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 Cauchy-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.

arXiv comment56 pages, 11 figures

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• Quantum spectral problems and isomonodromic deformations
• Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus
• On a class of elliptic orthogonal polynomials and their integrability
• Painlevé/CFT correspondence on a torus
• Nonlinear steepest descent on a torus: A case study of the Landau-Lifshitz equation
• Painlevé Kernels and Surface Defects at Strong Coupling