Characteristic Classes and Integrable Systems for Simple Lie Groups

TL;DR

This paper extends elliptic Calogero-Moser systems to simple Lie groups with non-trivial centers, constructing explicit Lax operators and Hamiltonians for these generalized integrable systems.

Contribution

It introduces a new class of elliptic Calogero-Moser systems for classical simply-connected Lie groups with non-trivial centers, including explicit Lax pairs and Hamiltonians.

Findings

Constructed special bases for non-trivial characteristic classes

Derived explicit forms of Lax operators and Hamiltonians

Generalized classical Calogero-Moser systems to new Lie group classes

Abstract

This paper is a continuation of our previous paper \cite{LOSZ}. For simple complex Lie groups with non-trivial center, i.e. classical simply-connected groups, $E_6$ and $E_7$ we consider elliptic Modified Calogero-Moser systems corresponding to the Higgs bundles with an arbitrary characteristic class. These systems are generalization of the classical Calogero-Moser (CM) systems related to a simple Lie groups and contain CM systems related to some (unbroken) subalgebras. For all algebras we construct a special basis, corresponding to non-trivial characteristic classes, the explicit forms of Lax operators and Hamiltonians.