The monodromy of real Bethe vectors for the Gaudin model

Abstract

The Bethe algebras for the Gaudin model act on the multiplicity space of tensor products of irreducible $\mathfrak{gl}_r$-modules and have simple spectrum over real points. This fact is proved by Mukhin, Tarasov and Varchenko who also develop a relationship to Schubert intersections over real points. We use an extension to $\overline{M}_{0,n+1}(\mathbb{R})$ of these Schubert intersections, constructed by Speyer, to calculate the monodromy of the spectrum of the Bethe algebras. We show this monodromy is described by the action of the cactus group $J_n$ on tensor products of irreducible $\mathfrak{gl}_r$-crystals.