AGT correspondence, (q-)Painlevè equations and matrix models

Abstract

Painlevè equation for conformal blocks is a combined corollary of integrability and Ward identities, which can be explicitly revealed in the matrix model realization of AGT relations. We demonstrate this in some detail, both for q-Painlevè equations for the q-Virasoro conformal block, or AGT dual gauge theory in 5d, and for ordinary Painlevè equations, or AGT dual gauge theory in 4d. Especially interesting is the continuous limit from 5d to 4d and its description at the level of equations for eight τ-functions. Half of these equations are governed by integrability and another half by Ward identities.