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

The universal enveloping algebra of ${\mathcal{W}}_{1+\mathrm{\infty }}$ is isomorphic to the affine Yangian of ${\mathfrak{g}\mathfrak{l}}_1$. We study the $\mathcal{N}=2$ 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.