On Painlevé/gauge theory correspondence

TL;DR

This work builds a concrete bridge between Painlevé transcendents and four-dimensional rank-one $\mathcal{N}=2$ gauge theories by identifying Painlevé isomonodromic problems with the oper limit of Hitchin flat connections. It shows that Seiberg–Witten curves arise as spectral curves of Hitchin systems and that Painlevé $\tau$-functions correspond to dual Nekrasov partition functions or $c=1$ irregular conformal blocks, linking RG flow and Whitham deformations to isomonodromic dynamics. The authors derive explicit long-distance expansions for PI, PII, and PIV tau-functions and compute magnetic/dyonic prepotentials for Argyres–Douglas theories $H_0$, $H_1$, and $H_2$, verifying the Painlevé/gauge theory dictionary and providing new AD$-$Painlevé expansions. Overall, the paper reveals a deep synergy between integrable systems, Hitchin moduli, and BPS partition functions, with potential generalizations to higher rank, higher genus, and quantum deformations via irregular CFT blocks and AGT-type correspondences.

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.

Figures (2)

• Figure 1: Coalescence diagram for Painlevé equations.
• Figure 2: Canonical rays for irregular expansions at $t=\infty$.