Liouville CFT, Matrix Models and constrained WZW

TL;DR

This paper connects Liouville conformal blocks with irregular operators to matrix models and WZW models, revealing new integral representations and differential equations that deepen understanding of these theories.

Contribution

It introduces a matrix model realization of Liouville conformal blocks with irregular operators derived from an $SL(2, ext{C})$ WZW model, linking different theoretical frameworks.

Findings

Matrix models with $eta$-deformed measures represent Liouville conformal blocks.

Irregular KZ equations are satisfied by the conformal blocks.

A matrix model realization using generalized resolvents is provided.

Abstract

In this paper we study matrix model realizations of Liouville conformal blocks with degenerate and irregular operators. The corresponding matrix model is Hermitian with a $\beta$-deformed measure and the degree of the potential corresponds to the degree of the irregular operator in the CFT conformal block. We then show how such matrix integrals can be obtained from an $SL(2,\mathbb{C})$ WZW model with a conformal constraint giving rise to Liouville theory. The corresponding conformal blocks satisfy irregular versions of Knizhnik-Zamolodchikov (KZ) equations and we provide a matrix model realization in terms of generalized resolvents.