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

Abstract: Painlev`e 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 $q$-Painlev`e equations for the $q$-Virasoro conformal block, or AGT dual gauge theory in $5d$, and for ordinary Painlev`e 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 $\tau$-functions. Half of these equations are governed by integrability and another half by Ward identities.