On Higher Spin Symmetries in AdS_{5}

TL;DR

This work addresses the problem of formulating interacting higher-spin gauge theories in AdS$_5$ by embedding the $SU(4)$ gravitational symmetry into a larger $SU(10)$ algebra to unify spin-2 and spin-3 symmetries. It provides an explicit embedding where the $SU(10)$ adjoint decomposes into a spin-2 ($15$) and a spin-3 ($84$) sector, with fields organized as symmetric pairs of $SU(4)$ indices, allowing extraction of linear and quadratic curvatures around an $AdS_5$ background. A cubic, Vasiliev-inspired action for spin-2 and spin-3 fields is constructed using this unified algebra, with a specific coefficient ensuring on-shell consistency, and the work outlines a generalization to higher spins via the sequence $SU({s+2\choose 3})$. The results illuminate the structure and potential algebras governing finite sets of symmetric higher-spin fields in five dimensions and highlight challenges in achieving fully nonlinear, finite-spin actions. The paper also connects to broader themes in higher-spin gravity, including MacDowell–Mansouri–type formulations and the quest for consistent interactions in AdS backgrounds.

Abstract

A special embedding of the SU(4) algebra in SU(10), including both spin two and spin three symmetry generators, is constructed. A possible five dimensional action for massless spin two and three fields with cubic interaction is constructed. The connection with the previously investigated higher spin theories in $AdS_{5}$ background is discussed. Generalization to the more general case of symmetries, including spins $2,3,\dots s$, is shown.