Sigma models from Gaudin spin chains

Abstract

We solve the classical and quantum problems for the 1D sigma model with target space the flag manifold $\frac{\mathrm{U}(3)}{\mathrm{U}(1{)}^3}$, 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.

43 pages, 4 figures