Characteristic Classes and Integrable Systems. General Construction

TL;DR

This paper constructs integrable systems from topologically non-trivial Higgs bundles over elliptic curves, introducing modified Calogero-Moser systems with unique phase space properties linked to bundle topology and characteristic classes.

Contribution

It introduces a new class of integrable systems called modified Calogero-Moser systems derived from Higgs bundles with non-trivial topology, expanding the understanding of their phase space and algebraic structure.

Findings

Modified Calogero-Moser systems have fewer particles but more spin variables.

The systems are constructed using Lax operators, Hamiltonians, and Poisson structures based on dynamical r-matrices.

The configuration space is the moduli space of holomorphic bundles with non-trivial characteristic classes.

Abstract

We consider topologically non-trivial Higgs bundles over elliptic curves with marked points and construct corresponding integrable systems. In the case of one marked point we call them the modified Calogero-Moser systems (MCM systems). Their phase space has the same dimension as the phase space of the standard CM systems with spin, but less number of particles and greater number of spin variables. Topology of the holomorphic bundles are defined by their characteristic classes. Such bundles occur if G has a non-trivial center, i.e. classical simply-connected groups, $E_6$ and $E_7$. We define the conformal version CG of G - an analog of GL(N) for SL(N), and relate the characteristic classes with degrees of CG-bundles. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, define the phase spaces and the Poisson structure using dynamical r-matrices. To describe the systems we use a special basis in the Lie algebras that generalizes the basis of t'Hooft matrices for sl(N). We find that the MCM systems contain the standard CM systems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the holomorphic bundles with non-trivial characteristic classes.