Gluing affine Yangians with bi-fundamentals

Abstract The affine Yangian of $$ {\mathfrak{gl}}_1 $$ gl 1 is isomorphic to the universal enveloping algebra of $$ {\mathcal{W}}_{1+\infty } $$ W 1 + ∞ and can serve as a building block in the construction of new vertex operator algebras. In [1], a two-parameter family generalization of $$ \mathcal{N} $$ N = 2 supersymmetric $$ {\mathcal{W}}_{\infty } $$ W ∞ algebra was constructed by “gluing” two affine Yangians of $$ {\mathfrak{gl}}_1 $$ gl 1 using operators that transform as (□, $$ \overline{\square} $$ □ ¯ ) and ( $$ \overline{\square} $$ □ ¯ , □) 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 (□, □) 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.

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