Classification of three-family grand unification in string theory. I. The SO(10) andE6models

We give a classification of three-family SO(10) and E${}_6$ grand unification in string theory within the framework of conformal (free) field theory and asymmetric orbifolds. We argue that the construction of such models in heterotic string theory requires certain ${\mathbf{Z}}_6$ asymmetric orbifolds that include a ${\mathbf{Z}}_3$ outerautomorphism, the latter yielding a level-three current algebra for the grand unification gauge group SO(10) or E${}_6$. We then classify all such ${\mathbf{Z}}_6$ asymmetric orbifolds that result in models with a non-Abelian hidden sector. All models classified in this paper have only one adjoint (but no other higher representation) Higgs field in the grand unified gauge group. In addition, all of them are completely anomaly-free. There are two types of such three-family models. The first type consists of the unique SO(10) model with SU(2) $\otimes$ SU(2) $\otimes$ SU(2) as its hidden sector (which is not asymptotically free at the string scale). This SO(10) model has four left-handed and one right-handed $16$’s. The second type is described by a moduli space containing 17 models (distinguished by their massless spectra). All these models have an SU(2) hidden sector, and five left-handed and two right-handed families in the grand unified gauge group. One of these models is the unique E${}_6$ model with an asymptotically free SU(2) hidden sector. The others are SO(10) models, eight of them with an asymptotically free hidden sector at the string scale.