A Universal Inequality for CFT and Quantum Gravity

TL;DR

This work proves a universal bound on the weight of the lightest primary operator in any unitary 2D CFT with c, c̃ > 1 and no extended chiral algebra, namely 0 < Δ1 < (c + c̃)/12 + 0.473695. By exploiting modular invariance and a medium-temperature expansion, the author derives a general inequality for primaries, independent of large central charge, and interprets the result in AdS3/CFT2 as a bound on the lightest massive bulk excitation M1 ≤ 1/(4 G_N) in the flat-space limit. The approach separates primaries from descendants via Virasoro representation theory, decomposes the partition function, and uses S-transform constraints to bound Δ1 without assuming holomorphic factorization or SUSY. The findings provide a non-asymptotic, universal constraint on quantum gravity in AdS3 and offer a quantitative target for the spectrum of consistent theories, with implications for the mass scale of bulk excitations and the structure of quantum gravity in three dimensions.

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 $Δ_1$ that satisfies $0 < Δ_1 < (c_L + c_R)/12 + 0.473695$. 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 $Λ< 0$ can be no heavier than $1/(4 G_N) + o(|Λ|^(1/2))$. 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.

Figures (4)

• Figure 1: One logical possibility is that there exists no "sharp" bound on $\Delta_1$ / $c_{\rm total}$, but rather only a statistical falloff at large values of the ratio.
• Figure 2: One logical possibility is that there is no limit on $\Delta_1$ whatsoever for any given central charge, even in the "statistical" sense.
• Figure 3: In the case where the Hilbert space factorizes completely as a product of purely left- and right-moving CFTs, it is possible to show that $\Delta_1$ can never be greater than ${{c_{\rm total}}\over{24}} + 1$. This is the case described by Höhn-Witten's conjectured "extremal" CFT. It is unknown whether or not CFT exist that saturate this bound for $c$ equal to any positive integer multiple of 24.
• Figure 4: In this paper we have proven that the distribution of unitary conformal field theories in two dimensions looks something like the scatter plot above, where $\Delta_1$ is the weight of the lowest primary operator. It is an open question whether there exist CFT that saturate the bound at leading order in $c_{\rm total}$, or whether further considerations could reduce the slope of the red bounding line from ${1\over{12}}$, perhaps to as low as ${1\over{24}}$.