Gluing affine Yangians with bi-fundamentals

Abstract

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

68 pages, 7 figures; v2: added discussion on relation to the box-antibox construction, published version