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

TL;DR

This paper explores the deep connections between irregular conformal blocks, Painlevé III equations, and blow-up equations, providing new insights into their structure and extending analysis into strong-coupling regimes with explicit representations.

Contribution

It introduces a novel approach linking irregular conformal blocks with Painlevé III and blow-up equations, including a construction in the strong-coupling regime and explicit integral representations.

Findings

Consistent functional representation with BPZ and Hamilton-Jacobi equations.

Derivation of limiting blow-up equations for Nekrasov partition functions.

Explicit integral representations for specific irregular blocks.

Abstract

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. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painlev\'e III$_3$ equation, and obtain in this way a general expression, reproducing $c=1$ and quasiclassical $c\to\infty$ 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.