# 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

${S}_{\mathrm{WZW}}={S}_{\mathrm{kin}}+{S}_{\mathrm{top}}\phantom{\rule{thinmathspace}{0ex}}.$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

Let $G$ be compact and simply connected.

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

${\mathrm{CS}}_{c}:B{G}_{\mathrm{conn}}\to {B}^{3}U\left(1{\right)}_{\mathrm{conn}}\phantom{\rule{thinmathspace}{0ex}}.$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

${\mathrm{WZW}}_{c}:G\to {B}^{2}U\left(1{\right)}_{\mathrm{conn}}\phantom{\rule{thinmathspace}{0ex}}.$WZW_{\mathbf{c}} : G \to \mathbf{B}^2 U(1)_{conn} \,.

The construction of the canonical morphism ${\mathrm{WZW}}_{c}$ goes as follows. Consider, inside $B{G}_{\mathrm{conn}}$ the substack $B{G}_{♭\mathrm{dR}}$ of trivialized principal $G$-bundles with flat connections. The morphism ${\mathrm{CS}}_{c}$, restricted to $B{G}_{♭\mathrm{dR}}$, factors as

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

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

$\begin{array}{ccc}B{G}_{♭\mathrm{dR}}& {\to }^{{\mathrm{CS}}_{c}}& {\Omega }^{3}\left(-\right)\\ ↓& & ↓\\ ♭BG& \to & {B}^{3}U\left(1{\right)}_{\mathrm{conn}}\end{array}$\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 ${\mathrm{WZW}}_{c}$ between the homotopy fibers over the distinguished points. Since we have homotopy pullbacks

$\begin{array}{ccc}G& \to & B{G}_{♭\mathrm{dR}}\\ ↓& & ↓\\ *& \to & ♭BG\end{array}$\array{ G&\to& \mathbf{B}G_{\flat dR}\\ \downarrow && \downarrow\\ {*}& \to & \flat\mathbf{B}G }

and

$\begin{array}{ccc}{B}^{2}U\left(1{\right)}_{\mathrm{conn}}& \to & {\Omega }^{3}\left(-\right)\\ ↓& & ↓\\ *& \to & {B}^{3}U\left(1{\right)}_{\mathrm{conn}}\end{array},$\array{ \mathbf{B}^2 U(1)_{conn}&\to & \Omega^3(-)\\ \downarrow && \downarrow\\ {*}& \to & \mathbf{B}^3 U(1)_{conn} },

the morphism ${\mathrm{WZW}}_{c}$ can naturally be seen as a morphism from $G$ (as a smooth manifold) to the 2-stack ${B}^{2}U\left(1{\right)}_{\mathrm{conn}}$ of circle 2-bundles with connection. In other words, if $G$ is a compact simply connected simple Lie group, the differential refinement ${\mathrm{CS}}_{c}$ of the degree 4 characteristic class $c$ 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:

$\mathrm{exp}\left(i{S}_{\mathrm{top}}\left(g\right)\right)={\mathrm{hol}}_{\Sigma }\left({g}^{*}{\mathrm{WZW}}_{c}\right)\phantom{\rule{thinmathspace}{0ex}}.$\exp(i S_{top}(g)) = hol_{\Sigma}( g^* WZW_{\mathbf{c}} ) \,.

## Properties

### Holography

By Chern-Simons holography (see there for more details) the 2d WZW CFT on $G$ is related to $G$-Chern-Simons theory in $3d$.

### D-branes

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 $n\in ℕ$ 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 $BU\left(1\right)$-principal 2-bundle ($U\left(1\right)$-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.

## References

### General

An introduction is in

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

A classical textbook account in the general context of 2d CFT is

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

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 $𝔤$.

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 $𝓁𝔤$ – the affine Lie algebra – of $𝔤$ as the evident analog that replaces $𝔤$ 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

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

### Relation to extended TQFT

Relation to extended TQFT is discussed in

## Discussion

Would someone translate the Topological term for an aging algebraic topologist? B^3U(1) aka K(Z,3) or is it 4? sub conn denotes the classifying space for bundles with connection? The morphism from G to B^2 is strict or up to homotopy?

Revised on May 21, 2013 23:53:39 by Urs Schreiber (89.204.130.130)