Berezin quantization of gauged WZW and coset models

Gauged WZW and coset models are known to be useful to prove holomorphic factorization of the partition function of WZW and coset models. In this note we show that these gauged models can be also important to quantize the theory in the context of the Berezin formalism. For gauged coset models Berezin quantization procedure also admits a further holomorphic factorization in the complex structure of the moduli space.This work is dedicated to Professor Michel Ryan on the occasion of his 60th birthday.     

Pohlmeyer reduction of superstring sigma model

Motivated by a desire to find a useful 2d Lorentz-invariant reformulation of the AdS₅×S⁵ superstring world-sheet theory in terms of physical degrees of freedom we construct the “Pohlmeyer-reduced” version of the corresponding sigma model. The Pohlmeyer reduction procedure involves several steps. Starting with a coset space string sigma model in the conformal gauge and writing the classical equations in terms of currents one can fix the residual conformal diffeomorphism symmetry and kappa-symmetry and introduce a new set of variables (related locally to currents but non-locally to the original string coordinate fields) so that the Virasoro constraints are automatically satisfied. The resulting equations can be obtained from a Lagrangian of a non-Abelian Toda type: a gauged WZW model with an integrable potential coupled also to a set of 2d fermionic fields. A gauge-fixed form of the Pohlmeyer-reduced theory can be found by integrating out the 2d gauge field of the gauged WZW model. The small-fluctuation spectrum near the trivial vacuum contains 8 bosonic and 8 fermionic degrees of freedom with equal mass. We conjecture that the reduced model has world-sheet supersymmetry and is ultraviolet-finite. We show that in the special case of the AdS₂×S² superstring model the reduced theory is indeed supersymmetric: it is equivalent to the N=2 supersymmetric extension of the sine-Gordon model.     

Algebraic Coset Conformal Field Theories

All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess–Zumino–Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic quantum field theory to approach coset Conformal Field Theories. Two conjectures are formulated and their consequences are discussed. Some results are presented which prove the conjectures in special cases. In particular, one of the results states that a class of representations of coset W _N (N≥ 3) algebras with critical parameters are irreducible, and under the natural compositions (Connes' fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a long-standing conjecture about the representations of these algebras. Received: 5 November 1998 / Accepted: 18 October 1999     

Global Gauge Anomalies in Coset Models of Conformal Field Theory

We study the occurrence of global gauge anomalies in the coset models of two-dimensional conformal field theory that are based on gauged WZW models. A complete classification of the non-anomalous theories for a wide family of gauged rigid adjoint or twisted-adjoint symmetries of WZW models is achieved with the help of Dynkin’s classification of Lie subalgebras of simple Lie algebras.     

On holomorphic factorization of WZW and coset models

It is shown how coupling to gauge fields can be used to explain the basic facts concerning holomorphic factorization of the WZW model of two dimensional conformal field theory, which previously have been understood primarily by using conformal field theory Ward identities. We also consider in a similar vein the holomorphic factorization ofG/H coset models. We discuss theG/G model as a topological field theory and comment on a conjecture by Spiegelglas.Research supported in part by NSF Grant PHY86-20266     

Fixed point resolution in extended WZW models

A formula is derived for the fixed point resolution matrices of simple current extended WZW models and coset conformal field theories. Unlike the analogous matrices for unextended WZW models, these matrices are in general not symmetric, and they may have field-dependent twists. They thus provide non-trivial realizations of the general conditions presented in earlier work with Fuchs and Schweigert.     

Realizations of the exceptional modular invariant A(1)1 partition functions

We show that the E₆ and E₈ modular invariant combinations of A⁽¹⁾₁ characters in the classification of Cappelli, Itzykson and Zuber can be realized as partition functions of k=1 conformally invariant WZW models on the group manifolds of Sp(4) and G₂, respectively. Together with the D₄ combination, which is known to be realized by the WZW model on SU(3), these are the only such cases where the SU(2) local symmetry extends to a larger one. The E₇ combination is briefly discussed.     

