The supersymmetric affine Yangian

Abstract

The affine Yangian of ${\mathfrak{gl}}_1$ is known to be isomorphic to ${\mathcal{W}}_{1+\infty }$, 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 $\mathcal{N}=2$ superconformal version of ${\mathcal{W}}_{1+\infty }$. Our construction is based on the observation that the $\mathcal{N}=2$ superconformal ${\mathcal{W}}_{1+\infty }$ algebra contains two commuting bosonic ${\mathcal{W}}_{1+\infty }$ 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.

35 pages, 5 figures