A Universal Inequality for CFT and Quantum Gravity

TL;DR

This paper proves a universal bound on the lightest primary operator dimension in 2D unitary CFTs, which translates into a mass limit for the lightest massive excitation in 3D quantum gravity with negative cosmological constant, independent of supersymmetry or string theory assumptions.

Contribution

It establishes a rigorous, universal inequality linking CFT operator dimensions to gravitational mass bounds without relying on large central charge or special symmetries.

Findings

Bound on the lightest primary operator: Δ₁ < (c_L + c_R)/12 + 0.473695.

Mass limit for the lightest excitation in 3D gravity: no heavier than 1/(4 G_N) + o(|Λ|^{1/2}).

Proof applies at finite central charge and requires only unitarity and modular invariance.

Abstract

We prove that every unitary two-dimensional conformal field theory (with no extended chiral algebra, and with central charges $c_L, c_R > 1$) contains a primary operator with dimension $\Delta_1$ that satisfies $0 < \Delta_1 < (c_L + c_R)/12 + 0.473695$. Translated into gravitational language using the AdS_3 /CFT_2 dictionary, our result proves rigorously that the lightest massive excitation in any theory of 3D gravity with cosmological constant $\Lambda < 0$ can be no heavier than $1/(4 G_N) + o(|\Lambda|^(1/2))$. In the flat-space approximation, this limiting mass is twice that of the lightest BTZ black hole. The derivation of the bound applies at finite central charge for the 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.