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Isomonodromic tau-functions on a torus as Fredholm determinants, and charged partitions

Harini Desiraju
Thursday, April 15, 2021 - 16:00 to 17:00

In the past five years, a surge of new techniques from different areas of mathematics and physics led to a rigorous study of the tau-functions of isomonodromic systems on a Riemann sphere and in particular, the Painlevé equations.  We know that the tau-functions of Painlevé VI, V, III can be described as a Fredholm determinant of a combination of Toeplitz operators called Widom constants and as a series of Conformal blocks or Nekrasov functions, the tau-function of Painlevé II can be written as a Fredholm determinant of an integrable operator, and the tau-function of Painlevé I is described by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves.In this talk I will show that the isomonodromic tau-function on a torus with Fuchsian singularities and generic monodromies can be written as a Fredholm determinant of Cauchy-Plemelj operators, and its minor expansion is a combinatorial series labeled by charged tuples of Young diagrams. The simplest example in this setting is a torus with one puncture associated to the formulation of the Painlevé VI equation as a time-dependent Hamiltonian system with an elliptic potential, the time being the modular parameter of the torus. I will show that the isomonodromic tau-function of such a system is a Fredholm determinant described solely by hypergeometric functions, and its combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function with a non-zero total charge.

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