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

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