Discrete matrix models for partial sums of conformal blocks associated to Painlev\'e transcendents

TL;DR

This paper introduces a discrete matrix model to compute partial sums of conformal blocks related to Painlevé transcendents, providing new integral and determinant representations that connect conformal field theory with matrix models.

Contribution

It proposes a novel discrete matrix model framework to evaluate partial sums of Painlevé conformal blocks, linking them to Hankel determinants and hypergeometric functions.

Findings

Derived multiple integral representations for partial sums.

Connected conformal blocks to Hankel determinants and hypergeometric functions.

Provided a new computational approach for Painlevé-related conformal blocks.

Abstract

A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives an asymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $\tau$-function for certain Painlev\'e transcendents. The coefficients in this expansion are given in terms of conformal blocks of a two-dimensional conformal field theory, which can be written as infinite sums over pairs of partitions. In this note a discrete matrix model is proposed on a lattice whose partition function can be used to obtain a multiple integral representation for the length restricted partial sums of the Painlev\'e conformal blocks. This leads to expressions of the partial sums involving H\"ankel determinants associated to the discrete measure of the matrix model, or equivalently, Wronskians of the corresponding moment generating function which is shown to be of the generalized hypergeometric type.