Vector Bundles and Lax Equations on Algebraic Curves

Abstract

The Hamiltonian theory of zero-curvature equations with spectral parameter on an arbitrary compact Riemann surface is constructed. It is shown that the equations can be seen as commuting flows of an infinite-dimensional field generalization of the Hitchin system. The field analog of the elliptic Calogero-Moser system is proposed. An explicit parameterization of Hitchin system based on the Tyurin parameters for stable holomorphic vector bundles on algebraic curves is obtained.

A new hierarchy of integrable systems associated to Hurwitz spaces

“…We expect our systems for gR 2 to be natural deformations of two-dimensional version of Hitchin systems proposed in Krichever (2002b). The isomonodromic deformations in higher genus briefly discussed here should be in close relation to existing frameworks (Korotkin & Samtleben 1997;Levin & Olshanetski 1997;Krichever 2002a), as well as to non-autonomous Hitchin systems (Hitchin 1987).…”

Classicalr-matrices and the Feigin–Odesskii algebra via Hamiltonian and Poisson reductions

“…In the general context of holomorphic vector bundles over Riemann surfaces this is known as the Tyurin parameterisation [30]. This parameterisation has found recent application in a general description of Hitchin systems and their r-matrices [31], [32], [33]. Expanding the functions g(z) and η(z) in a neighbourhood of the contour γ,…”

“…The Lax matrix for the spin Calogero-Moser system in the form (56) was originally obtained in the paper by Krichever [32], while the formula (57) for the corresponding r-matrix has been given recently in [33].…”

Integrable chains on algebraic curves

“…It is straightforward to check that, if we defineż s (n),α s (n) by the formulae (2.7), (2.8) in [2], i.e.ż…”

“…The Riemann-Roch theorem implies that naive direct generalization of equations (1.1, 1.2) for matrix functions, which are meromorphic on an algebraic curve Γ of genus g > 0, leads to an over-determined system of equations (see details in [2]). In [1] it was found for the continuous case, that if the matrix functions L and M have moving poles with special dependence on x and t besides fixed poles, then equation (1.1) is a well-defined system on the space of singular parts of L and M at fixed poles.…”