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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*{Wess-Zumino-Witten model} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{wesszuminowitten_theory}{}\paragraph*{{$\infty$-Wess-Zumino-Witten theory}}\label{wesszuminowitten_theory} [[!include infinity-Wess-Zumino-Witten theory - contents]] \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{ActionFunctional}{Action functional}\dotfill \pageref*{ActionFunctional} \linebreak \noindent\hyperlink{KineticTerm}{Kinetic term}\dotfill \pageref*{KineticTerm} \linebreak \noindent\hyperlink{TopologicalTerm}{Topological term -- WZW term}\dotfill \pageref*{TopologicalTerm} \linebreak \noindent\hyperlink{WZWTermFor2dModel}{For the 2d WZW model}\dotfill \pageref*{WZWTermFor2dModel} \linebreak \noindent\hyperlink{FormalizationGenerally}{Generally}\dotfill \pageref*{FormalizationGenerally} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{EquationsOfMotion}{Equations of motion}\dotfill \pageref*{EquationsOfMotion} \linebreak \noindent\hyperlink{HolographyAndRigorousConstruction}{Holography and Rigorous construction}\dotfill \pageref*{HolographyAndRigorousConstruction} \linebreak \noindent\hyperlink{DBranes}{D-branes for the WZW model}\dotfill \pageref*{DBranes} \linebreak \noindent\hyperlink{quantization}{Quantization}\dotfill \pageref*{quantization} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{original_references}{Original references}\dotfill \pageref*{original_references} \linebreak \noindent\hyperlink{introductions_and_surveys}{Introductions and surveys}\dotfill \pageref*{introductions_and_surveys} \linebreak \noindent\hyperlink{ReferencesRelationToGerbesAndCS}{Relation to gerbes and Chern-Simons theory}\dotfill \pageref*{ReferencesRelationToGerbesAndCS} \linebreak \noindent\hyperlink{partition_functions}{Partition functions}\dotfill \pageref*{partition_functions} \linebreak \noindent\hyperlink{ReferencesDBranes}{D-branes for the WZW model}\dotfill \pageref*{ReferencesDBranes} \linebreak \noindent\hyperlink{relation_to_dimensional_reduction_of_chernsimons}{Relation to dimensional reduction of Chern-Simons}\dotfill \pageref*{relation_to_dimensional_reduction_of_chernsimons} \linebreak \noindent\hyperlink{relation_to_extended_tqft}{Relation to extended TQFT}\dotfill \pageref*{relation_to_extended_tqft} \linebreak \noindent\hyperlink{in_solid_state_physics}{In solid state physics}\dotfill \pageref*{in_solid_state_physics} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \textbf{Wess-Zumino-Witten model} (or \textbf{WZW model} for short, also called \textbf{Wess-Zumino-Novikov-Witten} model, or short \textbf{WZNW} model) is a 2-dimensional [[sigma-model]] [[quantum field theory]] whose target space is a [[Lie group]]. This may be regarded as the boundary theory of [[Chern-Simons theory]] for Lie group $G$. The [[vertex operator algebra]]s corresponding to the WZW model are [[current algebra]]s. \hypertarget{ActionFunctional}{}\subsection*{{Action functional}}\label{ActionFunctional} For $G$ a [[Lie group]], the [[configuration space]] of the WZW over a 2-[[dimension]]al [[manifold]] $\Sigma$ is the space of [[smooth function]]s $g : \Sigma \to G$. The [[action functional]] of the WZW [[sigma-model]] is the sum of two terms, a kinetic term and a topological term \begin{displaymath} S_{WZW} = S_{kin} + S_{top} \,. \end{displaymath} \hypertarget{KineticTerm}{}\subsubsection*{{Kinetic term}}\label{KineticTerm} The Lie group canonically carries a [[Riemannian metric]] and the kinetic term is the standard one for [[sigma-model]]s on Riemannian [[target space]]s. \hypertarget{TopologicalTerm}{}\subsubsection*{{Topological term -- WZW term}}\label{TopologicalTerm} \hypertarget{WZWTermFor2dModel}{}\paragraph*{{For the 2d WZW model}}\label{WZWTermFor2dModel} In [[higher differential geometry]], then given any closed [[differential n-form|differential (p+2)-form]] $\omega \in \Omega^{p+2}_{cl}(X)$, it is natural to ask for a [[prequantization]] of it, namely for a [[circle n-bundle with connection|circle (p+1)-bundle with connection]] $\nabla$ (equivalently: [[cocycle]] in degree-$(p+2)$-[[Deligne cohomology]]) on $X$ whose [[curvature]] is $F_\nabla = \omega$. In terms of [[moduli stacks]] this means asking for lifts of the form \begin{displaymath} \itexarray{ && \mathbf{B}^{p+1}U(1)_{conn} \\ &{}^{\mathllap{\nabla}}\nearrow& \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} } \end{displaymath} in the [[homotopy theory]] of [[smooth homotopy types]]. This immediately raises the question for natural classes of examples of such prequantizations. One such class arises in [[infinity-Lie theory]], where $\omega$ is a [[left invariant form]] on a [[smooth infinity-group]] given by a [[cocycle]] in [[L-infinity algebra cohomology]]. The [[prequantum n-bundles]] arising this way are the higher [[WZW terms]] discussed here. In low degree of traditional [[Lie theory]] this appears as follows: On [[Lie groups]] $G$, those closed $(p+2)$-forms $\omega$ which are [[left invariant forms]] may be identified, via the general theory of [[Chevalley-Eilenberg algebras]], with degree $(p+2)$-[[cocycles]] $\mu$ in the [[Lie algebra cohomology]] of the [[Lie algebra]] $\mathfrak{g}$ corresponding to $G$. These in turn may arise, via the [[van Est map]], as the [[Lie differentiation]] of a degree-$(p+2)$-[[cocycle]] $\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^{p+2}U(1)$ in the [[Lie group cohomology]] of $G$ itself, with [[coefficients]] in the [[circle group]] $U(1)$. This happens to be the case notably for $G$ a [[simply connected topological space|simply connected]] [[compact Lie group|compact]] [[semisimple Lie group]] such as [[special unitary group|SU]] or [[spin group|Spin]], where $\mu = \langle -,[-,-]\rangle$ is the [[Lie algebra cohomology|Lie algebra 3-cocycle]] in [[transgression]] with the [[Killing form]] [[invariant polynomial]] $\langle -,-\rangle$. This is, up to normalization, a representative of the de Rham image of a generator $\mathbf{c}$ of $H^3(\mathbf{B}G, U(1)) \simeq H^4(B G, \mathbb{Z}) \simeq \mathbb{Z}$. Generally, by the discussion at \emph{[[geometry of physics -- principal bundles]]}, the cocycle $\mathbf{c}$ [[modulating morphism|modulates]] an [[infinity-group extension]] which is a [[circle n-group|circle p-group]]-[[principal infinity-bundle]] \begin{displaymath} \itexarray{ \mathbf{B}^p U(1) &\longrightarrow& \hat G \\ && \downarrow \\ && G &\stackrel{\Omega\mathbf{c}}{\longrightarrow}& \mathbf{B}^{p+1}U(1) } \end{displaymath} whose higher [[Dixmier-Douady class]] class $\int \Omega \mathbf{c} \in H^{p+2}(X,\mathbb{Z})$ is an integral lift of the real cohomology class encoded by $\omega$ under the [[de Rham isomorphism]]. In the example of [[spin group|Spin]] and $p = 1$ this extension is the [[string 2-group]]. Such a [[Lie theory|Lie theoretic]] situation is concisely expressed by a diagram of [[smooth homotopy types]] of the form \begin{displaymath} \itexarray{ && &\longrightarrow& \mathbf{B}^{p+1}U(1) \\ &{}^{\mathllap{\Omega \mathbf{c}}}\nearrow& &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}} \\ G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \,, \end{displaymath} where $\flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \simeq \flat_{dR}\mathbf{B}^{p+2}U(1)$ is the \href{cohesive+infinity-topos+--+structures#deRhamCohomology}{de Rham coefficients} (see also at \emph{[[geometry of physics -- de Rham coefficients]]}) and where the homotopy filling the diagram is what exhibits $\omega$ as a de Rham representative of $\Omega \mathbf{c}$. Now, by the very [[homotopy pullback]]-characterization of the [[Deligne complex]] $\mathbf{B}^{p+1}U(1)_{conn}$ (\href{Deligne+cohomology#TheExactDifferentialCohomologyHexagon}{here}), such a diagram is equivalently a [[prequantization]] of $\omega$: \begin{displaymath} \itexarray{ && \mathbf{B}^{p+1}U(1)_{conn} &\longrightarrow& \mathbf{B}^{p+1}U(1) \\ &{}^\mathllap{\nabla}\nearrow& \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}} \\ G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \,. \end{displaymath} For $\omega = \langle -,[-,-]\rangle$ as above, we have $p= 1$ and so $\nabla$ here is a [[circle n-bundle with connection|circle 2-bundle with connection]], often referred to as a [[bundle gerbe]] [[connection on a bundle gerbe|with connection]]. As such, this is also known as the \emph{WZW gerbe} or similar. This terminology arises as follows. In (\href{Wess-Zumino-Witten+model#WessZumino71}{Wess-Zumino 84}) the [[sigma-model]] for a [[string]] propagating on the [[Lie group]] $G$ was considered, with only the standard [[kinetic