  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.
