Crystals and monodromy of Bethe vectors

Abstract

Fix a semisimple Lie algebra $\mathfrak{g}$. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for $\mathfrak{g}$-representations. These algebras depend on a parameter in the Deligne–Mumford moduli space of marked stable genus $0$ curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. We study the monodromy of these eigenvectors as the parameter varies within the real locus, giving an action of the fundamental group of this moduli space called the cactus group. We prove that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of $\mathfrak{g}$-crystals (as conjectured by Etingof), and we prove that the coboundary category of normal $\mathfrak{g}$-crystals can be reconstructed using the coverings of the moduli spaces. To prove the conjecture, we construct a crystal structure on the set of eigenvectors for the shift of argument algebras, another family of commutative algebras acting on any irreducible $\mathfrak{g}$-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on $\mathfrak{g}$-crystals.