action]] term. Then in (\href{Wess-Zumino-Witten+model#Witten84}{Witten 84}) it was observed that for this [[action functional]] to give a [[conformal field theory]] after [[quantization]], a certain [[higher gauge theory|higher gauge]] [[interaction term]] has to the added. The resulting [[sigma-model]] came to be known as the \emph{[[Wess-Zumino-Witten model]]} or \emph{WZW model} for short, and the term that Witten added became the \emph{WZW term}. In terms of [[string theory]] it describes the propagation of the [[string]] on the group $G$ subject to a [[force]] of [[gravity]] given by the [[Killing form]] [[Riemannian metric]] and subject to a [[B-field]] [[higher gauge field|higher gauge force]] whose [[field strength]] is $\omega$. In (\href{Wess-Zumino-Witten+model#Gawedzki87}{Gawedzki 87}) it was observed that when formulated properly and generally, this WZW term is the [[surface holonomy]] functional of a [[connection on a bundle gerbe]] $\nabla$ on $G$. This is equivalently the $\nabla$ that we just motivated above. Later WZW terms, or at least their curvature forms $\omega$, were recognized all over the place in [[quantum field theory]]. For instance the [[Green-Schwarz sigma-models for super p-branes]] each have an [[action functional]] that is the sum of the standard [[kinetic action]] plus a WZW term of degree $p+2$. In general WZW terms are ``[[gauged WZW model|gauged]]'' which means, as we will see, that they are not defined on the give [[smooth infinity-group]] $G$ itself, but on a bundle $\tilde G$ of differential moduli stacks over that group, such that a map $\Sigma \to \tilde G$ is a pair consisting of a map $\Sigma \to G$ and of a [[higher gauge field]] on $\Sigma$ (a ``tensor multiplet'' of fields). \hypertarget{FormalizationGenerally}{}\paragraph*{{Generally}}\label{FormalizationGenerally} The following (\hyperlink{FSS12}{FSS 12}, \hyperlink{dcct}{dcct}) is a general axiomatization of WZW terms in [[cohesive (infinity,1)-topos|cohesive homotopy theory]]. In an ambient [[cohesive (∞,1)-topos]] $\mathbf{H}$, let $\mathbb{G}$ be a [[sylleptic ∞-group]], equipped with a [[Hodge filtration]], hence in particular with a chosen morphism \begin{displaymath} \iota \colon \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) \longrightarrow \flat_{dR} \mathbf{B}^2 \mathbb{G} \end{displaymath} to its \href{}{de Rham coefficients} \begin{defn} \label{RefinementOfHodgeFiltration}\hypertarget{RefinementOfHodgeFiltration}{} Given an [[∞-group]] object $G$ in $\mathbf{H}$ and given a [[group cohomology|group cocycle]] \begin{displaymath} \mathbf{c} \colon \mathbf{B}G \longrightarrow \mathbf{B}^2 \mathbb{G} \,, \end{displaymath} then a \emph{refinement of the [[Hodge filtration]]} of $\mathbb{G}$ along $\mathbf{c}$ is a completion of the [[cospan]] formed by $\flat_{dR}\mathbf{c}$ and by $\iota$ above to a [[diagram]] of the form \begin{displaymath} \itexarray{ \mathbf{\Omega}^1_{flat}(-,G) &\stackrel{\mu}{\longrightarrow}& \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) \\ \downarrow && \downarrow^{\mathrlap{\iota}} \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^2 \mathbb{G} } \,. \end{displaymath} We write $\tilde G$ for the [[homotopy pullback]] of this refinement along the [[Maurer-Cartan form]] $\theta_G$ of $G$ \begin{displaymath} \itexarray{ \tilde G &\stackrel{\theta_{\tilde G}}{\longrightarrow}& \mathbf{\Omega}^1_{flat}(-,G) \\ \downarrow && \downarrow \\ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR}\mathbf{B}G } \,. \end{displaymath} \end{defn} \begin{example} \label{}\hypertarget{}{} Let $\mathbf{H} =$ [[Smooth∞Grpd]] and $\mathbb{G} = \mathbf{B}^p U(1)$ the [[circle n-group|circle (p+1)-group]]. For $G$ an ordinary [[Lie group]], then $\mu$ may be taken to be the [[Lie algebra cohomology|Lie algebra cocycle]] corresponding to $\mathbf{c}$ and $\tilde G \simeq G$. On the opposite extreme, for $G = \mathbf{B}^p U(1)$ itself with $\mathbf{c}$ the identity, then $\tilde G = \mathbf{B}^p U (1)_{conn}$ is the [[coefficients]] for [[ordinary differential cohomology]] (the [[Deligne complex]] under [[Dold-Kan correspondence]] and [[infinity-stackification]]). Hence a more general case is a fibered product of these two, where $\tilde G$ is such that a map $\Sigma \longrightarrow \tilde G$ is equivalently a pair consisting of a