1D sigma models from Gaudin models

In this talk we will consider a one-dimensional sigma model with the target space being an $\mathsf{SU}(3)$ full flag manifold, equipped with an arbitrary invariant metric. 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 equations of a special type. These results are based on the previously discovered connection between sigma models and Gaudin models, which also holds in the $\mathsf{SU}(n)$ case.