The supersymmetric affine Yangian

The affine Yangian of ${gl}_1$ is known to be isomorphic to $W_{1+\infty }$, the $W$-algebra that characterizes the bosonic higher spin — CFT duality. In this paper we propose some of the defining relations of the Yangian that are relevant for the $N=2$ superconformal version of $W_{1+\infty }$. Our construction is based on the observation that the $N=2$ superconformal $W_{1+\infty }$ algebra contains two commuting bosonic $W_{1+\infty }$ 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 ${gl}_1$ by adding in generators that transform appropriately.