Calogero-Moser Models II: Symmetries and Foldings

TL;DR

This paper develops universal Lax pairs for Calogero-Moser models across all simply laced root systems, introduces twisted non-simply laced models via symmetries, and explores models with multiple coupling constants.

Contribution

It provides new Lax pairs for various Calogero-Moser models, including twisted and non-simply laced types, with detailed symmetry and coupling constant structures.

Findings

Universal Lax pairs for all simply laced root systems including E8.

Construction of twisted non-simply laced models from automorphisms.

Identification of models with multiple independent coupling constants.

Abstract

Universal Lax pairs (the root type and the minimal type) are presented for Calogero-Moser models based on simply laced root systems, including E_8. They exist with and without spectral parameter and they work for all of the four choices of potentials: the rational, trigonometric, hyperbolic and elliptic. For the elliptic potential, the discrete symmetries of the simply laced models, originating from the automorphism of the extended Dynkin diagrams, are combined with the periodicity of the potential to derive a class of Calogero-Moser models known as the `twisted non-simply laced models'. For untwisted non-simply laced models, two kinds of root type Lax pairs (based on long roots and short roots) are derived which contain independent coupling constants for the long and short roots. The BC_n model contains three independent couplings, for the long, middle and short roots. The G_2 model based on long roots exhibits a new feature which deserves further study.