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Abstract. We study the implications of 't Hooft anomaly (i.e. obstruction to gauging) on quantum field theory in the gapless phase, focusing on the case when the global symmetry is Z_2. Using the modular bootstrap method, universal bounds on (1+1)-dimensional bosonic conformal field theory 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 non-anomalous. We also consider theories with a U(1) global symmetry. We comment that there is no bound on the lightest U(1) charged operator if the symmetry is non-anomalous, and discuss its implications on the weak gravity conjecture in AdS3/CFT2.
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