Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero–Gaudin system

Takasaki, Kanehisa

doi:10.1063/1.1591053

Cited by 3 publications

(4 citation statements)

References 28 publications

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“…Isomonodromic deformations on a torus with n simple poles can be characterised by the following system of linear differential equations [20,24] ∂ ∂z Φ (z) = Φ (z) L z (z) ,…”

Section: Setupmentioning

confidence: 99%

Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

Monte¹,

Desiraju²,

Gavrylenko³

2023

J. Phys. A: Math. Theor.

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We compute the monodromy dependence of the isomonodromic tau function on a torus with n Fuchsian singularities and SL(N) residue matrices by using its explicit Fredholm determinant representation. We show that the exterior logarithmic derivative of the tau function defines a closed one-form on the space of monodromies and times, and identify it with the generating function of the monodromy symplectomorphism. As an illustrative example, we discuss the simplest case of the one-punctured torus in detail. Finally, we show that previous results obtained in the genus zero case can be recovered in a straightforward manner using the techniques presented here.

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“…Isomonodromic deformations on a torus with n simple poles can be characterised by the following system of linear differential equations [20,24] ∂ ∂z Φ (z) = Φ (z) L z (z) ,…”

Section: Setupmentioning

confidence: 99%

Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

Monte¹,

Desiraju²,

Gavrylenko³

2023

J. Phys. A: Math. Theor.

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“…, z n }, also called Fuchsian singularities. Differently from what happens on the sphere, L(z)dz is not a single-valued matrix differential, but rather has the following twist properties along the torus A-and B-cycles [48,[55][56][57]:…”

Section: General Fuchsian System On the Torusmentioning

confidence: 99%

“…These Hamiltonians can all be obtained as usual from the logarithmic derivative of a single tau function [48,[55][56][57]:…”

Section: )mentioning

confidence: 99%

“…As a byproduct, we will also obtain a direct link between the isomonodromic tau function and the SU (n), rather than U (n), gauge theory. However, note that in the case of more than one puncture, the Calogero-like variables Q i do not specify the whole system: there are additional spin variables satisfying the Kirillov-Kostant Poisson bracket for sl(N ) [57] that will not enter in the following discussion: while it may be that there is some further connection between Z D and these remaining dynamical variables, this does not seem evident at the moment.…”

Section: Solution Of the Elliptic Schlesinger Systemmentioning

confidence: 99%

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Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

et al. 2021

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We study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

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

Monte

Desiraju

Gavrylenko

2023

Commun. Math. Phys.

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We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $$GL(N,{\mathbb {C}})$$ G L ( N , 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 $$SL(2,{\mathbb {C}})$$ S L ( 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.

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Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero–Gaudin system

Cited by 3 publications

References 28 publications

Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

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

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