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

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 $c_{\rm tot}/12 + O(1)$. 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 $\Delta_n \leq c_{\rm tot}/12 + O(1)$ for large $c_{\rm tot}$ 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.