Spin Calogero models associated with Riemannian symmetric spaces of negative curvature

TL;DR

This paper explores the Hamiltonian symmetry reduction of geodesic systems on symmetric spaces of negative curvature, resulting in integrable spin Calogero models with hyperbolic interactions, classified for various momentum map values.

Contribution

It introduces a new geometric reduction approach to derive and classify spin Calogero models associated with symmetric spaces of negative curvature, including their quantized versions.

Findings

Reduction yields integrable spin Calogero models with hyperbolic potentials

Classification of special momentum map values leading to spinless models

Models reproduce known rational Calogero models in zero curvature limit

Abstract

The Hamiltonian symmetry reduction of the geodesics system on a symmetric space of negative curvature by the maximal compact subgroup of the isometry group is investigated at an arbitrary value of the momentum map. Restricting to regular elements in the configuration space, the reduction generically yields a spin Calogero model with hyperbolic interaction potentials defined by the root system of the symmetric space. These models come equipped with Lax pairs and many constants of motion, and can be integrated by the projection method. The special values of the momentum map leading to spinless Calogero models are classified under some conditions, explaining why the $BC_n$ models with two independent coupling constants are associated with $SU(n+1,n)/S(U(n+1)\times U(n))$ as found by Olshanetsky and Perelomov. In the zero curvature limit our models reproduce rational spin Calogero models studied previously and similar models correspond to other (affine) symmetric spaces, too. The construction works at the quantized level as well.