Irregular conformal blocks, Painleve III and the blow-up equations

We study the relation of irregular conformal blocks with the Painleve III$_3$ 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 Painleve III$_3$. 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 $ctoinfty$ 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 Matheiu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painleve III$_3$ equation, and obtain in this way a general expression, reproducing $c=1$ and quasiclassical $ctoinfty$ 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.