The monodromy of real Bethe vectors for the Gaudin model

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.