Lattice Wess-Zumino-Witten model and quantum groups

Quantum groups play the role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as Poisson-Lie symmetries of the corresponding phase spaces. We discuss specifically the Wess-Zumino-Witten conformally invariant quantum field model combining two chiral parts which describe the left- and right-moving degrees of freedom. On one hand, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling space-time and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice sl (2)-based current algebra and show how to use it to rigorously construct an exact solution of the quantum SL (2) WZW model on lattice.     

Modular invariance and the spectrum of two dimensional conformal field theories

Modular invariance has recently emerged as a powerful tool in conformal field theory. In conjunction with the representation theory of infinite dimensional Lie algebras, the study of modular invariance gave the spectrum of several families of theories. These include the minimal conformal models (Cardy and others), WZW theories which describe string propagation on group manifolds (Gepner and Witten) and parafermionic field theories (Gepner and Qiu). The minimal conformal models models were shown to be a product of two SU(2) WZW theories (Gepner). These results represent a step towards a complete classification of conformal field theories, an important goal both for the study of critical phenomena and string theory.     

Duals for Non-Abelian Lattice Gauge Theories by Categorical Methods

We introduce duals for non-Abelian lattice gauge theories in dimension at least three by using a categorical approach to the notion of duality in lattice theories. We first discuss the general concepts for the case of a dual-triangular lattice (i.e., the dual lattice is triangular) and find that the commutative tetrahedron condition of category theory can directly be used to define a gauge-invariant action for the dual theory. We then consider the cubic lattice (where the dual is cubic again). The case of the gauge group SU(2) is discussed in detail. We will find that in this case gauge connections of the dual theory correspond to SU(2) spin networks, suggesting that the dual is a discrete version of a quantum field theory of quantum simplicial complexes (i.e. the dual theory lives already on a quantized level in its classical form). We conclude by showing that our notion of duality leads to a hierarchy of extended lattice gauge theories closely resembling the one of extended topological quantum field theories. The appearance of this hierarchy can be understood by the quantum von Neumann hierarchy introduced by one of the authors in previous work.     

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.     

Weyl fields,quantum integrability and conformal invariant field theories

We show that some Weyl field theories arise as a quantum “linear” problem associated to some Kac-Moody algebras. We relate this quantum “linear” problem to the conformal invariant field theories studied by Dashen and Frishman and to the WZW field theory.     

Gauging of chiral bosonized actions

We present bosonic actions which are equivalent to various chiral fermion theories. For the case of one chiral fermion coupled to an abelian gauge field, we present two bosonized actions, one corresponding to regularizing in the vector conserving scheme and the other in the left-right scheme. We then propose an action for the non-abelian bosonization of Weyl fermions which is a WZW action coupled to a fixed curved background. The chiral WZW action is then coupled to non-abelian gauge fields. We derive the anomalies of the axial current (in the vector conserving scheme) and the left-right currents (in the left-right regularization scheme), both for the abelian and non-abelian bosonized actions. The expressions for the anomalies are identical to those derived in the corresponding fermionic theories.     

Simple WZW currents

A complete classification of simple currents of WZW theories is obtained. The proof is based on an analysis of the quantum dimensions of the primary fields. Simple currents are precisely the primaries with unit quantum dimension; for WZW theories explicit formulae for the quantum dimensions can be derived so that an identification of the fields with unit quantum dimension is possible.     

Effective action for D-branes on SU(2)/U(1) gauged WZW model

Dynamics of D-branes on SU(2)/U(1) gauged WZW model are investigated. We find the effective action for infinite k, where k is the level of WZW model. We also consider finite k correction to the effective action which is compatible with Fedosov's deformation quantization of the background.     

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.     

Notes on generalized global symmetries in QFT

It was recently argued that quantum field theories possess one‐form and higher‐form symmetries, labelled ‘generalized global symmetries.’ In this paper, we describe how those higher‐form symmetries can be understood mathematically as special cases of more general 2‐groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one‐form and higher‐form symmetries. We discuss analogues of topological defects for some of these higher symmetry groups, relating some of them to ordinary topological defects. We also discuss topological defects in cases in which the moduli ‘space’ (technically, a stack) admits an action of a higher symmetry group. Finally, we outline a proposal for how certain anomalies might potentially be understood as describing a transmutation of an ordinary group symmetry of the classical theory into a 2‐group or higher group symmetry of the quantum theory, which we link to WZW models and bosonization.     

Nested T-Duality

We identify the obstructions for Kiritsis–Obers T-duality of boundary WZW models. The open string duality pattern is much richer than in the closed strings case since it depends substantially on the geometry of branes. In particular, the duality obstructions disappear for certain brane configurations associated to non-regular elements of the Cartan torus. It is shown in this case that the boundary WZW model is “nested” in the twisted boundary WZW model as the dynamical subsystem of the latter     

New integrable lattice hierarchies

In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula.     

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.     

Entanglement and topological interfaces

In this paper we consider entanglement entropies in two‐dimensional conformal field theories in the presence of topological interfaces. Tracing over one side of the interface, the leading term of the entropy remains unchanged. The interface however adds a subleading contribution, which can be interpreted as a relative (Kullback‐Leibler) entropy with respect to the situation with no defect inserted. Reinterpreting boundaries as topological interfaces of a chiral half of the full theory, we rederive the left/right entanglement entropy in analogy with the interface case. We discuss WZW models and toroidal bosonic theories as examples.     

Form factors of exponential fields for two-parametric family of integrable models

A two-parametric family of integrable models (the SS model) that contains as particular cases several well known integrable quantum field theories is considered. After the quantum group restriction it describes a wide class of integrable perturbed conformal field theories. Exponential fields in the SS model are closely related to the primary fields in these perturbed theories. We use the bosonization approach to derive an integral representation for the form factors of the exponential fields in the SS model. The same representations for the sausage model and the cosine–cosine model are obtained as limiting cases. The results are tested at the special points, where the theory contains free particles.     

The N = 2 supersymmetric Toda lattice hierarchy

We propose a new integrable N = 2 supersymmetric Toda lattice hierarchy which may be relevant for constructing a supersymmetric one-matrix model. We define its first two Hamiltonian structures, the recursion operator and Lax-pair representation. We provide partial evidence for the existence of an infinite-dimensional N = 2 superalgebra of its flows. We study its bosonic limit and introduce new Lax-pair representations for the bosonic Toda lattice hierarchy. Finally we discuss the relevance this approach for constructing N = 2 supersymmetric generalized Toda lattice hierarchies.     

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