The supersymmetric affine Yangian

TL;DR

This paper introduces a supersymmetric affine Yangian structure related to the ${ m extbf{N}=2}$ superconformal ${ m extbf{W}}_{1+ extbf{ extit{ extbf{ extbf{infty}}}}}$ algebra, extending the known bosonic case and highlighting its relation to two commuting ${ m extbf{W}}_{1+ extbf{ extit{ extbf{ extbf{infty}}}}}$ algebras.

Contribution

It defines the relations of a supersymmetric affine Yangian that corresponds to the ${ m extbf{N}=2}$ superconformal ${ m extbf{W}}_{1+ extbf{ extit{ extbf{ extbf{infty}}}}}$ algebra, built from two affine Yangians of $rak{gl}_1$.

Findings

The ${ m extbf{N}=2}$ superconformal ${ m extbf{W}}_{1+ extbf{ extit{ extbf{ extbf{infty}}}}}$ algebra contains two commuting bosonic ${ m extbf{W}}_{1+ extbf{ extit{ extbf{ extbf{infty}}}}}$ algebras.

The additional generators in the supersymmetric case transform in bi-minimal representations.

The affine Yangian for the ${ m extbf{N}=2}$ case can be constructed from two affine Yangians of $rak{gl}_1$ with added generators.

Abstract

The affine Yangian of $\mathfrak{gl}_1$ is known to be isomorphic to ${\cal W}_{1+\infty}$, the $W$-algebra that characterizes the bosonic higher spin -- CFT duality. In this paper we propose defining relations of the Yangian that are relevant for the ${\cal N}=2$ superconformal version of ${\cal W}_{1+\infty}$. Our construction is based on the observation that the ${\cal N}=2$ superconformal ${\cal W}_{1+\infty}$ algebra contains two commuting bosonic ${\cal W}_{1+\infty}$ algebras, and that the additional generators transform in bi-minimal representations with respect to these two algebras. The corresponding affine Yangian can therefore be built up from two affine Yangians of $\mathfrak{gl}_1$ by adding in generators that transform appropriately.