Twin-plane-partitions and $\mathcal{N}=2$ affine Yangian

TL;DR

This paper explores the supersymmetric extension of the affine Yangian related to ${ m W}_{1+ olinebreak ext{infinity}}$, defining its relations and representations via twin-plane-partitions, advancing understanding of supersymmetric algebraic structures.

Contribution

It identifies the full set of relations for the ${ m N}=2$ supersymmetric affine Yangian and constructs its explicit representation on twin-plane-partitions.

Findings

Defined the supersymmetric affine Yangian relations.

Constructed twin-plane-partition representations.

Established the algebra's action explicitly.

Abstract

The universal enveloping algebra of ${\cal W}_{1+\infty}$ is isomorphic to the affine Yangian of $\mathfrak{gl}_1$. We study the ${\cal N}=2$ supersymmetric version of this correspondence, and identify the full set of defining relations of the supersymmetric affine Yangian. These relations can be deduced by demanding that the algebra has a representation on twin-plane-partitions, which we construct by gluing pairs of plane partitions. We define the action of the algebra on these twin-plane-partitions explicitly.