Universal Bounds in Even-Spin CFTs

TL;DR

The paper proves a universal bound on the lowest primary dimension $Δ_1$ in unitary 2D CFTs with only even-spin operators ($c,\tilde{c}>1$) by exploiting modular invariance under $S$ and $ST$. It constructs polynomial constraints from an intermediate-temperature expansion and derives an analytic bound $Δ_1 \le \frac{c_{ m tot}}{24} + 0.09270...$, with a numerically refined bound approaching $Δ_1 \le \frac{c_{ m tot}}{24} + 0.008554...$ in the large central charge limit. Via the AdS$_3$/CFT$_2$ dictionary, this bound translates into a bulk mass constraint for the lightest excitation, $M_1 \le \frac{1}{8G_N}$ (up to $O(\sqrt{-Λ})$ corrections) in the flat-space limit, corresponding to the lightest spinless BTZ black hole; the bound scales with the total central charge in AdS and yields insights into the dual gravitational spectrum. The work further discusses extending the method to bound higher conformal dimensions and to count primary operators within energy ranges, linking spectral data to bulk gravitational entropy and providing universal spectral constraints for even-spin CFTs and their gravitational duals.

Abstract

We prove using invariance under the modular $S$- and $ST$-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges $c,\tilde{c}>1$) contains a primary operator with dimension $Δ_1$ satisfying $0 < Δ_1 < (c+\tilde{c})/24 + 0.09280...$ After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS$_3$/CFT$_2$ dictionary, the bound on $Δ_1$ proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant $Λ< 0$ can be no heavier than $1/(8G_N)+O(\sqrt{-Λ})$; the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.