Current presentation for the double super-Yangian $DY(\mathfrak{gl}(m|n))$ and Bethe vectors

TL;DR

This paper constructs Bethe vectors for supersymmetric Yangian models using projections in the Yangian double algebra, providing explicit formulas and demonstrating their equivalence through recursion relations.

Contribution

It introduces a novel method of constructing Bethe vectors for $DY( ext{gl}(m|n))$ using projections onto Borel subalgebras, with two different realizations and proven equivalence.

Findings

Bethe vectors expressed via matrix elements of monodromy matrices

Two isomorphic realizations yield different presentations of Bethe vectors

Recursion relations confirm the equivalence of the constructed Bethe vectors

Abstract

We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians $Y(\mathfrak{gl}(m|n)$ in terms of the current generators of the Yangian double $DY(\mathfrak{gl}(m|n))$. More specifically, we use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian double $DY(\mathfrak{gl}(m|n))$ we obtain two different presentations for the Bethe vectors. These Bethe vectors are also shown to obey some recursion relations which prove their equivalence.