Calogero-Moser Models IV: Limits to Toda theory

TL;DR

This paper explores the relationship between Calogero-Moser and Toda integrable models, focusing on their Hamiltonians and Lax pairs, and identifies conditions under which Calogero-Moser models limit to Toda models.

Contribution

It clarifies the conditions and representations under which Calogero-Moser models reduce to Toda models, especially highlighting the role of minimal type Lax pairs.

Findings

Elliptic Calogero-Moser Hamiltonians tend to Toda Hamiltonians with proper scaling.

Most Calogero-Moser Lax pairs do not have Toda limits, except minimal type Lax pairs.

Minimal type Lax pairs correspond to minimal Lie algebra representations and tend to Toda Lax pairs.

Abstract

Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs of Calogero-Moser models are specified by the representations of the reflection groups, which are not the same as those of the corresponding Lie algebras. The latter specify the Lax pairs of Toda models. The Hamiltonians of the elliptic Calogero-Moser models tend to those of Toda models as one of the periods of the elliptic function goes to infinity, provided the dynamical variables are properly shifted and the coupling constants are scaled. On the other hand most of Calogero-Moser Lax pairs, for example, the root type Lax pairs, do not a have consistent Toda model limit. The minimal type Lax pairs, which corresponds to the minimal representations of the Lie algebras, tend to the Lax pairs of the corresponding Toda models.