map $\Sigma \to G$ and of differential $p$-form data on $\Sigma$. This is the case of relevance for WZW models of [[super p-branes]] with ``tensor multiplet'' fields on them, such as the [[D-branes]] and the [[M5-brane]]. \end{example} \begin{prop} \label{}\hypertarget{}{} In the situation of def. \ref{RefinementOfHodgeFiltration} there is an essentially unique [[prequantum n-bundle|prequantization]] \begin{displaymath} \mathbf{L}_{WZW} \colon \tilde G \longrightarrow \mathbf{B}^2 \mathbb{G}_{conn} \end{displaymath} of the closed differential form \begin{displaymath} \mu(\theta_{\tilde G}) \colon \tilde G \stackrel{\theta_{\tilde G}}{\longrightarrow} \mathbf{\Omega}^1_{flat}(-,G) \stackrel{\mu}{\longrightarrow} \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) \end{displaymath} whose underlying $\mathbb{G}$-[[principal ∞-bundle]] is [[modulating morphism|modulated]] by the [[looping and delooping|looping]] $\Omega \mathbf{c}$ of the original cocycle. This we call the \emph{WZW term} of $\mathbf{c}$ with respect to the chosen refinement of the Hodge structure. \end{prop} \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{EquationsOfMotion}{}\subsubsection*{{Equations of motion}}\label{EquationsOfMotion} The [[variational calculus|variational derivative]] of the WZW [[action functional]] is \begin{displaymath} \delta S_{WZW}(g) = -\frac{k}{2 \pi i } \int_\Sigma \langle (g^{-1}\delta g), \partial (g^{-1}\bar \partial g) \rangle \,. \end{displaymath} Therefore the classical [[equations of motion]] for function $g \colon \Sigma \to G$ are \begin{displaymath} \partial(g^{-1}\bar \partial g) = 0 \;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\; \bar \partial(g \partial g^{-1}) = 0 \,. \end{displaymath} The space of solutions to these equations is small. However, discussion of the [[quantization]] of the theory (\hyperlink{HolographyAndRigorousConstruction}{below}) suggests to consider these equations with the real [[Lie group]] $G$ replaced by its [[complexification]] to the [[complex Lie group]] $G({\mathbb{C}})$. Then the general solution to the equations of motion above has the form \begin{displaymath} g(z,\bar z) = g_{\ell}(z) g_r(\bar z)^{-1} \end{displaymath} where hence $g_{\ell} \colon \Sigma \to G(\mathbb{C})$ is any [[holomorphic function]] and $g_r$ similarly any anti-holomorphic function. (e.g. \hyperlink{Gawedzki99}{Gawedzki 99 (3.18), (3.19)}) \hypertarget{HolographyAndRigorousConstruction}{}\subsubsection*{{Holography and Rigorous construction}}\label{HolographyAndRigorousConstruction} By the [[AdS3-CFT2 and CS-WZW correspondence]] (see there for more details) the 2d WZW [[CFT]] on $G$ is related to $G$-[[Chern-Simons theory]] in $3d$. In fact a rigorous constructions of the $G$-WZW model as a [[rational 2d CFT]] is via the [[FRS-theorem on rational 2d CFT]], which constructs the model as a [[boundary field theory]] of the $G$-[[Chern-Simons theory]] as a [[3d TQFT]] incarnated via a [[Reshetikhin-Turaev construction]]. \hypertarget{DBranes}{}\subsubsection*{{D-branes for the WZW model}}\label{DBranes} The characterization of [[D-brane]] [[submanifolds]] for the [[open string]] WZW model on a [[Lie group]] $G$ comes from two consistency conditions: \begin{enumerate}% \item geometrical condition: For the open string [[CFT]] to still have [[current algebra]] [[worldsheet]] symmetry, hence for half the current algebra symmetry of the closed WZW string to be preserved, the [[D-brane]] [[submanifolds]] need to be [[conjugacy classes]] of the group manifold (see e.g. \hyperlink{AlekseevSchomerus}{Alekseev-Schomerus} for a brief review and further pointers). These conjugacy classes are therefore also called the \textbf{symmetric D-branes}. Notice that these conjugacy classes are equivalently the [[leaves]] of the [[foliation]] induced by the canonical [[Cartan-Dirac structure]] on $G$, hence (by the discussion at [[Dirac structure]]), the leaves induced by the [[Lagrangian dg-submanifold|Lagrangian sub-Lie 2-algebroids]] of the [[Courant Lie 2-algebroid]] which is the [[higher gauge groupoid]] (see there) of the background [[B-field]] on $G$.