Pure anti-de Sitter supergravity and the conformal bootstrap

We consider graviton scattering in maximal supergravity on Anti-de Sitter space (AdS) in $d+1$ dimensions for $d=3,4,\text{and $6$}$ with no extra compact spacetime factor. Holography suggests that this theory is dual to an exotic maximally supersymmetric conformal field theory (CFT) in $d$ dimensions whose only light single trace operator is the stress tensor. This contrasts with more standard cases like Type IIB string theory on $AdS_5\times S^5$ dual to $\mathcal{N}=4$ Super-Yang-Mills, where the CFT has light single trace operators for each Kaluza-Klein mode on $S^5$. We compute the 1-loop correction to the pure AdS$_{d+1}$ theory in a small Planck length expansion, which is dual to the large central charge expansion in the CFT. We find that this correction saturates the most general non-perturbative conformal bootstrap bounds on this correlator in the large central charge regime for $d=3,4,6$, while the 1-loop correction to CFTs with string/M-theory duals all lie inside the allowed region.