Abstract
We aim at formulating a higher-spin gravity theory around AdS2 relevant for holography. As a first step, we investigate its kinematics by identifying the low-dimensional cousins of the standard higher-dimensional structures in higher-spin gravity such as the singleton, the higher-spin symmetry algebra, the higher-rank gauge and matter fields, etc. In particular, the higher-spin algebra is given here by [λ] and parameterized by a real parameter λ. The singleton is defined to be a Verma module of the AdS2 isometry subalgebra so (2, 1) ⊂ [λ] with conformal weight Delta =frac{1pm lambda }{2}. On the one hand, the spectrum of local modes is determined by the Flato-Fronsdal theorem for the tensor product of two such singletons. It is given by an infinite tower of massive scalar fields in AdS2 with ascending masses expressed in terms of λ. On the other hand, the higher-spin fields arising through the gauging of [λ] algebra do not propagate local degrees of freedom. Our analysis of the spectrum suggests that AdS2 higher-spin gravity is a theory of an infinite collection of massive scalars with fine-tuned masses, interacting with infinitely many topological gauge fields. Finally, we discuss the holographic CFT1 duals of the kinematical structures identified in the bulk.
Highlights
Introduction and summaryThe recent surge of interest in AdS2/CFT1 correspondence prompted by the Sachdev-YeKitaev (SYK) model [1–4] motivates a thorough investigation of lowest-dimensional higher-spin (HS) holographic duality
We investigate its kinematics by identifying the low-dimensional cousins of the standard higher-dimensional structures in higher-spin gravity such as the singleton, the higher-spin symmetry algebra, the higher-rank gauge and matter fields, etc
It is given by an infinite tower of massive scalar fields in AdS2 with ascending masses expressed in terms of λ
Summary
The recent surge of interest in AdS2/CFT1 correspondence prompted by the Sachdev-YeKitaev (SYK) model [1–4] (see e.g. [5, 6] for reviews) motivates a thorough investigation of lowest-dimensional higher-spin (HS) holographic duality. Forms a higher-spin multiplet corresponding to the twisted-adjoint module (where RAdS stands for the curvature radius of the AdS2 spacetime) Applying the standard AdS2/CFT1 dictionary on (1.1), the boundary dual of this tower of scalar fields should be a collection of primary operators On with conformal weight ∆n = n−λ+1. The main result reviewed here is the Flato-Fronsdal type theorem which describes the tensor product of two singletons as an infinite collection of Verma modules with running conformal weights. These results allow to describe the higher-spin algebra as an infinite collection of Verma modules with running conformal dimensions and thereby identify the right-hand-side of the Flato-Fronsdal theorem as the twisted-adjoint representation. Appendix E builds the most general class of so(2, 1) actions on U so(2, 1) realized by inhomogeneous first-order differential operators in auxiliary variables
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