Abstract
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 function of the monodromy map. This, in particular, solves a recent conjecture by A. Its, O. Lisovyy and A. Prokhorov.
Highlights
We show that Fock–Goncharov coordinates are log-canonical for the symplectic form
Symplectic aspects of the monodromy map for the Fuchsian systems were studied starting from [3,25,33]; in these papers it was proved that the monodromy map is a symplectomorphism from a symplectic leaf in the space of coefficients of the system to a symplectic leaf in the monodromy manifold
Our interest in this subject stems from the study of monodromy map of a second order equation on a Riemann surface; such a map was proven to be a symplectomorphism [9,10,32,34]
Summary
Symplectic aspects of the monodromy map for the Fuchsian systems were studied starting from [3,25,33]; in these papers it was proved that the monodromy map is a symplectomorphism from a symplectic leaf in the space of coefficients of the system to a symplectic leaf in the monodromy manifold. The Poisson structure induced on A from the Poisson structure (1.12) on A0 via the reduction on the level set of the moment map, corresponding to the group action G j → SG j , is non-degenerate and the corresponding symplectic form is ωA = dθA, where the symplectic potential θA for ωA is given by θA = This symplectic structure appeared in [11] but the connection to dynamical r -matrix and associated Poisson structure was not known until now. 4) and “strong” (Theorem 3.2) versions of the Its– Lisovyy–Prokhorov conjecture about coincidence of the external derivative of the Malgrange form with the natural symplectic form on the monodromy manifold. In the S L(2) case we derive equations which define the monodromy dependence of , G j and tau-functions (Theorem 6.1, Corollary 6.1 and Proposition 6.3)
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