AGT correspondence, (q-)Painlev\`e equations and matrix models

TL;DR

This paper explores the deep connections between AGT correspondence, Painlevé equations, and matrix models, revealing how integrability and Ward identities underpin these relations in both 4d and 5d gauge theories.

Contribution

It explicitly demonstrates the realization of Painlevé equations within matrix models associated with AGT relations, including the continuous limit from 5d to 4d.

Findings

Explicit matrix model realization of Painlevé equations in AGT context

Description of the 5d to 4d limit at the level of tau-function equations

Identification of integrability and Ward identities governing these equations

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.