On Spectral Flow Symmetry and Knizhnik-Zamolodchikov Equation

TL;DR

This paper explores the spectral flow symmetry in SL(2,R)_k affine algebra using Liouville theory representations, rederives winding-violating three-point functions, and proves identities among solutions of the KZ equation.

Contribution

It introduces a novel method leveraging Liouville theory to analyze spectral flow symmetry and exact solutions of the KZ equation in WZNW models.

Findings

Rederived winding-violating three-point functions succinctly

Proved new identities among KZ solutions

Enhanced understanding of spectral flow symmetry in affine algebras

Abstract

It is well known that five-point function in Liouville field theory provides a representation of solutions of the SL(2,R)_k Knizhnik-Zamolodchikov equation at the level of four-point function. Here, we make use of such representation to study some aspects of the spectral flow symmetry of sl(2)_k affine algebra and its action on the observables of the WZNW theory. To illustrate the usefulness of this method we rederive the three-point function that violates the winding number in SL(2,R) in a very succinct way. In addition, we prove several identities holding between exact solutions of the Knizhnik-Zamolodchikov equation.