Monodromy of solutions of the Knizhnik-Zamolodchikov equation: SL(2,C)/SU(2) WZNW model

TL;DR

This paper explores the monodromy of conformal blocks in the SL(2,C)/SU(2) WZNW model, establishing connections with Liouville field theory and revealing two fusion matrices due to singularities in the conformal blocks.

Contribution

It provides three explicit, equivalent representations of the monodromy in the WZNW model using Liouville theory and identifies the existence of two fusion matrices.

Findings

Three equivalent monodromy representations in terms of Liouville theory

Identification of two possible fusion matrices in the model

Relation between conformal block asymptotics and Liouville correlation functions

Abstract

Three explicit and equivalent representations for the monodromy of the conformal blocks in the SL(2,C)/SU(2) WZNW model are proposed in terms of the same quantity computed in Liouville field theory. We show that there are two possible fusion matrices in this model. This is due to the fact that the conformal blocks, being solutions to the Knizhnik-Zamolodchikov equation, have a singularity when the SL(2,C) isospin coordinate x equals the worldsheet variable z. We study the asymptotic behaviour of the conformal block when x goes to z. The obtained relation inserted into a four point correlation function in the SL(2,C)/SU(2) WZNW model gives some expression in terms of two correlation functions in Liouville field theory.