  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.
