Sparseness bounds on local operators in holographic $CFT_d$

Understand

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 $\rho(\Delta) \lesssim \exp\left(\frac{2\pi\Delta}{d-1}\right)$ 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 $\rho(\Delta, J_i, Q)$ in $d=3$ through $6$.