Gaudin models associated to classical Lie algebras

Abstract

We study the Gaudin model associated to Lie algebras of classical types.

First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.

Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian.