Refined Painlevé/gauge theory correspondence and quantum tau functions

In this paper we study strong coupling asymptotic expansions of $\mathcal{N}=2$ $D=4$ $SU(2)$ gauge theory partition functions in general $\mathrm{\Omega }$-background. This is done by refining Painlevé/gauge theory correspondence in terms of quantum Painlevé equations, obtained from ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_2$ 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.