Refined Painlevé/gauge theory correspondence and quantum tau functions

TL;DR

This work extends the Painlevé/gauge theory correspondence to general Ω-backgrounds by formulating a quantum deformation of Painlevé equations via C^2/ℤ2 blowup relations. It provides a systematic framework to obtain strong-coupling tau-function expansions for N=2 SU(2) partition functions across N_f=4 down to Argyres–Douglas points, using both quantum tau-functions and Zak transforms. The authors validate their results through refined holomorphic anomaly calculations and through comparisons with irregular Liouville/CFT constructions, including AGT and X_2 vertex operator formalisms. The study reveals two distinct strong-coupling expansion patterns (linear and square exponential) corresponding to different light-particle spectra, and outlines coalescence structures among PV, PIII, PIV, PII, and PI in the quantum setting. The results open routes to nonperturbative insights into refined topological strings, AD theories, and potential q-difference and higher-rank generalizations, with several open questions highlighted for future exploration.

Abstract

In this paper we study strong coupling asymptotic expansions of ${\mathcal N}=2$ $D=4$ $SU(2)$ gauge theory partition functions in general $Ω$-background. This is done by refining Painlevé/gauge theory correspondence in terms of quantum Painlevé 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.

Figures (4)

• Figure 1: Painlevé coalescence diagram
• Figure 2: Gauge theory degeneration diagram
• Figure 3: Canonical rays around critical point $\infty$
• Figure 4: Painlevé coalescence diagram with symmetries