Generalized spin Sutherland systems revisited

TL;DR

This paper generalizes spin Sutherland systems by exploring their derivation from Yang--Mills theory and geodesic motion, revealing their connection to Lie group automorphisms and integrability.

Contribution

It introduces new generalized spin Sutherland systems derived via two methods, linking them to Dynkin diagram automorphisms and providing both finite- and infinite-dimensional perspectives.

Findings

Systems correspond to Dynkin diagram automorphisms.

Both finite- and infinite-dimensional derivations lead to the same integrable models.

Infinite-dimensional approach uses twisted current algebras.

Abstract

We present generalizations of the spin Sutherland systems obtained earlier by Blom and Langmann and by Polychronakos in two different ways: from SU(n) Yang--Mills theory on the cylinder and by constraining geodesic motion on the N-fold direct product of SU(n) with itself, for any N>1. Our systems are in correspondence with the Dynkin diagram automorphisms of arbitrary connected and simply connected compact simple Lie groups. We give a finite-dimensional as well as an infinite-dimensional derivation and shed light on the mechanism whereby they lead to the same classical integrable systems. The infinite-dimensional approach, based on twisted current algebras (alias Yang--Mills with twisted boundary conditions), was inspired by the derivation of the spinless Sutherland model due to Gorsky and Nekrasov. The finite-dimensional method relies on Hamiltonian reduction under twisted conjugations of N-fold direct product groups, linking the quantum mechanics of the reduced systems to representation theory similarly as was explored previously in the N=1 case.