Blowups in BPS/CFT correspondence, and Painlev\'e VI

We study four dimensional supersymmetric gauge theory in the presence of surface and point-like defects (blowups) and propose an identity relating partition functions at different values of $\Omega$-deformation parameters $({\varepsilon}_{1}, {\varepsilon}_{2})$. As a consequence, we obtain the formula conjectured in 2012 by O$.$Gamayun, N$.$Iorgov, and O$.$Lysovyy, relating the tau-function ${\tau}_{PVI}$ to $c=1$ conformal blocks of Liouville theory and propose its generalization for the case of Garnier-Schlesinger system. To this end we clarify the notion of the quasiclassical tau-function ${\tau}_{PVI}$ of Painlev\'e VI and its generalizations. We also make some remarks about the sphere partition functions, the boundary operator product expansion in the ${\mathcal{N}}=(4,4)$ sigma models related to four dimensional ${\mathcal{N}}=2$ theories on toric manifolds, discuss crossed instantons on conifolds, elucidate some aspects of the BPZ/KZ correspondence, and applications to quantization.