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

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