Sparseness bounds on local operators in holographic $CFT_d$

TL;DR

This paper derives bounds on the density of local operators in holographic conformal field theories using anti-de Sitter gravity thermodynamics, showing these bounds are saturated by black holes and exclude weakly coupled theories in certain dimensions.

Contribution

It establishes new sparseness bounds on operator spectra in holographic CFTs, including spins and charges, extending previous results and linking them to black hole thermodynamics.

Findings

Bound $ ho( riangle) \,\lesssim\, \exp(\frac{2\pi\triangle}{d-1})$ for CFT$_d$

Bounds are saturated by black holes at the Hawking-Page transition

Bounds exclude weakly coupled holographic theories in $d=2$

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 $\rho(\Delta) \lesssim \exp\left(\frac{2\pi\Delta}{d-1}\right)$ 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 $6$. All bounds are saturated by black holes at the Hawking-Page transition and vanish beyond the corresponding BPS bound.