Gluing affine Yangians with bi-fundamentals

Abstract

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

Keywords

• Conformal and W Symmetry
• Higher Spin Symmetry
• Integrable Hierarchies