Contents

# Contents

## Idea

The Wess-Zumino-Witten model (or WZW model for short, also called Wess-Zumino-Novikov-Witten model, or short 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 algebras corresponding to the WZW model are current algebras.

## Action functional

For $G$ a Lie group, the configuration space of the WZW over a 2-dimensional manifold $\Sigma$ is the space of smooth functions $g : \Sigma \to G$.

The action functional of the WZW sigma-model is the sum of two terms, a kinetic term and a topological term

$S_{WZW} = S_{kin} + S_{top} \,.$

### Kinetic term

The Lie group canonically carries a Riemannian metric and the kinetic term is the standard one for sigma-models on Riemannian target spaces.

### Topological term – WZW term

#### For the 2d WZW model

In higher differential geometry, then given any closed 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 (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

$\array{ && \mathbf{B}^{p+1}U(1)_{conn} \\ &{}^{\mathllap{\nabla}}\nearrow& \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} }$

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 compact semisimple Lie group such as SU or Spin, where $\mu = \langle -,[-,-]\rangle$ is the 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 geometry of physics – principal bundles, the cocycle $\mathbf{c}$ modulates an infinity-group extension which is a circle p-group-principal infinity-bundle

$\array{ \mathbf{B}^p U(1) &\longrightarrow& \hat G \\ && \downarrow \\ && G &\stackrel{\Omega\mathbf{c}}{\longrightarrow}& \mathbf{B}^{p+1}U(1) }$

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 and $p = 1$ this extension is the string 2-group.

Such a Lie theoretic situation is concisely expressed by a diagram of smooth homotopy types of the form

$\array{ && &\longrightarrow& \mathbf{B}^{p+1}U(1) \\ &{}^{\mathllap{\Omega \mathbf{c}}}\nearrow& &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}} \\ G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \,,$

where $\flat_{dR}\mathbf{B}^{p+2}\mathbb{R} \simeq \flat_{dR}\mathbf{B}^{p+2}U(1)$ is the de Rham coefficients (see also at 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 $\mathbf{B}^{p+1}U(1)_{conn}$ (here), such a diagram is equivalently a prequantization of $\omega$:

$\array{ && \mathbf{B}^{p+1}U(1)_{conn} &\longrightarrow& \mathbf{B}^{p+1}U(1) \\ &{}^\mathllap{\nabla}\nearrow& \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\theta_{\mathbf{B}^p U(1)}}} \\ G &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}_{cl}^{p+2} &\stackrel{}{\longrightarrow}& \flat_{dR}\mathbf{B}^{p+2}\mathbb{R} } \,.$

For $\omega = \langle -,[-,-]\rangle$ as above, we have $p= 1$ and so $\nabla$ here is a circle 2-bundle with connection, often referred to as a bundle gerbe with connection. As such, this is also known as the WZW gerbe or similar.

This terminology arises as follows. In (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 (Witten 84) it was observed that for this action functional to give a conformal field theory after quantization, a certain higher gauge interaction term has to the added. The resulting sigma-model came to be known as the Wess-Zumino-Witten model or WZW model for short, and the term that Witten added became the 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 force whose field strength is $\omega$. In (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” 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).

#### Generally

The following (FSS 12, dcct) is a general axiomatization of WZW terms in 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

$\iota \colon \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) \longrightarrow \flat_{dR} \mathbf{B}^2 \mathbb{G}$

to its de Rham coefficients

###### Definition

Given an ∞-group object $G$ in $\mathbf{H}$ and given a group cocycle

$\mathbf{c} \colon \mathbf{B}G \longrightarrow \mathbf{B}^2 \mathbb{G} \,,$

then a 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

$\array{ \mathbf{\Omega}^1_{flat}(-,G) &\stackrel{\mu}{\longrightarrow}& \mathbf{\Omega}^2_{cl}(-,\mathbb{G}) \\ \downarrow && \downarrow^{\mathrlap{\iota}} \\ \flat_{dR}\mathbf{B}G &\stackrel{\flat_{dR}\mathbf{c}}{\longrightarrow}& \flat_{dR}\mathbf{B}^2 \mathbb{G} } \,.$

We write $\tilde G$ for the homotopy pullback of this refinement along the Maurer-Cartan form $\theta_G$ of $G$

$\array{ \tilde G &\stackrel{\theta_{\tilde G}}{\longrightarrow}& \mathbf{\Omega}^1_{flat}(-,G) \\ \downarrow && \downarrow \\ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR}\mathbf{B}G } \,.$
###### Example

Let $\mathbf{H} =$ Smooth∞Grpd and $\mathbb{G} = \mathbf{B}^p U(1)$ the circle (p+1)-group.

For $G$ an ordinary Lie group, then $\mu$ may be taken to be the 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.

###### Proposition

In the situation of def. there is an essentially unique prequantization

$\mathbf{L}_{WZW} \colon \tilde G \longrightarrow \mathbf{B}^2 \mathbb{G}_{conn}$

of the closed differential form

$\mu(\theta_{\tilde G}) \colon \tilde G \stackrel{\theta_{\tilde G}}{\longrightarrow} \mathbf{\Omega}^1_{flat}(-,G) \stackrel{\mu}{\longrightarrow} \mathbf{\Omega}^2_{cl}(-,\mathbb{G})$

whose underlying $\mathbb{G}$-principal ∞-bundle is modulated by the looping $\Omega \mathbf{c}$ of the original cocycle.

