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

TL;DR

This paper uses the modular bootstrap to analyze (1+1)d conformal field theories with $Z_N$ symmetry, deriving bounds on operators and showing the necessity of certain relevant operators within specific central charge ranges.

Contribution

It proves the existence of $Z_N$-symmetric relevant/marginal operators in certain central charge ranges and refines bounds based on 't Hooft anomalies, advancing understanding of symmetry constraints in CFTs.

Findings

Existence of $Z_N$-symmetric relevant operators for $N-1 \\le c \\le 9-N$ when $N \\le 4$

Robust gapless fixed points are excluded in these ranges under $Z_N$ symmetry

Stronger bounds depending on the 't Hooft anomaly are obtained.

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 $N\leq4$, 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.