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

In this paper we study the extension of Painlev'e/gauge theory correspondence to circular quivers by focusing on the special case of $SU(2)$ $\mathcal{N}=2^{\ast }$ 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 $S{L}_2$ 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 $SU(2)$ $\mathcal{N}=2^{\ast }$ theory on self-dual $\mathrm{\Omega }$-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.