How instanton combinatorics solves Painlev\'e VI, V and III's

TL;DR

This paper connects Painlevé transcendents with 2D conformal field theory and instanton partition functions, providing combinatorial series solutions that enhance numerical computation and applications in integrable systems.

Contribution

It introduces a novel combinatorial series representation of Painlevé functions based on conformal blocks and instanton calculus, linking integrable systems with quantum field theory.

Findings

Series representations enable efficient numerical evaluation of Painlevé functions.

Connections established between Painlevé equations, conformal blocks, and instanton partition functions.

Applications include analysis of Fredholm determinants and correlation functions in integrable models.

Abstract

We elaborate on a recently conjectured relation of Painlev\'e transcendents and 2D CFT. General solutions of Painlev\'e VI, V and III are expressed in terms of $c=1$ conformal blocks and their irregular limits, AGT-related to instanton partition functions in $\mathcal{N}=2$ supersymmetric gauge theories with $N_f=0,1,2,3,4$. Resulting combinatorial series representations of Painlev\'e functions provide an efficient tool for their numerical computation at finite values of the argument. The series involve sums over bipartitions which in the simplest cases coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the GUE, and all-order conformal perturbation theory expansions of correlation functions in the sine-Gordon field theory at the free-fermion point.