On Painlev\'e/gauge theory correspondence

TL;DR

This paper explores the deep mathematical connection between Painlevé equations and four-dimensional ${ m N}=2$ gauge theories, linking isomonodromic problems with Hitchin systems and Nekrasov partition functions.

Contribution

It establishes a precise correspondence between Painlevé isomonodromic problems and the oper limit of Hitchin system connections in ${ m N}=2$ gauge theories, providing new insights into their structure.

Findings

Derived long-distance expansions for Painlevé functions using Nekrasov partition functions.

Connected Painlevé equations with $c=1$ irregular conformal blocks.

Clarified the role of the Omega background in the gauge theory correspondence.

Abstract

We elucidate the relation between Painlev\'e equations and four-dimensional rank one ${\cal N= 2}$ theories by identifying the connection associated to Painlev\'e isomonodromic problems with the oper limit of the flat connection of the Hitchin system associated to gauge theories and by studying the corresponding renormalisation group flow. Based on this correspondence we provide long-distance expansions at various canonical rays for all Painlev\'e functions in terms of magnetic and dyonic Nekrasov partition functions for ${\cal N= 2}$ SQCD and Argyres-Douglas theories at self-dual Omega background $\epsilon_1+\epsilon_2= 0$, or equivalently in terms of $c= 1$ irregular conformal blocks.