Quantum magnet chains and Kashiwara crystals

The talk is based on the joint paper with Iva Halacheva, Joel Kamnitzer, and Alex Weekes https://arxiv.org/abs/1708.05105 . Solutions of the algebraic Bethe ansatz for quantum magnet chains are, generally, multivalued functions of the parameters of the integrable system. I will explain how to compute some monodromies of solutions of Bethe ansatz for the Gaudin magnet chain assigned to a semisimple finite-dimensional Lie algebra g in terms of Kashiwara crystals (which are combinatorial objects modeling finite-dimensional representations of g). Namely, the Bethe eigenvectors in the Gaudin model can be regarded as a covering of the Deligne-Mumford moduli space of stable rational curves, which is unramified over the real locus of the Deligne-Mumford space. The monodromy action of the fundamental group of the real Deligne-Mumford space (called cactus group) on the eigenvectors is naturally equivalent to the action of the same group by commutors (i.e. combinatorial analog of a braiding) on a tensor product of Kashiwara crystals.
If time allows, I will also discuss the generalization of this monodromy theorem to the XXX Heisenberg magnet chain which involves Kirillov-Reshetikhin crystals.