Gluing affine Yangians with bi-fundamentals

TL;DR

This paper introduces a new gluing construction of affine Yangians using operators transforming as symmetric or antisymmetric pairs, providing a clearer geometric interpretation through pairs of plane partitions connected by Young diagram-shaped cross-sections.

Contribution

It presents a novel two-parameter gluing method of affine Yangians with symmetric and antisymmetric operators, expanding the algebraic and geometric understanding of these structures.

Findings

Constructed a non-isomorphic two-parameter gluing of affine Yangians.

Provided a geometric interpretation via pairs of plane partitions connected by Young diagrams.

Enhanced the understanding of the algebraic structure with a more transparent geometric picture.

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.