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.
Highlights
Global symmetries and their ‘t Hooft anomalies are central tools in analyzing strongly coupled quantum systems
We will apply the techniques of the conformal bootstrap, which exploits the internal consistency of conformal field theory (CFT), to derive general constraints on ð1 þ 1Þd unitary bosonic CFT with ZN global symmetry
The 0-form global symmetry in a general quantum field theory in d spacetime dimensions is implemented by a codimension-one topological defect [57,58]
Summary
Global symmetries and their ‘t Hooft anomalies (i.e., obstruction to gauging) are central tools in analyzing strongly coupled quantum systems. In the special case of a ZN global symmetry with small N, we will derive a universal upper bound ΔsQc1⁄4ala0r on the scaling dimension of the lightest symmetry-preserving scalar operator This generalizes the previous works when there is no symmetry [17] and when the symmetry is Z2 [1]. For small N and a given anomaly k, our bound further implies that there must be a symmetry-preserving relevant scalar operator for any CFT within a certain range of the central charge. × suð5Þ1 and Interestingly, our bootstrap bounds for nonanomalous Z5 at c 1⁄4 4 and that for Z6 with the k 1⁄4 3 anomaly at c 1⁄4 5 appear to be (almost) saturated by suð5Þ1 and suð6Þ1.5 We leave these curious observations and an analytic derivation of (1.1) for future investigations Some of these commutant pairs of ðe8Þ1 have recently been discussed in [50,51]. See [1,44,45,53,54,55,56] and references therein for recent discussions on this topic
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