] (regarded as a [[0-truncated]] groupoid). The first nonabelian gauge theory to receive attention was \emph{[[Yang-Mills theory]]} : in that model $\Sigma$ is a 4-dimensional [[pseudo-Riemannian manifold]] modelling [[spacetime]]. The exponentiated [[action functional]] is given by the integral of differential 4-forms naturally associated with a connection and a Riemannian structure: \begin{displaymath} \exp(i S_{YM}(-)) : (P, \nabla) \mapsto \exp(i \int_\Sigma \frac{1}{e^2} \langle F_\nabla \wedge \star F_\nabla\rangle + i \theta \langle F_\nabla \wedge F_\nabla \rangle) \,. \end{displaymath} Here \begin{itemize}% \item $P$ is any $G$-[[principal bundle]] and $\nabla$ a [[connection on a bundle|connection]] on it; \item $F_\nabla \in \Omega^2(P, \mathfrak{g})$ is the [[Lie algebra]]-valued [[curvature]] 2-form of this connection; \item $\langle -,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$ is an \emph{[[invariant polynomial]]} on the Lie algebra: a bilinear form that is gauge invariant when evaluated on curvature 2-forms -- for $\mathfrak{g}$ a [[semisimple Lie algebra]] this would be the \emph{[[Killing form]]} and for a [[matrix Lie algebra]] this is simply the [[trace]] operation on products of matrices; \item $\star$ is the [[Hodge star]] operator given by the pseudo-Riemannian metric structure on $\Sigma$. \item $e \in \mathbb{R}$ is some constant, called the [[coupling constant]] of the model; \item $\theta$ is another parameter called the [[theta-angle]]. \end{itemize} The first summand in the exponent, that depending on the pseudo-Riemannian structure, is the crucial term for the direct application of this as a model of phenomenologically observed physics: it controls the dynamics of three of the four [[force]] fields in the [[standard model of particle physics]]. Instead of investigating this further, we shall here look at the case where $\frac{1}{e^2}$ is set to 0. While not \emph{directly} of phenomenological relevance, this is of quite some interest for the general theoretical understanding of the space of all possible field theories. Since the resulting action functional \begin{displaymath} \exp(i S_{tYM}) : (P, \nabla) \mapsto \exp(i \int_\Sigma \langle F_\nabla \wedge F_\nabla \rangle) \end{displaymath} no longer depends on any extra (pseudo-Riemannian) structure on $\Sigma$ this may be interpreted as defining a [[topological quantum field theory]] : one speaks of \emph{[[topological Yang-Mills theory]]} . This is not quite a $\sigma$-model in the sense that we have been discussing: while the [[configuration space]] of [[topological Yang-Mills theory]] does consist of maps into the [[target space]] $X = \mathbf{B}G_{conn}$ (the smooth [[moduli stack]] of $G$-principal bundles with connection, as discussed \href{HigherGeometricTargetSpaces}{above}), there is no way that the above action functional is induced directly from the transgression of the [[higher parallel transport|higher holonomy]] of a [[circle n-bundle with connection]] on this target space. This is because, at least for [[semisimple Lie group]]s $G$, these are nontrivial only for odd $n$, whereas here we have $n = dim \Sigma = 4$. But something closely related is true: $\exp(i S_{tYM})$ is the integrated \emph{[[curvature]]} functional of a circle $3$-bundle with connection on $\mathbf{B}G_{conn}$: what we call the \emph{[[Chern-Simons circle 3-bundle]]} . This means the following: in generalization of how an ordinary [[circle bundle]] with connection $\nabla$ has a [[curvature]] 2-form, a [[circle n-bundle with connection]] $\nabla$ on a manifold $X$ has a curvature $(n+1)$-form $F_\nabla \in \Omega^{n+1}_{cl}(X)$. These curvature forms are closed, but not necessarily exact. Nevertheless, a generalization of the [[Stokes theorem]] holds true for them: for $\Sigma$ of dimension $n+1$ and denoting by $\partial \Sigma$ the [[boundary]] of $\Sigma$ and by $\gamma : \Sigma \to X$ a $\Sigma$-shaped trajectory in $X$, we have that the integral of the curvature over $\Sigma$ equals the [[higher parallel transport|higher holonomy]] of $\nabla$ over $\partial_\Sigma$: \begin{displaymath} \exp(i \int_\Sigma \phi^* F_\nabla) = hol(\nabla, \gamma|_{\partial \Sigma}) \,. \end{displaymath} This property in fact characterizes equivalence classes of circle $n$-bundles with connection. When conceiving of circle $n$-bundles with connection as rules for assigning higher holonomy that satisfy this property, one speaks of \emph{[[Cheeger-Simons differential characters]]} . Therefore, if we can find a circle 3-bundle with connection on the [[moduli stack]] $\mathbf{B}G_{conn}$ of $G$-principal bundles with connection whose [[curvature]] 4-form at $(P,\nabla)$ is $\langle F_\nabla \wedge F_\nabla \rangle$, then we can interpret [[topological Yang-Mills theory]] on a 4-dimensional $\Sigma$ with boundary as being given by a $\sigma$-model on $\partial \Sigma$ with [[background gauge field]] that circle 3-bundle. For $G$ a connected and [[simply connected]] Lie group, such a circle 3-bundle indeed exists. Its characteristic morphism \begin{displaymath} \frac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} \end{displaymath} from the smooth moduli stack of $G$-bundles with connection to the smooth moduli 3-groupoid of [[circle n-bundle with connection|circle 3-bundles with connection]] is constructed and discussed in (), see \emph{[[Chern-Simons circle 3-bundle]]} . This is the differential refinement of the [[differential string structure|smooth first fractional Pontryagin class]] \begin{displaymath} \frac{1}{2}\mathbf{p}_1 : \mathbf{B}G \to \mathbf{B}^3 U(1) \end{displaymath} which in turn is a smooth refinement of the fractional [[Pontryagin class]] \begin{displaymath} \frac{1}{2} p_1 : B G \to B^3 U(1) \simeq K(\mathbb{Z}, 4) \end{displaymath} of the [[classifying space]] $B G$. To get a feeling for what this circle 3-bundle is like, we look at what its pull-back $\frac{1}{2}\hat \mathbf{p}_1(\phi) : \Sigma \stackrel{\phi}{\to} \mathbf{B}G_{conn} \stackrel{\frac{1}{2} \hat \mathbf{p}_1}{\to} \mathbf{B}^3 U(1)_{conn}$ to $\Sigma$ along any field configuration $\phi : \Sigma \to X = \mathbf{B}G_{conn}$ is like. Notice that for simply conneced $G$ the [[classifying space]] $B G$ has vanishing [[homotopy group]]s in degree $k \leq 3$. Therefore every $G$-principal bundle $P$ on the 3-dimensional $\partial \Sigma$ is necessarily trivializable. In this case the [[configuration space]] of the $\sigma$-model is equivalent to the [[groupoid of Lie algebra valued forms]] \begin{displaymath} SmoothGrpd(\partial \Sigma, \mathbf{B}G_{conn}) \simeq \Omega^1(\partial \Sigma, \mathfrak{g})//C^{\infty}(\partial \Sigma,G) \end{displaymath} on $\partial \Sigma$. For $A \in \Omega^1(\Sigma, \mathrak{g})$ a field configuration and $F_A = d A + \frac{1}{2}[A \wedge A]$ the corresponding curvature 2-form, the curvature 4-form of $\frac{1}{2}\hat \mathbf{p}_1(\phi)$ is $\langle F_A \wedge F_A \rangle$. Its connection 3-form $C$ satisfying $d C = \langle F_A \wedge F_A \rangle$ is -- up to a closed 3-form -- the \emph{[[Chern-Simons form|Chern-Simons 3-form]]} \begin{displaymath} C = cs(A) = \langle A \wedge F_A \rangle + \frac{1}{6}\langle A \wedge [A \wedge A]\rangle \,. \end{displaymath} Therefore the [[action functional]] of the 3-dimensional $\sigma$-model given by the [[background gauge field]] $\frac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$ is given by \begin{displaymath} \exp(i S(-))_{\Sigma_3} : A \mapsto \exp(i \int_{\Sigma_3} cs(A)) \,. \end{displaymath} The [[quantum field theory]] defined by this action functional is known as \emph{[[Chern-Simons theory]]} . \hypertarget{AKSZSigmaModel}{}\subsubsection*{{AKSZ theory as a higher Chern-Simons $\sigma$-model}}\label{AKSZSigmaModel} \hypertarget{summary}{}\paragraph*{{Summary}}\label{summary} Every [[symplectic Lie n-algebroid]] $\mathfrak{P}$ serves as the [[target space]] of a canonically defined topological $\sigma$-model of dimension $n+1$. This is called the [[AKSZ theory|AKSZ sigma-model]] of $\mathcal{P}$. This subsumes the following examples: \begin{itemize}% \item for $\mathfrak{P}$ a [[semisimple Lie algebra]], the AKSZ $\sigma$-model is ordinary [[Chern-Simons theory]]; \item for $\mathfrak{P}$ a [[Poisson Lie algebroid]], the AKSZ $\sigma$-model is corresponding [[Poisson sigma-model]]; \item for $\mathfrak{P}$ a [[Courant Lie 2-algebroid]], the AKSZ $\sigma$-model is corresponding [[Courant sigma-model]]. \end{itemize} Also the [[A-model]] and the [[B-model]] 2-dimensional topological $\sigma$-models are examples. The AKSZ [[action functional]] turns out to be very fundamental: by [[∞-Chern-Weil theory]] every [[invariant polynomial]] on an [[L-∞ algebroid]] induces an [[∞-Chern-Weil homomorphism]] and the corresponding [[∞-Chern-Simons theory]] action functional. Moreover, every [[symplectic Lie n-algebroid]] canonically carries a binary invariant polynomial. The AKSZ $\sigma$-model action functional is precisely the value of the $\infty$-Chern-Weil homomorphism on this invariant polynomial. This is shown at \href{http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory+--+examples#ASKZTheory}{∞-Chern-Simons theory -- Examples -- AKSZ theory}. \hypertarget{definition}{}\paragraph*{{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 \, \pi(\phi\wedge \cdots \wedge \phi) \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{SigmaDW}{}\subsubsection*{{Dijkgraaf-Witten theory as a $\sigma$-model}}\label{SigmaDW} THe 3-dimensional [[TQFT]] [[Dijkgraaf-Witten theory]] can be understood as being the [[∞-Chern-Simons theory]]-$\sigma$-model whose [[target space]] is $\mathbf{B}G$ for $G$ a [[discrete group]]. See \href{http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory+--+examples#DiscreteTargets}{∞-Chern-Simons theory -- Examples -- Dijkgraaf-Witten theory} \hypertarget{the_yetter_model_as_a_model}{}\subsubsection*{{The Yetter model as a $\sigma$-model}}\label{the_yetter_model_as_a_model} (\ldots{}) \hypertarget{dijkgraafwitten_theory}{}\subsubsection*{{$\infty$-Dijkgraaf-Witten theory}}\label{dijkgraafwitten_theory} (\ldots{}) \hypertarget{chernsimons_theory_2}{}\subsubsection*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory_2} See [[schreiber:∞-Chern-Simons theory]]. \hypertarget{references}{}\subsection*{{References}}\label{references} The [[AKSZ sigma-model]] is discussed in \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} \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}). \end{itemize} General $\infty$-Chern-Simons theory is discussed in \begin{itemize}% \item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:infinity-Chern-Simons theory]]} \end{itemize} \end{document}