(It has been suggested by [[Chris Rogers]] that such a foliation be thought of as a higher real [[polarization]].) \item cohomological condition: In order for the Kapustin-part of the [[Freed-Witten-Kapustin anomaly]] of the [[worldsheet]] [[action functional]] of the open WZW string to vanish, the D-brane must be equipped with a [[Chan-Paton gauge field]], hence a [[twisted unitary bundle]] ([[bundle gerbe module]]) of some rank $n \in \mathbb{N}$ for the restriction of the ambient [[B-field]] to the brane. For [[simply connected topological space|simply connected]] Lie groups only the rank-1 Chan-Paton gauge fields and their direct sums play a role, and their existence corresponds to a trivialization of the underlying $\mathbf{B}U(1)$-[[principal 2-bundle]] ($U(1)$-[[bundle gerbe]]) of the restriction of the [[B-field]] to the brane. There is then a discrete finite collection of symmetric D-branes = [[conjugacy classes]] satisfying this condition, and these are called the \textbf{integral symmetric D-branes}. (\hyperlink{AlekseevSchomerus}{Alekseev-Schomerus}, \hyperlink{GW}{Gawedzki-Reis}). As observed in \hyperlink{AlekseevSchomerus}{Alekseev-Schomerus}, this may be thought of as identifying a D-brane as a variant kind of a [[Bohr-Sommerfeld leaf]]. More generally, on non-simply connected group manifolds there are nontrivial higher rank [[twisted unitary bundles]]/[[Chan-Paton gauge fields]] over conjugacy classes and hence the cohomological ``integrality'' or ``Bohr-Sommerfeld''-condition imposed on symmetric D-branes becomes more refined (\hyperlink{Gawedzki04}{Gawedzki 04}). \end{enumerate} In summary, the [[D-brane]] [[submanifolds]] in a Lie group which induce an [[open string]] WZW model that a) has one [[current algebra]] symmetry and b) is [[Freed-Witten-Kapustin anomaly|Kapustin-anomaly]]-free are precisely the [[conjugacy class]]-submanifolds $G$ equipped with a [[twisted unitary bundle]] for the restriction of the background [[B-field]] to the conjugacy class. \hypertarget{quantization}{}\subsubsection*{{Quantization}}\label{quantization} on [[quantization]] of the WZW model, see at \begin{itemize}% \item [[quantization of loop groups]], \item [[equivariant elliptic cohomology]] \item [[quantization of Chern-Simons theory]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[geometry of physics -- WZW terms]] \item [[current algebra]], [[affine Lie algebra]] \item [[Knizhnik-Zamolodchikov equation]] \item [[coset WZW model]] \item [[gauged WZW model]] \item [[parameterized WZW model]] \item [[higher dimensional WZW model]] \item [[Green-Schwarz action functional]] \item [[analytically continued Wess-Zumino-Witten theory]] \item [[Gepner model]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{original_references}{}\subsubsection*{{Original references}}\label{original_references} The Wess-Zumino gauge-coupling term goes back to \begin{itemize}% \item [[Julius Wess]], [[Bruno Zumino]], \emph{Consequences of anomalous ward identities}. Phys. Lett. 37B, 95 (1971) \end{itemize} and was understood as yielding a 2-dimensional [[conformal field theory]] in \begin{itemize}% \item [[Edward Witten]], \emph{Non-Abelian bosonization in two dimensions} Commun. Math. Phys. 92, 455 (1984) \item [[Vadim Knizhnik]], [[Alexander Zamolodchikov]], \emph{Current algebra and Wess-Zumino model in two dimensions}, Nucl. Phys. B 247, 83-103 (1984) \end{itemize} and hence (a possible part of) a [[string theory]] [[vacuum]]/[[target space]] in \begin{itemize}% \item [[Doron Gepner]], [[Edward Witten]], \emph{String theory on group manifolds}, Nucl. Phys. B 278, 493-549 (1986) (\href{http://inspirehep.net/record/230076}{spire:230076}) \end{itemize} The WZ term on $\Sigma_2$ was understood in terms of an integral of a 3-form over a cobounding manifold $\Sigma_3$ in \begin{itemize}% \item [[Edward Witten]], \emph{Global aspects of current algebra}, Nucl. Phys. B223, 422 (1983) (\href{http://inspirehep.net/record/13234}{spire:13234}, , \href{https://www.phys.sinica.edu.tw/~spring8/users/jychen/pub/reference/NPB_v223_422.pdf}{pdf}) \end{itemize} for the case that $\Sigma_2$ is [[closed manifold|closed]], and generally, in terms of [[surface holonomy]] of [[bundle gerbes]]/[[circle 2-bundles with connection]] in \begin{itemize}% \item [[Krzysztof Gawedzki]], \emph{Topological Actions in two-dimensional Quantum Field Theories}, in [[Gerard `t Hooft]] et. al (eds.) \emph{Nonperturbative quantum field theory} Cargese 1987 proceedings, (\href{http://inspirehep.net/record/257658?ln=de}{web}) \item [[Giovanni Felder]] , [[Krzysztof Gawedzki]], A. Kupianen, \emph{Spectra of Wess-Zumino-Witten models with arbitrary simple groups}. Commun. Math. Phys. 117, 127 (1988) \item [[Krzysztof Gawedzki]], \emph{Topological actions in two-dimensional quantum field theories}. In: Nonperturbative quantum field theory. `tHooft, G. et al. (eds.). London: Plenum Press 1988 \end{itemize} as the [[surface holonomy]] of a [[circle 2-bundle with connection]]. See also the references at \emph{[[B-field]]} and at \emph{[[Freed-Witten anomaly cancellation]]}. See also \begin{itemize}% \item [[Pierre Deligne]], [[Daniel Freed]], chapter 6 of \emph{Classical field theory} (1999) (\href{https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf}{pdf}) this is a chapter in [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], J. Morgan, D.R. Morrison, [[Edward Witten|E. Witten]] (eds.) \emph{[[Quantum Fields and Strings]], A course for mathematicians}, 2 vols. Amer. Math. Soc. Providence 1999. (\href{http://www.math.ias.edu/qft}{web version}) \end{itemize} For the fully general understanding as the [[surface holonomy]] of a [[circle 2-bundle with connection]] see the references \hyperlink{ReferencesRelationToGerbesAndCS}{below}. See also \begin{itemize}% \item [[Edward Witten]], \emph{On holomorphic factorization of WZW and coset models}, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. (\href{http://projecteuclid.org/euclid.cmp/1104249222}{Euclid}) \end{itemize} \hypertarget{introductions_and_surveys}{}\subsubsection*{{Introductions and surveys}}\label{introductions_and_surveys} An survey of and introduction to the topic is in \begin{itemize}% \item Patrick Meessen, \emph{Strings Moving on Group Manifolds: The WZW Model} (\href{http://www.unioviedo.es/hepth/people/Patrick/fysica/zooi/WZW_ClassMunoz.pdf}{pdf}) \end{itemize} A classical textbook accounts include \begin{itemize}% \item [[Bojko Bakalov]], [[Alexander Kirillov]], chapter 7 (\href{http://www.math.sunysb.edu/~kirillov/tensor/chapter7.ps.gz}{ps.gz}) of \emph{Lectures on tensor categories and modular functor} (\href{http://www.math.sunysb.edu/~kirillov/tensor/tensor.html}{web}) \item P. Di Francesco, P. Mathieu, D. S\'e{}n\'e{}chal, \emph{Conformal field theory}, Springer 1997 \end{itemize} A basic introduction also to the super-WZW model (and with an eye towards the corresponding [[2-spectral triple]]) is in \begin{itemize}% \item [[Jürg Fröhlich]], [[Krzysztof Gawedzki]], \emph{Conformal Field Theory and Geometry of Strings}, extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 (\href{http://arxiv.org/abs/hep-th/9310187}{arXiv:hep-th/9310187}) \end{itemize} A useful account of the WZW model that encompasses both its [[action functional]] and [[path integral]] quantization as well as the [[current algebra]] aspects of the QFT is in \begin{itemize}% \item [[Krzysztof Gawedzki]], \emph{Conformal field theory: a case study} in Y. Nutku, C. Saclioglu, T. Turgut (eds.) \emph{Frontier in Physics} 102, Perseus Publishing (2000) (\href{http://xxx.lanl.gov/abs/hep-th/9904145}{hep-th/9904145}) \end{itemize} This starts in section 2 as a warmup with describing the 1d QFT of a particle propagating on a group manifold. The [[Hilbert space]] of [[states]] is expressed in terms of the [[Lie theory]] of $G$ and its [[Lie algebra]] $\mathfrak{g}$. In section 4 the [[quantization]] of the 2d WZW model is discussed in analogy to that. In lack of a full formalization of the quantization procedure, the author uses the loop algebra $\mathcal{l} \mathfrak{g}$ -- the [[affine Lie algebra]] -- of $\mathfrak{g}$ as the evident analog that replaces $\mathfrak{g}$ and discusses the [[Hilbert space]] [[space of states|of states]] in terms of that. He also indicates how this may be understood as a space of [[sections]] of a ([[prequantum line bundle|prequantum]]) [[line bundle]] over the [[loop group]]. See also \begin{itemize}% \item L. Feh\'e{}r, L. O'Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf, \emph{On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories}, Phys. Rep. \textbf{222} (1992), no. 1, 64 pp. \href{http://www.ams.org/mathscinet-getitem?mr=1192998}{MR93i:81225}, \item [[Krzysztof Gawedzki]], Rafal Suszek, [[Konrad Waldorf]], \emph{Global gauge anomalies in two-dimensional bosonic sigma