Abstract
The relation between the bosonic higher spin {mathcal{W}}_{infty}left[lambda right] algebra, the affine Yangian of mathfrak{g}{mathfrak{l}}_1 , and the SHc algebra is established in detail. For generic λ we find explicit expressions for the low-lying {mathcal{W}}_{infty}left[lambda right] 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 {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 SHc 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.
Highlights
The tensionless point of string theory on AdS is dual to the free limit of the dual field theory, at which the integrable structure of the field theory should have an explicit realization
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
Yangian algebras appear naturally in spin-chain models and are a hallmark of integrability. We shall explore this question for the original bosonic version of the higher spin — CFT duality [9]. Some time ago it was noted [10, 11] that the affine Yangian of gl1 is isomorphic to W1+∞[λ], the asymptotic symmetry algebra of the bosonic higher spin theory on AdS3.1 This isomorphism arises as the rational limit of the equivalence between the quantum-deformed W1+∞ algebra and the quantum toroidal algebra of gl1 [10], generalizing the construction of [13] to the toroidal case
Summary
(Note that in eq (2.1) the generators of the algebra are all associated to singular terms of the spectral parameter.) In addition we have the identity. There is a very natural class of representations of the affine Yangian of gl which are of interest in the connection to the W∞ algebra.2 These representations are best viewed in terms of plane partitions, i.e. three-dimensional box stacking configurations.. The highest weight state of a representation (labelled by three Young tableaux) is the plane partition configuration with the minimum number of boxes consistent with the specified asymptotics (the blue boxes in figure 1). The other states in the representation are obtained by the repeated action of the Yangian generators on this representation (given by the yellow boxes in figure 1) This action is given in terms of adding/removing boxes from a given valid stacking configuration. The resulting generators are compatible with eqs. (3.3) and (2.30)
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