On Bethe vectors in the slN+1 Gaudin model

This paper deals with the Gaudin model associated with the tensor product of n irreducible finite-dimensional slN+1-modules marked by distinct complex numbers z1, …, zn. The Bethe Ansatz is a method to construct common eigenvectors of the Gaudin Hamiltonians by means of chosen singular vectors in the factors and zj's. These vectors are called Bethe vectors. The question if the Bethe vectors are nonzero vectors is open. By the moment, the only way to verify that was based on a relation to critical points of the master function of the Gaudin model, and nontriviality of a Bethe vector was proved only in the case when the corresponding critical point is nondegenerate. However degenerate critical points do appear in the Gaudin model. We believe that the Bethe vectors never vanish, and suggest an approach that does not depend on nondegeneracy of the corresponding critical point. The idea is for a Bethe vector to choose a suitable subspace in the weight space and to check that the projection of the Bethe vector to this subspace is nonzero. We apply this approach to verify nontriviality of Bethe vectors in new examples.