models} (\href{http://arxiv.org/abs/1003.4154}{arXiv:1003.4154}) \item Paul de Fromont, [[Krzysztof Gaw?dzki]], Cl\'e{}ment Tauber, \emph{Global gauge anomalies in coset models of conformal field theory} (\href{http://arxiv.org/abs/1301.2517}{arXiv:1301.2517}) \end{itemize} \hypertarget{ReferencesRelationToGerbesAndCS}{}\subsubsection*{{Relation to gerbes and Chern-Simons theory}}\label{ReferencesRelationToGerbesAndCS} Discussion of [[circle n-bundle with connection|circle 2-bundles with connection]] (expressed in terms of [[bundle gerbes]]) and discussion of the WZW-background [[B-field]] ([[WZW term]]) in this language is in \begin{itemize}% \item [[Krzysztof Gawedzki]], Nuno Reis, \emph{WZW branes and gerbes}, Rev.Math.Phys. 14 (2002) 1281-1334 (\href{http://arxiv.org/abs/hep-th/0205233}{arXiv:hep-th/0205233}) \end{itemize} \begin{itemize}% \item [[Christoph Schweigert]], [[Konrad Waldorf]], \emph{Gerbes and Lie Groups}, in \emph{Trends and Developments in Lie Theory}, Progress in Math., Birkh\"a{}user (\href{http://arxiv.org/abs/0710.5467}{arXiv:0710.5467}) \end{itemize} Discussion of how this 2-bundle arises from the [[Chern-Simons circle 3-bundle]] is in \begin{itemize}% \item [[Alan Carey]], Stuart Johnson, [[Michael Murray]], [[Danny Stevenson]], [[Bai-Ling Wang]], \emph{Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories} Commun.Math.Phys. 259 (2005) 577-613 (\href{http://arxiv.org/abs/math/0410013}{arXiv:math/0410013}) \end{itemize} and related discussion is in \begin{itemize}% \item [[Konrad Waldorf]], \emph{Multiplicative Bundle Gerbes with Connection} , Differential Geom. Appl. 28(3), 313-340 (2010) (\href{http://arxiv.org/abs/0804.4835}{arXiv:0804.4835}) \end{itemize} See also Section 2.3.18 and section 4.7 of \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{http://ncatlab.org/schreiber/files/Erlangen2011Schreiber.pdf}{pdf slides}). \end{itemize} \hypertarget{partition_functions}{}\subsubsection*{{Partition functions}}\label{partition_functions} \begin{itemize}% \item [[Terry Gannon]], \emph{Partition Functions for Heterotic WZW Conformal Field Theories}, Nucl.Phys. B402 (1993) 729-753 (\href{http://arxiv.org/abs/hep-th/9209042}{arXiv:hep-th/9209042}) \end{itemize} \hypertarget{ReferencesDBranes}{}\subsubsection*{{D-branes for the WZW model}}\label{ReferencesDBranes} A characterization of [[D-branes]] in the WZW model as those [[conjugacy classes]] that in addition satisfy an integrality ([[Bohr-Sommerfeld quantization|Bohr-Sommerfeld]]-type) condition missed in other parts of the literature is stated in \begin{itemize}% \item [[Anton Alekseev]], [[Volker Schomerus]], \emph{D-branes in the WZW model}, Phys.Rev.D60:061901,1999 (\href{http://arxiv.org/abs/hep-th/9812193v2}{arXiv:hep-th/9812193v2}) \end{itemize} The refined interpretation of the integrality condition as a choice of trivialization of the underling [[principal 2-bundle]]/[[bundle gerbe]] of the [[B-field]] over the brane was then noticed in section 7 of \begin{itemize}% \item [[Krzysztof Gawedzki]], Nuno Reis, \emph{WZW branes and gerbes}, Rev.Math.Phys. 14 (2002) 1281-1334 (\href{http://arxiv.org/abs/hep-th/0205233}{arXiv:hep-th/0205233}) \end{itemize} The observation that this is just the special rank-1 case of the more general structure provided by a [[twisted unitary bundle]] of some rank $n$ on the D-brane ([[gerbe module]]) which is twisted by the restriction of the [[B-field]] to the D-brane -- the [[Chan-Paton gauge field]] -- is due to \begin{itemize}% \item [[Krzysztof Gawedzki]], \emph{Abelian and non-Abelian branes in WZW models and gerbes}, Commun.Math.Phys. 258 (2005) 23-73 (\href{http://arxiv.org/abs/hep-th/0406072}{arXiv:hep-th/0406072}). \end{itemize} The observation that the ``multiplicative'' structure of the WZW-[[B-field]] (induced from it being the [[transgression]] of the [[Chern-Simons circle 3-bundle|Chern-Simons circle 3-connection]] over the [[moduli stack]] of $G$-[[principal connections]]) induces the [[Verlinde ring]] fusion product structure on symmetric D-branes equipped with [[Chan-Paton gauge fields]] is discussed in \begin{itemize}% \item [[Alan Carey]], [[Bai-Ling Wang]], \emph{Fusion of symmetric $D$-branes and Verlinde rings}, Commun. Math. Phys.277:577-625 (2008) (\href{http://arxiv.org/abs/math-ph/0505040}{arXiv:math-ph/0505040}) \end{itemize} The image in [[K-theory]] of