Classification of three-family grand unification in string theory. II. The SU(5) and SU(6) models

Requiring that supersymmetric SU(5) and SU(6) grand unifications in the heterotic string theory must have three chiral families, adjoint (or higher representation) Higgs fields in the grand unified gauge group, and a non-Abelian hidden sector, we construct such string models within the framework of free conformal field theory and asymmetric orbifolds. Within this framework, we construct all such string models via ${\mathbf{Z}}_6$ asymmetric orbifolds that include a ${\mathbf{Z}}_3$ outerautomorphism, the latter yielding a level-three current algebra for the grand unification gauge group SU(5) or SU(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. This Higgs field is neutral under all other gauge symmetries. The list of hidden sectors for three-family SU(6) string models are SU(2), SU(3), and SU(2)$\otimes$SU(2). In addition to these, three-family SU(5) string models can also have an SU(4) hidden sector. Some of the models have an apparent anomalous U(1) gauge symmetry.