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) having only even-spin primary operators (with no extended chiral algebra and with right- and left-central charges c,c˜>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{c}>1 $$\end{document}) contains a primary operator with dimension Δ1 satisfying 0<Δ1<c+c˜24+0.09280....\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0<{\varDelta}_1<\frac{c+\tilde{c}}{24}+0.09280.\;.\;.\;. $$\end{document} 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 AdS3/CFT2 dictionary, the bound on Δ1 proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than 1/8GN+O−Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 1/8{G}_N+O\left(\sqrt{-\varLambda}\right) $$\end{document}; the bounds on the number of 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.