<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> symmetries, anomalies, and the modular bootstrap
Ying-Hsuan Lin, Shu-Heng Shao
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 $N\ensuremath{-}1\ensuremath{\le}c\ensuremath{\le}9\ensuremath{-}N$ for $N\ensuremath{\le}4$, with the end points saturated by various Wess-Zumino-Witten 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.