Gaudin model and Cactus group

Cactus group is the fundamental group of the real locus of the Deligne- Mumford moduli space of stable rational curves. We define an action of this group on the set of Bethe vectors of the Gaudin magnet chain (for Lie algebra $\mathfrak{sl}(2)$) and relate this to the Berenstein-Kirillov group of piecewise linear transformations of the Gelfand-Tsetlin polytope. Some conjectures generalizing this construction will be discussed.