Sigma models from Gaudin spin chains

TL;DR

This paper solves classical and quantum problems for a 1D sigma model with a flag manifold target space, using a mapping to Gaudin models to find geodesics and spectra explicitly.

Contribution

It introduces a novel mapping between the sigma model on the flag manifold and Gaudin spin chains, enabling explicit solutions for geodesics and spectra.

Findings

Explicit description of geodesics via elliptic functions

Spectrum of Laplace-Beltrami operator obtained from polynomial equations

Mapping applicable to general U(n) cases

Abstract

We solve the classical and quantum problems for the 1D sigma model with target space the flag manifold $\mathrm{U}(3)\over \mathrm{U}(1)^3$, equipped with the most general invariant metric. In particular, we explicitly describe all geodesics in terms of elliptic functions and demonstrate that the spectrum of the Laplace-Beltrami operator may be found by solving polynomial (Bethe) equations. The main technical tool that we use is a mapping between the sigma model and a Gaudin model, which is also shown to hold in the $\mathrm{U}(n)$ case.