Abstract

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 $ρ(Δ) \lesssim \exp\left(\frac{2πΔ}{d-1}\right)$ for CFT$_d$. 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 $J_i$ and $U(1)$ charge $Q$, obtaining bounds on $ρ(Δ, J_i, Q)$ in $d=3$ through $6$. All bounds are saturated by black holes at the Hawking-Page transition and vanish beyond the corresponding BPS bound.