Liouville CFT, Matrix Models and constrained WZW

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.