nLab
Wess-Zumino-Witten model

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

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

$${\mathrm{CS}}_c:B{G}_{\mathrm{conn}}\to B^{3}U(1{)}_{\mathrm{conn}}.$$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

$${\mathrm{WZW}}_c:G\to B^{2}U(1{)}_{\mathrm{conn}}.$$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

$$B{G}_{♭\mathrm{dR}}\to {\Omega }^3(-)\to B^{3}U(1{)}_{\mathrm{conn}}$$\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 $B{G}_{♭\mathrm{dR}}\to B{G}_{\mathrm{conn}}$ factors through the stack $♭BG$ of principal $G$-bundles with flat conenctions, we have a homotopy commutative diagram

$$\begin{array}{ccc}B{G}_{♭\mathrm{dR}}& {\to }^{{\mathrm{CS}}_c}& {\Omega }^3(-)\\ \downarrow & & \downarrow \\ ♭BG& \to & B^{3}U(1{)}_{\mathrm{conn}}\end{array}$$\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

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

and

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

$$\exp (i{S}_{\mathrm{top}}(g))={\mathrm{hol}}_{\Sigma }(g^{*}{\mathrm{WZW}}_c).$$\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 $n\in \mathbb{N}$ 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 $BU(1)$-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.

Related concepts

• coset WZW model

• higher dimensional WZW model

• Green-Schwarz action functional

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.

See also

• 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 Gawȩdzki, 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

• Krzysztof Gawedzki, Nuno Reis, WZW branes and gerbes, Rev.Math.Phys. 14 (2002) 1281-1334 (arXiv:hep-th/0205233)

• Christoph Schweigert, Konrad Waldorf, Gerbes and Lie Groups, in Trends and Developments in Lie Theory, Progress in Math., Birkhäuser (arXiv:0710.5467)

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

• Alan Carey, Stuart Johnson, Michael Murray, Danny Stevenson, Bai-Ling Wang, Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories Commun.Math.Phys. 259 (2005) 577-613 (arXiv:math/0410013)

and related discussion is in

• Konrad Waldorf, Multiplicative Bundle Gerbes with Connection , Differential Geom. Appl. 28(3), 313-340 (2010) (arXiv:0804.4835)

See also Section 2.3.18 and section 4.7 of

• Urs Schreiber, differential cohomology in a cohesive topos (pdf slides).

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

• Anton Alekseev, Volker Schomerus, D-branes in the WZW model, Phys.Rev.D60:061901,1999 (arXiv:hep-th/9812193v2)

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

• Krzysztof Gawedzki, Nuno Reis, WZW branes and gerbes, Rev.Math.Phys. 14 (2002) 1281-1334 (arXiv:hep-th/0205233)

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

• Krzysztof Gawedzki, Abelian and non-Abelian branes in WZW models and gerbes, Commun.Math.Phys. 258 (2005) 23-73 (arXiv:hep-th/0406072).

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

• Alan Carey, Bai-Ling Wang, Fusion of symmetric $D$-branes and Verlinde rings, Commun. Math. Phys.277:577-625 (2008) (arXiv:math-ph/0505040)

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

• Eckard Meinrenken, On the quantization of conjugacy classes, Enseign. Math. (2) 55 (2009), no. 1-2, 33-75 (arXiv:0707.3963)

Relation to extended TQFT

Relation to extended TQFT is discussed in

• Dan Freed, 4-3-2 8-7-6

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?