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Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus

We study the relation between class S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two dimensional torus with punctures. Turning on the self dual $Ω$-background corresponds to a deautonomization of the Seiberg-Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $τ$-function is proportional to the dual gauge theory partition function, the proportionality factor being a non trivial function of the solution of the deautonomized Seiberg-Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $W_N$ free fermion correlators on the torus.