Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces

TL;DR

This paper explores solutions to Bethe ansatz equations for orthosymplectic Lie superalgebras, introducing a reproduction procedure to generate solution families and linking them to self-dual superspaces and isotropic superflags.

Contribution

It introduces a reproduction procedure for Bethe ansatz solutions and establishes a correspondence with self-dual superspaces and isotropic superflags, extending understanding of these algebraic structures.

Findings

Defined a reproduction procedure for solutions.

Constructed a family of solutions called a population.

Linked populations to self-dual superspaces and superflags.

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 $\mathfrak{osp}_{2m|0}=\mathfrak{so}_{2m}$.