Spectral curves for the triple reduced product of coadjoint orbits for SU(3)

TL;DR

This paper explores the geometric and spectral properties of the triple reduced product of coadjoint orbits in SU(3), linking it to Hitchin systems, spectral curves, and Hamiltonian dynamics.

Contribution

It provides a novel identification of the reduced product with Hitchin pairs and derives explicit spectral and Hamiltonian descriptions.

Findings

Identification of the reduced product with Hitchin pairs over a punctured sphere

Spectral curve analysis for the associated Hitchin system

Explicit Darboux coordinates and Hamiltonian differential equation

Abstract

We give an identification of the triple reduced product of three coadjoint orbits in SU(3) with a space of Hitchin pairs over a genus 0 curve with three punctures, where the residues of the Higgs field at the punctures are constrained to lie in fixed coadjoint orbits. Using spectral curves for the corresponding Hitchin system, we identify the moment map for a Hamiltonian circle action on the reduced product. Finally, we make use of results of Adams, Harnad, and Hurtubise to find Darboux coordinates and a differential equation for the Hamiltonian.