Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions

Abstract

We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in GL(N,{mathbb {C}}) can be written in terms of a Fredholm determinant of Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of SL(2,{mathbb {C}}) this combinatorial expression takes the form of a dual Nekrasov–Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results we also propose a definition of the tau function of the Riemann–Hilbert problem on a torus with generic jump on the A-cycle.