Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions

TL;DR

This paper demonstrates that the isomonodromic tau function on a torus with Fuchsian singularities can be expressed as a Fredholm determinant and expanded via charged partitions, linking to Nekrasov-Okounkov functions and free fermion conformal blocks.

Contribution

It establishes a novel representation of the tau function as a Fredholm determinant and connects it to combinatorial charged partitions and conformal blocks on a torus.

Findings

Tau function expressed as a Fredholm determinant.

Minor expansion described by charged partitions.

Connection to Nekrasov-Okounkov functions and free fermion conformal blocks.

Abstract

We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $SL(2,\mathbb{C})$ this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results, we also propose a definition of the tau function of the Riemann-Hilbert problem on a torus with generic jump on the A-cycle.