Cactus group and monodromy of Bethe vectors

TL;DR

This paper explores the action of the cactus group on Bethe vectors in Gaudin models, establishing a conjectural isomorphism with crystal tensor product actions and proving it for f1=f1f1_2, linking algebraic and geometric structures.

Contribution

It defines and investigates the cactus group action on Bethe vectors, conjectures its isomorphism with crystal actions, and proves this for f1=f1f1_2, connecting monodromy, Galois groups, and crystal bases.

Findings

Cactus group acts on Bethe vectors of Gaudin models.

Conjecture: this action is isomorphic to crystal tensor product action.

Proven case: f1=f1f1_2, linking monodromy and crystal bases.

Abstract

Cactus group is the fundamental group of the real locus of the Deligne-Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to arbitrary semisimple Lie algebra $\mathfrak{g}$. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the tensor product of crystals coming from the general coboundary category formalism. We prove this conjecture in the case $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_2$ (in fact, for this case the conjecture almost immediately follows from the results of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We also present some conjectures generalizing this result to Bethe vectors of shift of argument subalgebras and relating the cactus group with the Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand-Tsetlin polytope.