Calogero-Moser Models: A New Formulation

TL;DR

This paper introduces a new root system-based formulation for Calogero-Moser models, providing novel Lax pairs that establish integrability for all simply-laced cases, including the long-unsolved E8 model.

Contribution

It presents a new root type Lax pair construction and a unified minimal type framework applicable to all simply-laced Calogero-Moser models, solving longstanding integrability issues.

Findings

Root type Lax pair is new and resembles the adjoint representation.

The formulation applies to all four potential types: rational, trigonometric, hyperbolic, elliptic.

Successfully establishes integrability for the E8 model.

Abstract

A new formulation of Calogero-Moser models based on root systems and their Weyl group is presented. The general construction of the Lax pairs applicable to all models based on the simply-laced algebras (ADE) are given for two types which we call `root' and `minimal'. The root type Lax pair is new; the matrices used in its construction bear a resemblance to the adjoint representation of the associated Lie algebra, and exist for all models, but they do not contain elements associated with the zero weights corresponding to the Cartan subalgebra. The root type provides a simple method of constructing sufficiently many number of conserved quantities for all models, including the one based on $E_{8}$, whose integrability had been an unsolved problem for more than twenty years. The minimal types provide a unified description of all known examples of Calogero-Moser Lax pairs and add some more. In both cases, the root type and the minimal type, the formulation works for all of the four choices of potentials: the rational, trigonometric, hyperbolic and elliptic.