Universal Bounds in Even-Spin CFTs

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 ${\mathrm{\Delta }}_1$ satisfying $0<{\mathrm{\Delta }}_1<(c+\tilde{c})\mathrm{/}24+\mathrm{0.09280...}$ 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 ${\mathrm{\Delta }}_1$ proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant $\mathrm{\Lambda }<0$ can be no heavier than $1\mathrm{/}(8G_N)+O(\sqrt{-\mathrm{\Lambda }})$; 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.