Anomalies and bounds on charged operators

We study the implications of the 't Hooft anomaly (i.e., obstruction to gauging) on conformal field theory, focusing on the case when the global symmetry is $Z_2$. Using the modular bootstrap, universal bounds on ($1+1$)-dimensional bosonic conformal field theories with an internal $Z_2$ global symmetry are derived. The bootstrap bounds depend dramatically on the 't Hooft anomaly. In particular, there is a universal upper bound on the lightest $Z_2$ odd operator if the symmetry is anomalous, but there is no bound if the symmetry is nonanomalous. In the nonanomalous case, we find that the lightest $Z_2$ odd state and the defect ground state cannot both be arbitrarily heavy. We also consider theories with a $U\left(1\right)$ global symmetry, and comment that there is no bound on the lightest $U\left(1\right)$ charged operator if the symmetry is nonanomalous.