Bethe Ansatz Equations for Orthosymplectic Lie Superalgebras and Self-dual Superspaces

Abstract

We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $$\mathfrak {osp}_{2m+1|2n}$$and $$\mathfrak {osp}_{2m|2n}$$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $$\mathcal R$$. Under some technical assumptions, we show that the superkernel W of $$\mathcal R$$is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in W and with the set of symmetric complete factorizations of $$\mathcal R$$. In particular, our results apply to the case of even Lie algebras of type D$${}_m$$corresponding to $$\mathfrak {osp}_{2m|0}=\mathfrak {so}_{2m}$$.