# Higher Spins and Yangian Symmetries

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Matthias R. Gaberdiel, Rajesh Gopakumar, Wei Li, Cheng Peng

**16/2/17**Published in : arXiv:1702.05100

The relation between the bosonic higher spin {\cal W}_\infty[\lambda] algebra, the affine Yangian of \mathfrak{gl}_{1}, and the SHc algebra is established in detail. For generic \lambda we find explicit expressions for the low-lying {\cal W}_\infty[\lambda] modes in terms of the affine Yangian generators, and deduce from this the precise identification between \lambda and the parameters of the affine Yangian. Furthermore, for the free field cases corresponding to \lambda=0 and \lambda=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.