Canonical structure and symmetries of the Schlesinger equations

Details Canonical structure and symmetries of the Schlesinger equations Canonical structure and symmetries of the Schlesinger equations

2003-11-16

arxiv.org/abs/math/0311261v4

Mathematics

Differential Geometry

92 pages, no figures. Theorem 1.2 corrected, other misprints removed. To appear on Comm. Math. Phys

Scientific paper

The Schlesinger equations $S_{(n,m)}$ describe monodromy preserving deformations of order $m$ Fuchsian systems with $n+1$ poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of $n$ copies of $m\times m$ matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations $S_{(n,m)}$ for all $n$, $m$ and we compute the action of the symmetries of the Schlesinger equations in these coordinates.

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