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


Taronna,  Massimo
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Boulanger, N., Ponomarev, D., Skvortsov, E. D., & Taronna, M. (2013). On the uniqueness of higher-spin symmetries in AdS and CFT. International Journal of Modern Physics A, 28(31): 1350162. doi:10.1142/S0217751X13501625.

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, 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.