# Gluing two affine Yangians of $\mathfrak{gl}_1$

## TL;DR

This paper constructs a new family of affine Yangian algebras by gluing two affine Yangians of , generalizing the  algebra and relating it to twin plane partitions and toric Calabi-Yau geometries.

## Contribution

It introduces a four-parameter family of affine Yangian algebras via a novel gluing construction, extending the  algebra and providing natural representations.

## Key findings

- Construction of a four-parameter family of affine Yangian algebras.
- Representation of these algebras via twin plane partitions.
- Geometric similarities to toric Calabi-Yau threefolds.

## Abstract

We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of $\mathfrak{gl}_1$. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the $\mathcal{N}=2$ affine Yangian, which is isomorphic to the universal enveloping algebra of $\mathfrak{u}(1)\oplus \mathcal{W}^{\mathcal{N}=2}_{\infty}[\lambda]$. All algebras that we construct have natural representations in terms of "twin plane partitions", a pair of plane partitions appropriately joined along one common leg. We observe that the geometry of twin plane partitions, which determines the algebra, bears striking similarities to the geometry of certain toric Calabi-Yau threefolds.