Sparseness bounds on local operators in holographic CFT d

We use the thermodynamics of anti-de Sitter gravity to derive sparseness bounds on the spectrum of local operators in holographic conformal field theories. The simplest such bound is $$ \rho \left(\varDelta \right)\lesssim \exp \left(\frac{2\pi \Delta}{d-1}\right) $$ for CFTd. Unlike the case of d = 2, this bound is strong enough to rule out weakly coupled holographic theories. We generalize the bound to include spins Ji and U(1) charge Q, obtaining bounds on ρ(Δ, Ji, Q) in d = 3 through 6. All bounds are saturated by black holes at the Hawking-Page transition and vanish beyond the corresponding BPS bound.