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

TL;DR

This paper proves the uniqueness of the Eastwood-Vasiliev higher-spin algebra in certain dimensions by analyzing the Jacobi identity, showing that only one consistent algebra exists for higher-spin symmetries in AdS and CFT.

Contribution

It demonstrates that the Eastwood-Vasiliev algebra is uniquely consistent for higher-spin symmetries in specific dimensions, with a classification of possible algebras and their relation to cubic vertices.

Findings

Eastwood-Vasiliev algebra is unique in d=4 and d>6.

In 5d, a one-parameter family of algebras exists.

Only one cubic vertex passes the Jacobi consistency test.

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.