Chern-Simons and WZW anomaly cancelations across dimensions

The WZW functional in D=4 can be derived directly from the Chern-Simons functional of a compactified D=5 gauge theory and the boundary fermions it supplants. A simple pedagogical model based on U(1) gauge groups illustrates how this works. A bulk-boundary system with the fermions eliminated manifestly evinces anomaly cancelations between CS and WZW terms.     

Lattice conformal theories and their integrable perturbations

We study the lattice analogues of the Wess-Zumino-Witten (WZW) and Toda conformal field theories. We describe discrete versions of the Drinfel'd-Sokolov reduction and the Sugawara construction for the WZW model, and show how to formulate a perturbation theory in the chiral sector. We describe the spaces of integrals of motion of the perturbed theories. We interpret the perturbed WZW model in terms of NLS hierarchy and obtain an embedding of this model into the lattice KP hierarchy.     

K-matrices for 2D conformal field theories

In this paper we examine fermionic type characters (Universal Chiral Partition Functions) for general 2D conformal field theories with a bilinear form given by a matrix of the form . We provide various techniques for determining these K-matrices, and apply these to a variety of examples including (higher level) WZW and coset conformal field theories. Applications of our results to fractional quantum Hall systems and (level restricted) Kostka polynomials are discussed.     

Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models

We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.     

Conserved charges and supersymmetry in principal chiral and WZW models

Conserved and commuting charges are investigated in both bosonic and supersymmetric classical chiral models, with and without Wess–Zumino terms. In the bosonic theories, there are conserved currents based on symmetric invariant tensors of the underlying algebra, and the construction of infinitely many commuting charges, with spins equal to the exponents of the algebra modulo its Coxeter number, can be carried out irrespective of the coefficient of the Wess–Zumino term. In the supersymmetric models, a different pattern of conserved quantities emerges, based on antisymmetric invariant tensors. The current algebra is much more complicated than in the bosonic case, and it is analysed in some detail. Two families of commuting charges can be constructed, each with finitely many members whose spins are exactly the exponents of the algebra (with no repetition modulo the Coxeter number). The conserved quantities in the bosonic and supersymmetric theories are only indirectly related, except for the special case of the WZW model and its supersymmetric extension.     

Dyonic integrable models

A class of non-abelian affine Toda models arising from the axial gauged two-loop WZW model is presented. Their zero curvature representation is constructed in terms of a graded Kac–Moody algebra. It is shown that the discrete multivacua structure of the potential together with non-abelian nature of the zero grade subalgebra allows soliton solutions with non-trivial electric and topological charges. The dressing transformation is employed to explicitly construct one and two soliton solutions and their bound states in terms of the tau functions. A discussion of the classical spectra of such solutions and the time delays are given in detail.     

Twisted Wess-Zumino-Witten Models on Elliptic Curves

Investigated is a variant of the Wess-Zumino-Witten model called a twisted WZW model, which is associated to a certain Lie group bundle on a family of elliptic curves. The Lie group bundle is a non-trivial bundle with flat connection and related to the classical elliptic r-matrix. (The usual (non-twisted) WZW model is associated to a trivial group bundle with trivial connection on a family of compact Riemann surfaces and a family of its principal bundles.) The twisted WZW model on a fixed elliptic curve at the critical level describes the XYZ Gaudin model. The elliptic Knizhnik-Zamolodchikov equations associated to the classical elliptic r-matrix appear as flat connections on the sheaves of conformal blocks in the twisted WZW model. Received: 21 January 1997 / Accepted: 1 April 1997     

On Chern–Simons and WZW Partition Functions

Direct analysis of the path integral reduces partition functions in Chern-Simons theory on a three-manifold M with group G to partition functions in a WZW model of maps from a Riemann surface ‡ to G. In particular, Chern-Simons theory on S³, S¹ 2 ‡, B³ and the solid torus correspond, respectively, to the WZW model of maps from S² to G, the G/G model for ‡, and Witten's gauged WZW path integral Ansatz for Chern-Simons states using maps from S² and from the torus to G. The reduction hinges on the characterization of {\cal A / G}_{n}$, the space of connections modulo those gauge transformations which are the identity at a point n, as itself a principal fiber bundle with affine-linear fiber.