Vector bundles and Lax equations on algebraic curves

TL;DR

This paper develops a Hamiltonian framework for zero-curvature equations with spectral parameters on Riemann surfaces, linking them to Hitchin systems and proposing a field analog of elliptic Calogero-Moser systems.

Contribution

It introduces a Hamiltonian theory for zero-curvature equations on algebraic curves and provides explicit parameterizations of Hitchin systems using Tyurin parameters.

Findings

Constructed Hamiltonian theory for zero-curvature equations on Riemann surfaces

Connected zero-curvature equations to Hitchin systems as commuting flows

Proposed a field analog of the elliptic Calogero-Moser system

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.