Higher Spin Superalgebras in any Dimension and their Representations

TL;DR

The work establishes a universal oscillator-based framework for higher spin algebras in arbitrary dimensions, showing that singleton representations of $o(d-1,2)$ underlie HS gauge theories in $AdS_d$ and that tensor products of these singletons reproduce the full spectrum of bulk massless fields, thereby generalizing the Flato–Fronsdal theorem beyond $d=4$. It constructs explicit HS algebras and their supersymmetric extensions using two-boson and spinor-extended oscillator realizations, together with projection methods to isolate the physical spectrum and ensure admissibility. The analysis introduces unfolded formulations for conformal fields, clarifies the AdS/CFT interpretation, and demonstrates isomorphisms with well-known lower-dimensional HS structures in $AdS_3$, $AdS_4$, and $AdS_5$. Overall, the paper provides a comprehensive group-theoretical basis for consistent HS gauge theories in any dimension, including a supersymmetric extension that acts on boundary conformal scalars and spinors as well as on bulk HS fields.

Abstract

Fock module realization for the unitary singleton representations of the $d-1$ dimensional conformal algebra $o(d-1,2)$, which correspond to the spaces of one-particle states of massless scalar and spinor in $d-1$ dimensions, is given. The pattern of the tensor product of a pair of singletons is analyzed in any dimension. It is shown that for $d>3$ the tensor product of two boson singletons decomposes into a sum of all integer spin totally symmetric massless representations in $AdS_d$, the tensor product of boson and fermion singletons gives a sum of all half-integer spin symmetric massless representations in $AdS_d$, and the tensor product of two fermion singletons in $d>4$ gives rise to massless fields of mixed symmetry types in $AdS_d$ depicted by Young tableaux with one row and one column together with certain totally antisymmetric massive fields. In the special case of $o(2,2)$, tensor products of 2d massless scalar and/or spinor modules contain infinite sets of 2d massless conformal fields of different spins. The obtained results extend the 4d result of Flato and Fronsdal \cite{FF} to any dimension and provide a nontrivial consistency check for the recently proposed higher spin model in $AdS_d$ \cite{d}. We define a class of higher spin superalgebras which act on the supersingleton and higher spin states in any dimension. For the cases of $AdS_3$, $AdS_4$, and $AdS_5$ the isomorphisms with the higher spin superalgebras defined earlier in terms of spinor generating elements are established.