Higher spins and Yangian symmetries

The relation between the bosonic higher spin W_∞[λ] algebra, the affine Yangian of gl₁, and the SH^c algebra is established in detail. For generic λ we find explicit expressions for the low-lying W_∞[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 W_∞ 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.