Hitchin system on singular curves I

TL;DR

This paper explores the Hitchin integrable system on singular algebraic curves, providing explicit formulas for Hitchin Hamiltonians and demonstrating how to derive known integrable systems like Calogero models from these geometric constructions.

Contribution

It introduces explicit descriptions of Hitchin systems on singular curves, including formulas for Hamiltonians and methods to obtain these systems via Hamiltonian reduction, extending previous work with new techniques.

Findings

Explicit formulas for Hitchin Hamiltonians on singular curves

Method to derive Hitchin systems via Hamiltonian reduction

Reproduction of Calogero systems from Hitchin systems on singular curves

Abstract

In this paper we study Hitchin system on singular curves. Some examples of such system were first considered by N. Nekrasov (hep-th/9503157), but our methods are different. We consider the curves which can be obtained from the projective line by gluing several points together or by taking cusp singularities. (More general cases of gluing subschemas will be considered in the next paper). It appears that on such curves all ingredients of Hitchin integrable system (moduli space of vector bundles, dualizing sheaf, Higgs field etc.) can be explicitly described, which may deserve independent interest. As a main result we find explicit formulas for the Hitchin hamiltonians. We also show how to obtain the Hitchin integrable system on such a curve as a hamiltonian reduction from a more simple system on some finite-dimensional space. In this paper we also work out the case of a degenerate curve of genus two and find the analogue of the Narasimhan-Ramanan parameterization of SL(2)-bundles. We describe the Hitchin system in such coordinates. As a demonstration of the efficiency of our approach we also rederive the rational and trigonometric Calogero systems from the Hitchin system on cusp and node with a marked point.