Calogero-Moser Models III: Elliptic Potentials and Twisting

TL;DR

This paper constructs universal Lax pairs for all Calogero-Moser models based on root systems, including twisted and extended models with elliptic, hyperbolic, trigonometric, and rational potentials, advancing integrability understanding.

Contribution

It introduces new Lax pairs for twisted non-simply laced Calogero-Moser models, including extended models with independent coupling constants, completing the universal Lax pair framework.

Findings

Constructed Lax pairs for twisted models based on B_n, C_n, BC_n root systems.

Introduced extended twisted models with additional potential terms.

Derived Lax pairs for all elliptic, hyperbolic, trigonometric, and rational potentials as limits.

Abstract

Universal Lax pairs of the root type with spectral parameter and independent coupling constants for twisted non-simply laced Calogero-Moser models are constructed. Together with the Lax pairs for the simply laced models and untwisted non-simply laced models presented in two previous papers, this completes the derivation of universal Lax pairs for all of the Calogero-Moser models based on root systems. As for the twisted models based on B_n, C_n and BC_nroot systems, a new type of potential term with independent coupling constants can be added without destroying integrability. They are called extended twisted models. All of the Lax pairs for the twisted models presented here are new, except for the one for the F_4 model based on the short roots. The Lax pairs for the twisted G_2 model have some novel features. Derivation of various functions, twisted and untwisted, appearing in the Lax pairs for elliptic potentials with the spectral parameter is provided. The origin of the spectral parameter is also naturally explained. The Lax pairs with spectral parameter, twisted and untwisted, for the hyperbolic, the trigonometric and the rational potential models are obtained as degenerate limits of those for the elliptic potential models.