This we call the WZW term of $\mathbf{c}$ with respect to the chosen refinement of the Hodge structure.

## Properties

### Equations of motion

The variational derivative of the WZW action functional is

$\delta S_{WZW}(g) = -\frac{k}{2 \pi i } \int_\Sigma \langle (g^{-1}\delta g), \partial (g^{-1}\bar \partial g) \rangle \,.$

Therefore the classical equations of motion for function $g \colon \Sigma \to G$ are

$\partial(g^{-1}\bar \partial g) = 0 \;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\; \bar \partial(g \partial g^{-1}) = 0 \,.$

The space of solutions to these equations is small. However, discussion of the quantization of the theory (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

$g(z,\bar z) = g_{\ell}(z) g_r(\bar z)^{-1}$

where hence $g_{\ell} \colon \Sigma \to G(\mathbb{C})$ is any holomorphic function and $g_r$ similarly any anti-holomorphic function.

### Holography and Rigorous construction

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.

### D-branes for the WZW model

The characterization of D-brane submanifolds for the open string WZW model on a Lie group $G$ comes from two consistency conditions:

1. 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. Alekseev-Schomerus for a brief review and further pointers). These conjugacy classes are therefore also called the 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 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.)

2. 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 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 integral symmetric D-branes. (Alekseev-Schomerus, Gawedzki-Reis). As observed in 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 (Gawedzki 04).

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 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.

### Quantization

on quantization of the WZW model, see at

## References

### Original references

The Wess-Zumino gauge-coupling term goes back to

and was understood as yielding a 2-dimensional conformal field theory in

and hence (a possible part of) a string theory vacuum/target space in

The WZ term on $\Sigma_2$ was understood in terms of an integral of a 3-form over a cobounding manifold $\Sigma_3$ in

for the case that $\Sigma_2$ is closed, and generally, in terms of surface holonomy of bundle gerbes/circle 2-bundles with connection in

• Krzysztof Gawedzki, Topological Actions in two-dimensional Quantum Field Theories, in Gerard 't Hooft et. al (eds.) Nonperturbative quantum field theory Cargese 1987 proceedings, (web)

• Giovanni Felder , Krzysztof Gawedzki, A. Kupianen, Spectra of Wess-Zumino-Witten models with arbitrary simple groups. Commun. Math. Phys. 117, 127 (1988)

• Krzysztof Gawedzki, Topological actions in two-dimensional quantum field theories. In: Nonperturbative quantum field theory. ‘tHooft, G. et al. (eds.). London: Plenum Press 1988

as the surface holonomy of a circle 2-bundle with connection. See also the references at B-field and at Freed-Witten anomaly cancellation.

For the fully general understanding as the surface holonomy of a circle 2-bundle with connection see the references below.

• Edward Witten, On holomorphic factorization of WZW and coset models, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. (Euclid)

### Introductions and surveys

An survey of and introduction to the topic is in

• Patrick Meessen, Strings Moving on Group Manifolds: The WZW Model (pdf)

A classical textbook accounts include

• Bojko Bakalov, Alexander Kirillov, chapter 7 (ps.gz) of Lectures on tensor categories and modular functor (web)

• P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal field theory, Springer 1997

A basic introduction also to the super-WZW model (and with an eye towards the corresponding 2-spectral triple) is in

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

• Krzysztof Gawedzki, Conformal field theory: a case study in Y. Nutku, C. Saclioglu, T. Turgut (eds.) Frontier in Physics 102, Perseus Publishing (2000) (hep-th/9904145)

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 of states in terms of that. He also indicates how this may be understood as a space of sections of a (prequantum) line bundle over the loop group.

• L. Fehér, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf, On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. Rep. 222 (1992), no. 1, 64 pp. MR93i:81225, doi

• Krzysztof Gawedzki, Rafal Suszek, Konrad Waldorf, Global gauge anomalies in two-dimensional bosonic sigma models (arXiv:1003.4154)

• Paul de Fromont, Krzysztof Gaw?dzki?, Clément Tauber, Global gauge anomalies in coset models of conformal field theory (arXiv:1301.2517)

### Relation to gerbes and Chern-Simons theory

Discussion of 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

Discussion of how this 2-bundle arises from the Chern-Simons circle 3-bundle is in

and related discussion is in

### D-branes for the WZW model

A characterization of D-branes in the WZW model as those conjugacy classes that in addition satisfy an integrality (Bohr-Sommerfeld-type) condition missed in other parts of the literature is stated in

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

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

The observation that the “multiplicative” structure of the WZW-B-field (induced from it being the transgression of the 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

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

Formalization of WZW terms in cohesive homotopy theory is in

### Relation to dimensional reduction of Chern-Simons

One can also obtain the WZW-model by KK-reduction from Chern-Simons theory.

E.g.

A discussion in higher differential geometry via transgression in ordinary differential cohomology is in

### Relation to extended TQFT

Relation to extended TQFT is discussed in

The formulation of the Green-Schwarz action functional for superstrings (and other branes of string theory/M-theory) as WZW-models (and ∞-WZW models) on (super L-∞ algebra L-∞ extensions of) the super translation group is in

### In solid state physics

The low-energy physics of a Heisenberg antiferromagnetic spin chain is argued to be described by a WZW model in

• Zheng-Xin Liu, Guang-Ming Zhang, Classification of quantum critical states of integrable antiferromagnetic spin chains and their correspondent two-dimensional topological phases (arXiv:1211.5421)