Classification of symmetric toroidal orbifolds

Abstract

Abstract We provide a complete classification of six-dimensional symmetric toroidal orbifolds which yield $ \mathcal{N}\geq 1 $ supersymmetry in 4D for the heterotic string. Our strategy is based on a classification of crystallographic space groups in six dimensions. We find in total 520 inequivalent toroidal orbifolds, 162 of them with Abelian point groups such as $ {{\mathbb{Z}}_3},{{\mathbb{Z}}_4},{{\mathbb{Z}}_6}\hbox{-}\mathrm{I} $ etc. and 358 with non-Abelian point groups such as S 3, D 4, A 4 etc. We also briefly explore the properties of some orbifolds with Abelian point groups and $ \mathcal{N}=1 $ , i.e. specify the Hodge numbers and comment on the possible mechanisms (local or non-local) of gauge symmetry breaking.