# Tau-Functions and Monodromy Symplectomorphisms

@article{Bertola2019TauFunctionsAM,
title={Tau-Functions and Monodromy Symplectomorphisms},
author={Marco Bertola and Dmitry Korotkin},
journal={Communications in Mathematical Physics},
year={2019},
volume={388},
pages={245 - 290}
}
• Published 8 October 2019
• Mathematics
• Communications in Mathematical Physics
We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical r-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock–Goncharov coordinates are log-canonical for the symplectic form. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the Jimbo–Miwa–Ueno tau-function as the generating…
• Mathematics
• 2022
: 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
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• 2022
. We study the WKB expansion of 2 × 2 system of linear diﬀerential equations with four fuchsian singularities. The main focus is on the generating function of the monodromy symplectomorphism which,
• Mathematics
Journal of Physics A: Mathematical and Theoretical
• 2022
Connection formulas relating Frobenius solutions of linear ODEs at different Fuchsian singular points can be expressed in terms of the large order asymptotics of the corresponding power series. We
We study symplectic properties of the monodromy map of the Schrodinger equation on a Riemann surface with a meromorphic potential having second order poles. At ﬁrst, we discuss the conditions for the
• Mathematics
Theoretical and Mathematical Physics
• 2021
We study symplectic properties of the monodromy map of second-order equations on a Riemann surface whose potential is meromorphic with double poles. We show that the Poisson bracket defined in terms
• Mathematics
• 2019
In this paper we consider the symplectic properties of the monodromy map of second order equations on a Riemann surface whose potential is meromorphic with second order poles. We show that the
• Mathematics
• 2019
Given an oriented graph on a punctured Riemann surface of arbitrary genus, we define a canonical symplectic structure over the set of flat connections on the dual graph, and show that it is invariant

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Given an oriented graph on a punctured Riemann surface of arbitrary genus, we define a canonical symplectic structure over the set of flat connections on the dual graph, and show that it is invariant

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Abstract We study symplectic properties of the monodromy map of second-order equations on a Riemann surface whose potential is meromorphic with double poles. We show that the Poisson bracket defined
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• Mathematics, Physics
Communications in Mathematical Physics
• 2021
The note corrects the aforementioned paper (Bertola in Commun Math Phys 294(2):539–579, 2010). The consequences of the correction are traced and the examples updated.
• Mathematics
• 2019
In this paper we consider the symplectic properties of the monodromy map of second order equations on a Riemann surface whose potential is meromorphic with second order poles. We show that the