We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in GL(N, C) can be written in terms of a Fred holm 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, 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.

Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions / Del Monte, Fabrizio; Desiraju, Harini; Gavrylenko, Pavlo. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 398:3(2023), pp. 1029-1084. [10.1007/s00220-022-04458-y]

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

Del Monte, Fabrizio;Desiraju, Harini;Gavrylenko, Pavlo
2023-01-01

Abstract

We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in GL(N, C) can be written in terms of a Fred holm 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, 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.
2023
398
3
1029
1084
https://doi.org/10.1007/s00220-022-04458-y
https://arxiv.org/abs/2011.06292
Del Monte, Fabrizio; Desiraju, Harini; Gavrylenko, Pavlo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135599
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