Higher Spin Superalgebras in any Dimension and their Representations

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 $Ad{S}_d$, the tensor product of boson and fermion singletons gives a sum of all half-integer spin symmetric massless representations in $Ad{S}_d$, and the tensor product of two fermion singletons in $d>4$ gives rise to massless fields of mixed symmetry types in $Ad{S}_d$ 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 $Ad{S}_d$ \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 $Ad{S}_3$, $Ad{S}_4$, and $Ad{S}_5$ the isomorphisms with the higher spin superalgebras defined earlier in terms of spinor generating elements are established.

LaTeX, 53 pages, no figures; v2: 54 pages, notations for higher spin algeras are simplified, a comment on tensor products of 2d massless fields and references added, typos corrected; v3: references added, typos corrected; v4: references added, notation simplified, clarification on finite dimensional subsuperalgebras of higher spin superalgebras added in section 5. The version published in JHEP

References in corpus (22)

• Nonlinear Equations for Symmetric Massless Higher Spin Fields in $(A)d{S}_d$
• On higher spins and the tensionless limit of String Theory
• On stringy AdS_5 x S^5 and higher spin holography
• Holography of the N=1 Higher-Spin Theory on AdS4
• On geometric equations and duality for free higher spins
• On the Tensionless Limit of Bosonic Strings, Infinite Symmetries and Higher Spins
• From Free Fields to $AdS$
• Large-N Collective Fields and Holography
• On the Frame-Like Formulation of Mixed-Symmetry Massless Fields in (A)dS(d)
• SL(2,Z) Action on Three-Dimensional CFTs and Holography
• Evaluating the AdS dual of the critical O(N) vector model
• Tensionless Strings, WZW Models at Critical Level and Massless Higher Spin Fields
• On the covariant quantization of tensionless bosonic strings in AdS spacetime
• Geometric Second Order Field Equations for General Tensor Gauge Fields
• Confinement/Deconfinement Transition of Large N Gauge Theories with N_f Fundamentals: N_f/N Finite
• Two-column higher spin massless fields in AdS(d)
• Massive gauge-invariant field theories on spaces of constant curvature
• Construction of bosonic string theory on infinitely curved Anti-de Sitter space
• Conformal Coupling of Higher Spin Gauge Fields to a Scalar Field in $Ad{S}_4$ and Generalized Weyl Invariance
• Gauged Vector Models and Higher-Spin Representations in AdS_5
• The minimal conformal O(N) vector sigma model at d=3
• The Holographic Dual of a SUSY Vector Model and Tensionless Open Strings

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