Irregular conformal blocks, Painlev\'e III and the blow-up equations

Understand

We study the relation of irregular conformal blocks with the Painlev\'e 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 Painlev\'e 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 $c\to\infty$ 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.