The supersymmetric affine Yangian

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.