Holomorphic Bundles and Many-Body Systems

TL;DR

This paper demonstrates that certain many-body integrable systems, including elliptic Calogero-Moser and Gaudin models, can be derived as degenerations of Hitchin systems, with applications to quantum integrability and explicit formulas on higher genus surfaces.

Contribution

It establishes a unifying framework connecting these systems to Hitchin moduli spaces and provides explicit Lax operators and quantum degenerations related to conformal field theory.

Findings

Degeneration of Hitchin systems yields elliptic Calogero-Moser and Gaudin models.

Explicit Lax operators on higher genus surfaces are derived.

Quantum counterparts relate to Knizhnik-Zamolodchikov-Bernard equations.

Abstract

We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins can be treated as a degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. Explicit formulas for the Lax operator on the higher genus surfaces are obtained in the Shottky parameterization. The constructions are motivated by the Conformal Field Theory, and their quantum counterpart can be treated as a degeneration of the critical level Knizhnik-Zamolodchikov-Bernard equations.