Wess-Zumino-Witten model


\infty-Wess-Zumino-Witten theory

Quantum field theory


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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 GG.

The vertex operator algebras corresponding to the WZW model are current algebras.

Action functional

For GG a Lie group, the configuration space of the WZW over a 2-dimensional manifold Σ\Sigma is the space of smooth functions g:ΣGg : \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. 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 GG be compact and simply connected.

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

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

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

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

The construction of the canonical morphism WZW cWZW_{\mathbf{c}} goes as follows. Consider, inside BG conn\mathbf{B}G_{conn} the substack BG dR\mathbf{B}G_{\flat dR} of trivialized principal GG-bundles with flat connections. The morphism CS cCS_{\mathbf{c}}, restricted to BG dR\mathbf{B}G_{\flat dR}, factors as

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

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

BG dR CS c Ω 3() BG B 3U(1) conn \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 cWZW_{\mathbf{c}} between the homotopy fibers over the distinguished points. Since we have homotopy pullbacks

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


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

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

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

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


Holography and Rigorous construction

By the AdS3-CFT2 and CS-WZW correspondence (see there for more details) the 2d WZW CFT on GG is related to GG-Chern-Simons theory in 3d3d.

In fact a rigorous constructions of the GG-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 GG-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 GG 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 GG, 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 GG.(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 nn \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)\mathbf{B}U(1)-principal 2-bundle (U(1)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 GG equipped with a twisted unitary bundle for the restriction of the background B-field to the conjugacy class.


on quantization of the WZW model, see at


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 Σ 2\Sigma_2 was understood in terms of an integral of a 3-form over a cobounding manifold Σ 3\Sigma_3 in

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

for the case that Σ 2\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.

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 account in the general context of 2d CFT is

  • 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 GG and its Lie algebra 𝔤\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 𝓁𝔤\mathcal{l} \mathfrak{g} – the affine Lie algebra – of 𝔤\mathfrak{g} as the evident analog that replaces 𝔤\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.

See also

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

See also Section 2.3.18 and section 4.7 of

Partition functions

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 nn 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 GG-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 dimensional reduction of Chern-Simons

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


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 March 21, 2014 09:49:08 by Urs Schreiber (