# $\mathcal{N}=2^*$ gauge theory, free fermions on the torus and Painlev\'e VI

## TL;DR

This paper connects $ ext{SU}(2)$ $ ext{N}=2^*$ gauge theory with Painlevé VI equations by expressing the tau-function through free fermions and Fredholm determinants, revealing new insights into isomonodromic deformations and RG flow.

## Contribution

It extends the Painlevé/gauge theory correspondence to circular quivers, providing explicit combinatorial and determinant formulas for the tau-function of isomonodromic deformations.

## Key findings

- Explicit combinatorial expression for the tau-function.
- Fredholm determinant formula for isomonodromic deformations.
- Exact solution of the RG flow in self-dual $ ext{Omega}$-background.

## Abstract

In this paper we study the extension of Painlev\'e/gauge theory correspondence to circular quivers by focusing on the special case of $SU(2)$ $\mathcal{N}=2^*$ theory. We show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of $SL_2$ flat connections on the one-punctured torus. This is achieved by reformulating the Riemann-Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the $SU(2)$ $\mathcal{N}=2^*$ theory on self-dual $\Omega$-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.