Higher Spins and Yangian Symmetries

Abstract

The relation between the bosonic higher spin ${\cal W}_\infty[λ]$ algebra, the affine Yangian of $\mathfrak{gl}_{1}$, and the SH$^c$ algebra is established in detail. For generic $λ$ we find explicit expressions for the low-lying ${\cal W}_\infty[λ]$ modes in terms of the affine Yangian generators, and deduce from this the precise identification between $λ$ and the parameters of the affine Yangian. Furthermore, for the free field cases corresponding to $λ=0$ and $λ=1$ we give closed-form expressions for the affine Yangian generators in terms of the free fields. Interestingly, the relation between the ${\cal W}_\infty$ modes and those of the affine Yangian is a non-local one, in general. We also establish the explicit dictionary between the affine Yangian and the SH$^c$ generators. Given that Yangian algebras are the hallmark of integrability, these identifications should pave the way towards uncovering the relation between the integrable and the higher spin symmetries.

31 pages, 1 figure