7D Bosonic Higher Spin Theory: Symmetry Algebra and Linearized Constraints

TL;DR

The paper constructs the minimal bosonic higher-spin extension of the AdS$_7$ algebra $SO(6,2)$, denoted $hs(8^*)$, via Grassmann-even spinor oscillators and imposes irreducibility by modding out a trace-containing ideal with an internal $SU(2)_K$; gauging yields a spectrum of massless fields with spins $s=0,2,4,...$ arising from the symmetric product of two 6D scalar doubletons. A master one-form $A$ and a master zero-form $\,oldsymbol{ ilde{ abla}}$ in a quasi-adjoint representation form the bulk degrees of freedom, with linearized curvature constraints around AdS$_7$ determining the spectrum and masses, notably $m^2=-8-2s$ and lowest energy $E_0=s+4$ for spin $s$. The independent fields are encoded in $oldsymbol{ ilde{ abla}}^{(s,0;0)}$ corresponding to Weyl tensors and their derivatives, while the scalar sector acquires a $K^2$-dressing yielding the correct AdS masses. The resulting HS theory provides a concrete minimal bosonic truncation of M-theory on $AdS_7 imes S^4$ in an unbroken phase, dual to $N$ free $(2,0)$ tensor multiplets at large $N$, and points to rich avenues for SUSY extensions, interactions, and inclusion of massive HS multiplets within a holographic framework.

Abstract

We construct the minimal bosonic higher spin extension of the 7D AdS algebra SO(6,2), which we call hs(8*). The generators, which have spin s=1,3,5,..., are realized as monomials in Grassmann even spinor oscillators. Irreducibility, in the form of tracelessness, is achieved by modding out an infinite dimensional ideal containing the traces. In this a key role is played by the tree bilinear traces which form an SU(2)_K algebra. We show that gauging of hs(8*) yields a spectrum of physical fields with spin s=0,2,4,...which make up a UIR of hs(8*) isomorphic to the symmetric tensor product of two 6D scalar doubletons. The scalar doubleton is the unique SU(2)_K invariant 6D doubleton. The spin s\geq 2 sector comes from an hs(8*)-valued one-form which also contains the auxiliary gauge fields required for writing the curvature constraints in covariant form. The physical spin s=0 field arises in a separate zero-form in a `quasi-adjoint' representation of hs(8*). This zero-form also contains the spin s\geq 2 Weyl tensors, i.e. the curvatures which are non-vanishing on-shell. We suggest that the hs(8*) gauge theory describes the minimal bosonic, massless truncation of M theory on AdS_7\times S^4 in an unbroken phase where the holographic dual is given by N free (2,0) tensor multiplets for large N.