Scalar fields and three-point functions in D = 3 higher spin gravity

Abstract

We compute boundary three-point functions involving two scalars and a gauge field of arbitrary spin in the AdS vacuum of Vasiliev's higher spin gravity, for any deformation parameter λ. In the process, we develop tools for extracting scalar field equations in arbitrary higher spin backgrounds. We work in the context of hs[λ]⊕hs[λ] Chern-Simons theory coupled to scalar fields, and make efficient use of the associative lone-star product underlying the hs[λ] algebra. Our results for the correlators precisely match expectations from CFT; in particular they match those of any CFT with W ∞ [λ] symmetry at large central charge, and with primary operators dual to the scalar fields. As this is expected to include the 't Hooft limit of the W N minimal model CFT, our results serve as further evidence of the conjectured AdS/CFT duality between these two theories.

On locality, holography and unfolding

“…attention beyond the free theory regime, with the exception of a handful of works [20][21][22][23][24][25][26][27]. A better understanding of the bulk side of the correspondence is however desirable in order to achieve a better grasp of Vasiliev's equations and also of the proposed dualities.…”

“…Here we use the notation: 25) and 26) for the corresponding components of the current according to the grading induced by …”

Supersymmetric holography on AdS3

“…(2.23) that is relevant for us is now 35) where sdet denotes the superdeterminant, and U(h) is an GL(∞|∞) + diagonal matrix with matrix elements…”

“…A concrete proposal was then made in [29], relating the large N 't Hooft like limit of the W N level k minimal models to a family of bosonic higher spin theories on AdS 3 . By now quite some evidence has been found in favour of this proposal [30,31,32,33,34,35,36]. The proposal is the natural analogue of the Klebanov-Polyakov duality since, for vanishing 't Hooft coupling, the CFT can be described as the singlet sector of a free theory [37].…”

Matching four-point functions in higher spin AdS3/CFT2

“…First, the fact that the symmetry algebras of the bulk and boundary theories agree [6,5,7] guarantees that correlators involving just the higher spin currents will match, since these are fixed by the higher spin symmetry. Second, a class of threepoint functions, OOJ (s) , has been shown to match [13,14], where O denotes a primary operator dual to a perturbative scalar field in the bulk, and J (s) is the spin-s current. Of course, conformal invariance fixes the functional form of three-point functions, and so this match is just a check of the overall normalization.…”