Universal Bounds on Operator Dimensions in General 2D Conformal Field Theories

Abstract

We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d conformal field theories with discrete spectra using modular invariance, including CFTs with chiral currents. We derive a bound on the conformal dimensions ${\Delta }_1$ and ${\Delta }_2$ going as $c_{\mathrm{tot}}/12+O(1)$. The bound is of the same form found for CFTs without chiral currents in arXiv:0902.2790 and arXiv:1312.0038. We then prove the inequality ${\Delta }_n\le c_{\mathrm{tot}}/12+O(1)$ for large $c_{\mathrm{tot}}$ and with appropriate restrictions on $n$. Using the AdS${}_3$/CFT${}_2$ correspondence, our bounds correspond to upper bounds on the masses of the lightest few states and a lower bound on the number of states. We conclude by checking our results against several candidate conformal field theories.

24 pages; extension of (and thus overlap with) arXiv:0902.2790 and arXiv:1312.0038