On Painlevé/gauge theory correspondence

Abstract

We elucidate the relation between Painlevé equations and four-dimensional rank one ${\cal N= 2}$ theories by identifying the connection associated to Painlevé 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 Painlevé functions in terms of magnetic and dyonic Nekrasov partition functions for ${\cal N= 2}$ SQCD and Argyres-Douglas theories at self-dual Omega background $ε_1+ε_2= 0$, or equivalently in terms of $c= 1$ irregular conformal blocks.

51 pages, 2 figures