Principal $W$-algebras for $\operatorname{GL}(m\vert n)$

Abstract

We consider the (finite) $W$-algebra $W_{m|n}$ attached to the principal nilpotent orbit in the general linear Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{m|n}\left(ℂ\right)$. Our main result gives an explicit description of $W_{m|n}$ as a certain truncation of a shifted version of the Yangian $Y\left({\mathfrak{g}\mathfrak{l}}_{1|1}\right)$. We also show that $W_{m|n}$ admits a triangular decomposition and construct its irreducible representations.