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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 $G$.
The vertex operator algebras corresponding to the WZW model are current algebras.
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
The Lie group canonically carries a Riemannian metric and the kinetic term is the standard one for sigma-models on Riemannian target spaces.
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.
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
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
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
and so a canonically induced morphism $WZW_{\mathbf{c}}$ between the homotopy fibers over the distinguished points. Since we have homotopy pullbacks
and
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:
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
to its de Rham coefficients
Given an ∞-group object $G$ in $\mathbf{H}$ and given a group cocycle
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
We write $\tilde G$ for the homotopy pullback of this refinement along the Maurer-Cartan form $\theta_G$ of $G$
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.
In the situation of def. 1 there is an essentially unique prequantization
of the closed differential form
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.
The variational derivative of the WZW action functional is
Therefore the classical equations of motion for function $g \colon \Sigma \to G$ are
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
where hence $g_{\ell} \colon \Sigma \to G(\mathbb{C})$ is any holomorphic function and $g_r$ similarly any anti-holomorphic function.
(e.g. Gawedzki 99 (3.18), (3.19))
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.
The characterization of D-brane submanifolds for the open string WZW model on a Lie group $G$ comes from two consistency conditions:
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.)
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.
on quantization of the WZW model, see at
The Wess-Zumino gauge-coupling term goes back to
and was understood as yielding a 2-dimensional conformal field theory in
Edward Witten, Non-Abelian bosonization in two dimensions Commun. Math. Phys. 92, 455 (1984)
Vadim Knizhnik, Alexander Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nucl. Phys. B 247, 83-103 (1984)
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 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.
See also
An survey of and introduction to the topic is in
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
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.
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)
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
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
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 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