these [[Chan-Paton gauge fields]] over conjugacy classes is shown to generate the [[Verlinde ring]]/the [[positive energy representations]] of the [[loop group]] in \begin{itemize}% \item [[Eckhard Meinrenken]], \emph{On the quantization of conjugacy classes}, Enseign. Math. (2) 55 (2009), no. 1-2, 33-75 (\href{http://arxiv.org/abs/0707.3963}{arXiv:0707.3963}) \end{itemize} Formalization of WZW terms in [[cohesive (infinity,1)-topos|cohesive homotopy theory]] is in \begin{itemize}% \item \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} \hypertarget{relation_to_dimensional_reduction_of_chernsimons}{}\subsubsection*{{Relation to dimensional reduction of Chern-Simons}}\label{relation_to_dimensional_reduction_of_chernsimons} One can also obtain the WZW-model by [[KK-reduction]] from [[Chern-Simons theory]]. E.g. \begin{itemize}% \item [[Matthias Blau]], G. Thompson, \emph{Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model}, Nucl.Phys. B408 (1993) 345-390 (\href{http://arxiv.org/abs/hep-th/9305010}{arXiv:hep-th/9305010}) \end{itemize} A discussion in [[higher differential geometry]] via [[transgression]] in [[ordinary differential cohomology]] is in \begin{itemize}% \item \emph{[[schreiber:Extended higher cup-product Chern-Simons theories]]} \item \emph{[[schreiber:A higher stacky perspective on Chern-Simons theory]]} \end{itemize} \hypertarget{relation_to_extended_tqft}{}\subsubsection*{{Relation to extended TQFT}}\label{relation_to_extended_tqft} Relation to [[extended TQFT]] is discussed in \begin{itemize}% \item [[Dan Freed]], \emph{[[4-3-2 8-7-6]]} \end{itemize} The formulation of the [[Green-Schwarz action functional]] for [[superstrings]] (and other [[branes]] of [[string theory]]/[[M-theory]]) as WZW-models (and [[schreiber:∞-Wess-Zumino-Witten theory|∞-WZW models]]) on ([[super L-∞ algebra]] [[L-∞ extensions]] of) the [[super translation group]] is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]}, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264}) \end{itemize} \hypertarget{in_solid_state_physics}{}\subsubsection*{{In solid state physics}}\label{in_solid_state_physics} The low-energy physics of a Heisenberg antiferromagnetic spin chain is argued to be described by a WZW model in \begin{itemize}% \item Zheng-Xin Liu, Guang-Ming Zhang, \emph{Classification of quantum critical states of integrable antiferromagnetic spin chains and their correspondent two-dimensional topological phases} (\href{http://arxiv.org/abs/1211.5421}{arXiv:1211.5421}) \end{itemize} See also section 7.10 of Fradkin's book. Discussion of [[symmetry protected topological order]] phases of matter in [[solid state physics]] via [[higher dimensional WZW models]] is in \begin{itemize}% \item Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, [[Xiao-Gang Wen]], \emph{Symmetry protected topological orders and the group cohomology of their symmetry group}, Phys. Rev. B 87, 155114 (2013) \href{http://arxiv.org/abs/1106.4772}{arXiv:1106.4772}; A short version in Science \textbf{338}, 1604-1606 (2012) \href{http://dao.mit.edu/~wen/pub/dDSPTsht.pdf}{pdf} \end{itemize} [[!redirects WZW model]] [[!redirects WZW-model]] [[!redirects Wess-Zumino-Witten-model]] [[!redirects WZW models]] [[!redirects WZW-models]] [[!redirects Wess-Zumino-Witten-models]] [[!redirects Wess-Zumino model]] [[!redirects Wess-Zumino-model]] [[!redirects Wess-Zumino models]] [[!redirects Wess-Zumino-models]] [[!redirects WZW theory]] [[!redirects WZW-theory]] [[!redirects Wess-Zumino-Novikov-Witten model]] [[!redirects Wess-Zumino-Novikov-Witten models]] [[!redirects WZNW model]] [[!redirects WZNW models]] [[!redirects WZNW-model]] [[!redirects WZNW-models]] [[!redirects WZNW theory]] [[!redirects WZNW-theory]] [[!redirects Wess-Zumino-Witten-theory]] [[!redirects WZW theories]] [[!redirects WZW-theories]] [[!redirects Wess-Zumino-Witten-theories]] [[!redirects Wess-Zumino theory]] [[!redirects Wess-Zumino-theory]] [[!redirects Wess-Zumino theories]] [[!redirects Wess-Zumino-theories]] [[!redirects 2d WZW model]] [[!redirects Wess-Zumino-Witten theory]] [[!redirects WZW term]] [[!redirects WZW terms]] [[!redirects WZW-term]] [[!redirects WZW-terms]] [[!redirects WZW gerbe]] [[!redirects WZW gerbes]] [[!redirects Wess-Zumino-Witten sigma model]] [[!redirects Wess-Zumino-Witten sigma models]] \end{document}