The monodromy of real Bethe vectors for the Gaudin model

TL;DR

This paper investigates the monodromy of real Bethe vectors in the Gaudin model, linking it to the action of the cactus group on tensor products of irreducible modules, extending previous work on spectrum simplicity.

Contribution

It extends the understanding of Bethe algebra monodromy by connecting it to the cactus group action via Schubert intersections on the moduli space.

Findings

Monodromy is described by the cactus group action.

Extended Schubert intersections to real moduli space.

Confirmed simple spectrum over real points.

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.