Twisted spin Sutherland models from quantum Hamiltonian reduction

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).

💡 Research Summary

The paper develops a systematic construction of integrable many‑body systems with internal spin degrees of freedom by applying quantum Hamiltonian reduction to the free particle on a compact, connected, simple Lie group (G). The starting point is the Laplace–Beltrami operator (\Delta_G) on (G), which serves as the kinetic term of a free quantum particle. The authors exploit the theory of polar group actions, which guarantees the existence of a global section (a “polar slice”) intersecting each orbit orthogonally. This structure allows one to introduce coordinates on a Weyl chamber of a maximal torus (T\subset G) and to rewrite (\Delta_G) as a sum of a flat Laplacian on the chamber plus a set of inverse‑square potentials of the form (\sin^{-2}(\alpha!\cdot! q)), where (\alpha) runs over the positive roots and (q) denotes the chamber coordinates.

A crucial novelty is the use of twisted conjugation as the symmetry under which the reduction is performed. For a non‑trivial Dynkin‑diagram automorphism (\sigma) of (G) the twisted action is \