In the past five years, a surge of new techniques from different areas of mathematics and physics led to a rigorous study of the taufunctions of isomonodromic systems on a Riemann sphere and in particular, the Painlevé equations. We know that the taufunctions 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 taufunction of Painlevé II can be written as a Fredholm determinant of an integrable operator, and the taufunction 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 taufunction on a torus with Fuchsian singularities and generic monodromies can be written as a Fredholm determinant of CauchyPlemelj 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 timedependent Hamiltonian system with an elliptic potential, the time being the modular parameter of the torus. I will show that the isomonodromic taufunction of such a system is a Fredholm determinant described solely by hypergeometric functions, and its combinatorial expression takes the form of a dual NekrasovOkounkov partition function with a nonzero total charge.
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Isomonodromic taufunctions on a torus as Fredholm determinants, and charged partitions
Research Group:
Harini Desiraju
Institution:
Birmingham
Schedule:
Thursday, April 15, 2021  16:00 to 17:00
Location:
Online
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