Even spin minimal model holography

Abstract

The even spin W^e_\infty algebra that is generated by the stress energy tensor together with one Virasoro primary field for every even spin s \geq 4 is analysed systematically by studying the constraints coming from the Jacobi identities. It is found that the algebra is characterised, in addition to the central charge, by one free parameter that can be identified with the self-coupling constant of the spin 4 field. We show that W^e_\infty can be thought of as the quantisation of the asymptotic symmetry algebra of the even higher spin theory on AdS_3. On the other hand, W^e_\infty is also quantum equivalent to the so(N) coset algebras, and thus our result establishes an important aspect of the even spin minimal model holography conjecture. The quantum equivalence holds actually at finite central charge, and hence opens the way towards understanding the duality beyond the leading 't Hooft limit.Comment: 32 pages, v2: reference added, minor changes in tex

Even spin N = 4 $$ \mathcal{N}=4 $$ holography

“…It is interesting to notice that, in contrast with previous even spin constructions [8,19,20,38,39], the N = 4 version is obtained using a Sp(2N ) rather than a O(N ) model. In the 2d bosonic case [19,20], the bosonic even spin W-algebra was related to different orthogonal and symplectic models [38], which are described at finite coupling by various cosets whose chiral algebra is freely generated.…”

“…The chiral algebra of the coset in the 't Hooft limit, and correspondingly the gauge sector of the dual higher spin theory, is spanned by one superprimary field per even conformal weight, thus extending the previous bosonic even spin construction [19,20,38], the N = 2 case [39], and a recent N = 1 proposal [21]. In the k → ∞ limit, the coset (1.2) is described by an Sp(2N ) vector model.…”

“…In the 2d bosonic case [19,20], the bosonic even spin W-algebra was related to different orthogonal and symplectic models [38], which are described at finite coupling by various cosets whose chiral algebra is freely generated. Even though an analoguous analysis of the quantum N = 4 even spin W-algebra is beyond the scope of this paper, we will briefly comment on this issue.…”

“…for finite N and k (and therefore finite central charge c), the model is not expected to be a mere truncation of the original [40]. In [38] it was found that there are two natural ways in which the free parameter γ of the quantum bosonic even spin W ∞ -algebra can be identified with λ at finite c. These two ways agree in the classical limit c → ∞, and they correspond to two different quantisations of the classical DS reduction of the even spin bosonic algebra hs e [λ]. This was seen as a reflection of the fact that hs e [µ] truncates for µ = N to either sp(N ) if N is even, or so(N )…”

The higher spin rectangle

“…This algebra, its supersymmetric extensions and holographic duals have been explored in [26][27][28][29]. In the other direction, W ∞ [λ], as also W N algebras, can be extended by adding a spin one field: the resulting algebra is known as W 1+∞ [λ].…”