Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions

TL;DR

This paper presents a Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems, linking them to Nekrasov sums and providing explicit formulas for special cases like Painlevé VI.

Contribution

It introduces a novel Fredholm determinant framework for isomonodromic tau functions, connecting them to fundamental solutions of elementary Fuchsian systems and explicit series representations.

Findings

Derived Fredholm determinant representation for tau functions.

Expressed kernel in terms of elementary Fuchsian system solutions.

Obtained explicit series for Painlevé VI solutions.

Abstract

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ gives a series representation of the general solution to Painlev\'e VI equation.