  Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras
acting on tensor product multiplicity spaces for g-representations. These
algebras depend on a parameter which is a point 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. In this paper, we study the
monodromy of these eigenvectors as the parameter varies within the real locus;
this gives an action of the fundamental group of this moduli space, which is
called the cactus group.
  We prove a conjecture of Etingof which states that the monodromy of
eigenvectors for Gaudin algebras agrees with the action of the cactus group on
tensor products of g-crystals. In fact, we prove that the coboundary category
of normal g-crystals can be reconstructed using the coverings of the moduli
spaces.
  Our main tool is the construction of a crystal structure on the set of
eigenvectors for shift of argument algebras, another family of commutative
algebras which act on any irreducible g-representation. We also prove that the
monodromy of such eigenvectors is given by the internal cactus group action on
g-crystals.
