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arXiv · 2209.06150

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

A. Mironov·V. Mishnyakov·A. Morozov·Z. Zakirova
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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.

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A. Mironov, V. Mishnyakov, A. Morozov, Z. Zakirova. 2022-09-13. AGT correspondence, (q-)Painlev\`e equations and matrix models. https://doi.org/10.1016/j.nuclphysb.2022.116022

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