W $$ \mathcal{W} $$ -symmetry, topological vertex and affine Yangian

Abstract

Abstract:We discuss the representation theory of the non-linear chiral algebra W 1+∞ of Gaberdiel and Gopakumar and its connection to the Yangian of u(1) whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of W 1+∞ are given by the topological vertex. The Yangian picture provides an infinite number of commuting charges which can be explicitly diagonalized in W 1+∞ highest weight representations. Many properties that are difficult to study in the W 1+∞ picture turn out to have a simple combinatorial interpretation, once translated to the Yangian picture.

Higher spins and Yangian symmetries

“…For example, multiplying the first equation (2.6) by the denominator of (2.4) we obtain σ 3 e(z), e(w) = (z − w) 3 + σ 2 (z − w) e(z), e(w) , (2.12) which upon expanding out in terms of modes leads to the relation The other cases work similarly, and we find in addition the identities (see also [11])…”

“…This allows us to determine the two structure constants that characterize the W ∞ [λ] algebra [20] -the central charge c and the OPE coefficient C 4 33 describing the coupling of two spin-3 fields to the spin-4 field -and thereby check the proposed dictionary. We also establish this identification for the two free field cases, the free fermion case corresponding to λ = 0 (which was already analysed in [11]) as well as the the free boson construction leading to the algebra with λ = 1.…”

“…More specifically, 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 gl 1 (as defined in [10,12]) is isomorphic to W 1+∞ [λ], the asymptotic symmetry algebra of the bosonic higher spin theory on AdS 3 . 1 This isomorphism arises as the rational limit of the equivalence between the quantum-deformed W 1+∞ algebra and the quantum toroidal algebra of gl 1 [10], generalizing the construction of [13] to the toroidal case.…”

“…In [11] Prochazka proposed a concrete dictionary for how the parameters of the affine Yangian of gl 1 and W 1+∞ [λ] are related to each other, and gave circumstantial evidence for this by comparing the structure of some representations. In this paper we confirm his claim independently by constructing the low-lying W ∞ [λ] generators explicitly in terms of the affine Yangian generators.…”

“…three-dimensional box stacking configurations. 3 They are special in that they have a finite number of states at every level and thus possess infinitely many null states; they are discussed in some detail in [11,25], and we only summarize the salient aspects which we will draw upon later. A generic plane partition representation is labelled by three Young tableaux.…”