# 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

Let $G$ be compact and simply connected.

Then by infinity-Chern-Weil theory the Killing form invariant polynomial on the Lie algebra $\mathfrak{g}$ induces a circle 3-bundle with connection on the smooth moduli stack $\mathbf{B}G_{conn}$ of $G$-principal bundles with connection.

$CS_{\mathbf{c}} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} \,.$

This is the Lagrangian for $G$-Chern-Simons theory. By looping this, including a differential twist, there is induced canonically a circle 2-bundle with connection

$WZW_{\mathbf{c}} : G \to \mathbf{B}^2 U(1)_{conn} \,.$

The construction of the canonical morphism $WZW_{\mathbf{c}}$ goes as follows. Consider, inside $\mathbf{B}G_{conn}$ the substack $\mathbf{B}G_{\flat dR}$ of trivialized principal $G$-bundles with flat connections. The morphism $CS_{\mathbf{c}}$, restricted to $\mathbf{B}G_{\flat dR}$, factors as

$\mathbf{B}G_{\flat dR}\to \Omega^3(-) \to \mathbf{B}^3 U(1)_{conn}$

where $\Omega^3(-)$ is identified with the 3-stack of trivialized circle bundles with connection whose connection form lives entirely in degree 3. Since $\mathbf{B}G_{\flat dR}\to \mathbf{B}G_{conn}$ factors through the stack $\flat\mathbf{B}G$ of principal $G$-bundles with flat conenctions, we have a homotopy commutative diagram

$\array{ \mathbf{B}G_{\flat dR}&\to^{CS_{\mathbf{c}}}& \Omega^3(-)\\ \downarrow && \downarrow\\ \flat\mathbf{B}G& \to & \mathbf{B}^3 U(1)_{conn} }$

and so a canonically induced morphism $WZW_{\mathbf{c}}$ between the homotopy fibers over the distinguished points. Since we have homotopy pullbacks

$\array{ G&\to& \mathbf{B}G_{\flat dR}\\ \downarrow && \downarrow\\ {*}& \to & \flat\mathbf{B}G }$

and

$\array{ \mathbf{B}^2 U(1)_{conn}&\to & \Omega^3(-)\\ \downarrow && \downarrow\\ {*}& \to & \mathbf{B}^3 U(1)_{conn} },$

the morphism $WZW_{\mathbf{c}}$ can naturally be seen as a morphism from $G$ (as a smooth manifold) to the 2-stack $\mathbf{B}^2 U(1)_{conn}$ of circle 2-bundles with connection. In other words, if $G$ is a compact simply connected simple Lie group, the differential refinement $CS_{\mathbf{c}}$ of the degree 4 characteristic class $\mathbf{c}$ provided by Chern-Simons theory naturally induces a circle 2-bundle with connection over the smooth manifold underlying the Lie group $G$.

The surface holonomy of this is the topological part of the WZW action functional:

$\exp(i S_{top}(g)) = hol_{\Sigma}( g^* WZW_{\mathbf{c}} ) \,.$

#### 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

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

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 $\tilde G = \mathbf{B}^pU (1)_{conn}$ 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. 1 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

• Edward Witten, Global aspects of current algebra. Nucl. Phys. B223, 422 (1983)

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 Gawędzki 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 Gawędzki, A. Kupianen, Spectra of Wess-Zumino-Witten models with arbitrary simple groups. Commun. Math. Phys. 117, 127 (1988)

• Krzysztof Gawędzki, 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.

### 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

Revised on April 17, 2015 11:50:27 by Anonymous Coward (156.35.89.187)