Gluing affine Yangians with bi-fundamentals

The affine Yangian of ${\mathfrak{g}\mathfrak{l}}_1$ is isomorphic to the universal enveloping algebra of ${\mathcal{W}}_{1+\mathrm{\infty }}$ and can serve as a building block in the construction of new vertex operator algebras. In [1], a two-parameter family generalization of $\mathcal{N}=2$ supersymmetric ${\mathcal{W}}_{\mathrm{\infty }}$ algebra was constructed by “gluing” two affine Yangians of ${\mathfrak{g}\mathfrak{l}}_1$ using operators that transform as $(\square, \overline{\square})$and$(\overline{\square}, \square)$ w.r.t. the two affine Yangians. In this paper we realize a similar (but non-isomorphic) two-parameter gluing construction where the gluing operators transform as $(\square, \square)$and$(\overline{\square}, \overline{\square})$ w.r.t. the two affine Yangians. The corresponding representation space consists of pairs of plane partitions connected by a common leg whose cross-section takes the shape of Young diagrams, offering a more transparent geometric picture than the previous construction.