Superintegrability of Calogero-Moser systems associated with the cyclic quiver

TL;DR

This paper demonstrates the superintegrability of complex Calogero-Moser systems linked to cyclic quivers, including their spin extensions and stability under harmonic potentials, expanding understanding of integrable models.

Contribution

It explicitly constructs first integrals for these systems, showing their superintegrability and connecting them to multicomponent KP hierarchy generalizations.

Findings

Superintegrability proven for systems on cyclic quiver varieties.

Construction of explicit first integrals for these systems.

Superintegrability preserved with added harmonic oscillator potential.

Abstract

We study complex integrable systems on quiver varieties associated with the cyclic quiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero-Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type $A_{n-1}$ and $B_n$, as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added.