Lattice Wess-Zumino-Witten Model and Quantum Groups

TL;DR

This paper explores the role of quantum groups as symmetries in the lattice-regularized Wess-Zumino-Witten model, providing a free field approach to its exact solution and revealing the coupling of space-time and internal symmetries.

Contribution

It introduces a lattice regularization of the WZW model with quantum group symmetries and develops a free field method for its exact solution.

Findings

Quantum group symmetry acts on chiral components.

Lattice regularization preserves quantum group symmetry.

Exact solution of the lattice $SL(2)$ WZW model achieved.

Abstract

Quantum groups play a 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 side, 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 the 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.