Calogero-Moser and Toda Systems for Twisted and Untwisted Affine Lie Algebras

TL;DR

This paper demonstrates how elliptic Calogero-Moser systems associated with simple Lie algebras scale to affine Toda systems under specific limits, revealing connections between these integrable models for twisted and untwisted affine Lie algebras.

Contribution

It introduces a scaling limit framework that links elliptic Calogero-Moser systems to affine Toda systems, including twisted and untwisted cases, based on the elliptic modulus and coupling parameters.

Findings

Scaling limits produce affine Toda systems from elliptic Calogero-Moser models.

Critical scaling corresponds to affine Lie algebras $ ext{G}^{(1)}$ and their duals.

Different parameter regimes yield Toda or Calogero-Moser systems.

Abstract

The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a general simple Lie algebra $\G$ are shown to scale to the (affine) Toda Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus $\tau$ and the Calogero-Moser couplings $m$ to infinity, while keeping fixed the combination $M = m e^{i \pi \delta \tau}$ for some exponent $\delta$. Critical scaling limits arise when $1/\delta$ equals the Coxeter number or the dual Coxeter number for the untwisted and twisted Calogero-Moser systems respectively; the limit consists then of the Toda system for the affine Lie algebras $\G^{(1)}$ and $(\G ^{(1)})^\vee$. The limits of the untwisted or twisted Calogero-Moser system, for $\delta$ less than these critical values, but non-zero, consists of the ordinary Toda system, while for $\delta =0$, it consists of the trigonometric Calogero-Moser systems for the algebras $\G$ and $\G^\vee$ respectively.