Two-toroidal Lie Algebra as Current Algebra of Four-dimensional K\"ahler WZW Model

TL;DR

This paper explores the infinite-dimensional symmetry algebra of a four-dimensional K"ahler WZW model, identifying it as a two-toroidal Lie algebra, extending the well-known affine Kac-Moody algebra from two dimensions.

Contribution

It introduces a novel two-toroidal Lie algebra structure as the current algebra of the four-dimensional K"ahler WZW model, extending the algebraic framework of the two-dimensional case.

Findings

Derived the two-toroidal Lie algebra as the current algebra

Expressed the energy-momentum tensor in terms of currents

Extended the affine Kac-Moody algebra to four dimensions

Abstract

We investigate the structure of an infinite-dimensional symmetry of the four-dimensional K\"ahler WZW model, which is a possible extension of the two-dimensional WZW model. We consider the SL(2,R) group and, using the Gauss decomposition method, we derive a current algebra identified with a two-toroidal Lie algebra, a generalization of the affine Kac-Moody algebra. We also give an expression of the energy-momentum tensor in terms of currents and extra terms.