Refined Painlev\'e/gauge theory correspondence and quantum tau functions

TL;DR

This paper refines the Painlevé/gauge theory correspondence using quantum Painlevé equations to analyze strong coupling expansions of ${ m N}=2$ $SU(2)$ gauge theory partition functions, revealing new insights into their asymptotic behavior.

Contribution

It introduces a systematic approach to study strong coupling expansions via quantum Painlevé equations derived from blowup relations, extending the existing correspondence.

Findings

Derived strong coupling asymptotic expansions of gauge theory partition functions.

Connected quantum Painlevé equations with holomorphic anomaly and conformal blocks.

Analyzed expansions around Argyres-Douglas points and singularities.

Abstract

In this paper we study strong coupling asymptotic expansions of ${\mathcal N}=2$ $D=4$ $SU(2)$ gauge theory partition functions in general $\Omega$-background. This is done by refining Painlev\'e/gauge theory correspondence in terms of quantum Painlev\'e equations, obtained from $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations. We present a general ansatz and a systematic analysis of the expansions of the gauge theory partition functions by solving the above equations around the strong coupling singularities, including Argyres-Douglas points. We compare our results with refined holomorphic anomaly equations and irregular Virasoro conformal blocks.