N = 2 ^∗ Gauge Theory, Free Fermions on the Torus and Painlevé VI

In this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of SU(2) 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 SL₂ 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) N= 2 ^∗ theory on self-dual Ω -background and, in the Seiberg–Witten limit, an elegant relation between the IR and UV gauge couplings.