# 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.

## Key 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.

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Source: https://tomesphere.com/paper/2508.20889