Higher Spins and Yangian Symmetries

Abstract

The relation between the bosonic higher spin ${\mathcal{W}}_{\infty }[\lambda ]$ algebra, the affine Yangian of ${\mathfrak{gl}}_1$, and the SH${}^c$ algebra is established in detail. For generic $\lambda$ we find explicit expressions for the low-lying ${\mathcal{W}}_{\infty }[\lambda ]$ 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 ${\mathcal{W}}_{\infty }$ 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

References in corpus (4)

• Higher Spins & Strings
• The ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathrm{S}}^3\times {\mathrm{S}}^1$ worldsheet S matrix
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Twisted sectors from plane partitions

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