Bilinear tau forms of quantum Painlevé equations and ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_2$ blowup relations in SUSY gauge theories

We derive bilinear tau forms of the canonically quantized Painlevé equations, thereby relating them to those previously obtained from the ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_2$ blowup relations for the $\mathcal{N}=2$ supersymmetric gauge theory partition functions on a general $\mathrm{\Omega }$-background. We fully fix the refined Painlevé/gauge theory dictionary by formulating the proper equations for the quantum nonautonomous Painlevé Hamiltonians. We also describe the symmetry structure of the quantum Painlevé tau functions and, as a byproduct of this analysis, obtain the ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_2$ blowup relations in the nontrivial holonomy sector of the gauge theory.