ZN symmetries, anomalies, and the modular bootstrap

We explore constraints on (1+1)d unitary conformal field theory with an internal ZN global symmetry, by bounding the lightest symmetry-preserving scalar primary operator using the modular bootstrap. Among the other constraints we have found, we prove the existence of a ZN-symmetric relevant/marginal operator if N-1≤c≤9-N for N≤4, with the end points saturated by various Wess-Zumino-Witten models that can be embedded into (e8)1. Its existence implies that robust gapless fixed points are not possible in this range of c if only a ZN symmetry is imposed microscopically. We also obtain stronger, more refined bounds that depend on the 't Hooft anomaly of the ZN symmetry.