A Classification of Three-Family $SO(10)$ and $E_6$ Grand Unification in String Theory

We give a classification of $3$-family $SO(10)$ and $E_6$ grand unification in string theory within the framework of conformal field theory and asymmetric orbifolds. We argue that the construction of such models in heterotic string theory requires certain ${\bf Z}_6$ asymmetric orbifolds that include a ${\bf Z}_3$ outer-automorphism, the latter yielding a level-3 current algebra for the grand unification gauge group $SO(10)$ or $E_6$. We then classify all such ${\bf Z}_6$ 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 $3$-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. None of these three $SU(2)$s is asymptotically-free at the string scale. This $SO(10)$ model has $4$ left-handed and $1$ right-handed ${\bf 16}$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 $5$ left-handed and $2$ right-handed families in the grand unified gauge group. One of these models is the unique $E_6$ model with an asymptotically-free hidden sector. The other 16 are $SO(10)$ models, 8 of them with an asymptotically free hidden sector at the string scale.