Irregular conformal blocks, Painlevé III and the blow-up equations
We study the relation of irregular conformal blocks with the Painlevé III3 equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painlevé III3. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both c = 1 and c → ∞ conformal blocks. We extend this analysis to the domain of strong-coupling regime where original definition of conformal blocks and Nekrasov functions is not known and apply the results to spectral problem of the Mathieu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painlevé III3 equation, and obtain in this way a general expression, reproducing c = 1 and quasiclassical c → ∞ results as its particular cases. We have also found explicit integral representations for c = 1 and c = −2 irregular blocks at infinity for some special points.