Refined Painlevé/Gauge Theory Correspondence and Quantum Tau Functions

Abstract In this paper we study strong coupling asymptotic expansions of $$\mathcal {N}=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> four-dimensional SU (2) gauge theory partition functions in general $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> -background. This is done by refining the Painlevé/gauge theory correspondence in terms of quantum Painlevé equations, obtained from $$\mathbb {C}^2/\mathbb {Z}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> 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 check our results via refined holomorphic anomaly equations and compare with irregular Virasoro conformal blocks.