On the uniqueness of higher-spin symmetries in AdS and CFT

TL;DR

The paper investigates how many higher-spin algebras exist in $AdS_d$ and how many $CFT_{d-1}$ theories with exactly conserved higher-spin tensors can occur. It translates the problem into enforcing the Jacobi identity for the HS gauge algebra (and the dual charge algebra in the CFT), effectively a deformation-quantization problem. The main results are that the Eastwood–Vasiliev algebra is the unique HS algebra for $d=4$ and $d>6$, while a one-parameter family appears in $d=5$; the $d=6$ case is left out due to technical issues. Importantly, introducing a single HS field forces the entire infinite tower, and among admissible non-Abelian cubic vertices in $AdS_d$ only one survives Jacobi consistency, corresponding to the germ of Eastwood–Vasiliev's algebra. These findings constrain HS theories and reinforce the tight link between AdS HS symmetries and dual CFT Ward identities.

Abstract

We study the uniqueness of higher-spin algebras which are at the core of higher-spin theories in AdS and of CFTs with exact higher-spin symmetry, i.e. conserved tensors of rank greater than two. The Jacobi identity for the gauge algebra is the simplest consistency test that appears at the quartic order for a gauge theory. Similarly, the algebra of charges in a CFT must also obey the Jacobi identity. These algebras are essentially the same. Solving the Jacobi identity under some simplifying assumptions spelled out, we obtain that the Eastwood-Vasiliev algebra is the unique solution for d=4 and d>6. In 5d there is a one-parameter family of algebras that was known before. In particular, we show that the introduction of a single higher-spin gauge field/current automatically requires the infinite tower of higher-spin gauge fields/currents. The result implies that from all the admissible non-Abelian cubic vertices in AdS(d), that have been recently classified for totally symmetric higher-spin gauge fields, only one vertex can pass the Jacobi consistency test. This cubic vertex is associated with a gauge deformation that is the germ of the Eastwood-Vasiliev's higher-spin algebra.