Twisted spin Sutherland models from quantum Hamiltonian reduction

TL;DR

This paper applies Hamiltonian reduction techniques to derive and analyze twisted spin Sutherland models, a class of integrable many-body systems with spin, expanding the understanding of their spectra and symmetries.

Contribution

It introduces a systematic approach to construct twisted spin Sutherland models from quantum Hamiltonian reduction using twisted conjugations and describes their spectral properties.

Findings

Derived explicit spectra for models related to symmetric tensor powers of SU(N).

Established the connection between twisted conjugations and integrable Sutherland models.

Provided detailed descriptions of reduced systems for arbitrary irreducible representations.

Abstract

Recent general results on Hamiltonian reductions under polar group actions are applied to study some reductions of the free particle governed by the Laplace-Beltrami operator of a compact, connected, simple Lie group. The reduced systems associated with arbitrary finite dimensional irreducible representations of the group by using the symmetry induced by twisted conjugations are described in detail. These systems generically yield integrable Sutherland type many-body models with spin, which are called twisted spin Sutherland models if the underlying twisted conjugations are built on non-trivial Dynkin diagram automorphisms. The spectra of these models can be calculated, in principle, by solving certain Clebsch-Gordan problems, and the result is presented for the models associated with the symmetric tensorial powers of the defining representation of SU(N).