A universal inequality for CFT and quantum gravity

Abstract

We prove that every unitary two-dimensional conformal field theory (with no extended chiral algebra, and with c, $\tilde{c} > 1$) contains a primary operator with dimension ∆₁ that satisfies $0 < {\Delta_1} < \frac{{c + \tilde{c}}}{{12}} + 0.473695$. Translated into gravitational language using the AdS₃/CFT₂ dictionary, our result proves rigorously that the lightest massive excitation in any theory of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than ${{1} \left/ {{\left( {4{G_N}} \right)}} \right.} + o\left( {\sqrt {{ - \Lambda }} } \right)$. In the flat-space approximation, this limiting mass is twice that of the lightest BTZ black hole. The derivation applies at finite central charge for the boundary CFT, and does not rely on an asymptotic expansion at large central charge. Neither does our proof rely on any special property of the CFT such as supersymmetry or holomorphic factorization, nor on any bulk interpretation in terms of string theory or semiclassical gravity. Our only assumptions are unitarity and modular invariance of the dual CFT. Our proof demonstrates for the first time that there exists a universal center-of-mass energy beyond which a theory of ”pure” quantum gravity can never consistently be extended.