Integrability of supersymmetric Calogero-Moser models

TL;DR

This paper investigates the integrability of supersymmetric Calogero-Moser models, explicitly constructs their Lax pairs, and identifies conserved charges demonstrating superintegrability, including for the ${ m N}=2$ case.

Contribution

It provides the first explicit Lax pair construction for supersymmetric Calogero-Moser models and identifies all conserved charges up to fifth order, establishing superintegrability.

Findings

Constructed the Lax pair for the supersymmetric model.

Derived the complete set of Liouville charges up to fifth order.

Identified additional conserved charges for superintegrability.

Abstract

We analyze the integrability of the ${\cal N}$-extended supersymmetric Calogero-Moser model. We explicitly construct the Lax pair $\{L,A\}$ for this system, which properly reproduces all equations of motion. After adding a supersymmetric oscillator potential we reduce the latter to solving $\dot{U}\,{=}\,A\,U$ for the time evolution operator $U(t)$. The bosonic variables, however, evolve independently of $U$ on closed trajectories, as is required for superintegrability. To visualize the structure of the conserved currents we derive the complete set of Liouville charges up to the fifth power in the momenta, for the ${\cal N}{=}\,2$ supersymmetric model. The additional, non-involutive, conserved charges needed for a maximal superintegrability of this model are also found.