${\mathbb{Z}}_N$ Symmetries, Anomalies, and the Modular Bootstrap

Abstract

We explore constraints on (1+1)$d$ unitary conformal field theory with an internal ${\mathbb{Z}}_N$ 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 ${\mathbb{Z}}_N$-symmetric relevant/marginal operator if $N-1\le c\le 9-N$ for $N\le 4$, with the endpoints saturated by various WZW models that can be embedded into $({\mathfrak{e}}_8{)}_1$. Its existence implies that robust gapless fixed points are not possible in this range of $c$ if only a ${\mathbb{Z}}_N$ symmetry is imposed microscopically. We also obtain stronger, more refined bounds that depend on the 't Hooft anomaly of the ${\mathbb{Z}}_N$ symmetry.

arXiv comment26+13 pages, 10 figures, 2 tables; v2: minor revision

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