Blowups in BPS/CFT Correspondence, and Painlevé 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 Ω-deformation parameters (ε₁,ε₂). As a consequence, we obtain the formula conjectured in 2012 by O. Gamayun, N. Iorgov, and O. Lysovyy, relating the tau-function τ_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 τ_PVI of Painlevé VI and its generalizations. We also make some remarks about the sphere partition functions, the boundary operator product expansion in the N=(4,4) sigma models related to four-dimensional N=2 theories on toric manifolds, discuss crossed instantons on conifolds, elucidate some aspects of the BPZ/KZ correspondence, and applications to quantization.