For the first time we systematically discuss the $N=1$ supersymmetric $SU(12{)}_C\times SU(2{)}_L\times SU(2{)}_R$ models, $SU(4{)}_C\times SU(6{)}_L\times SU(2{)}_R$ models, and $SU(4{)}_C\times SU(2{)}_L\times SU(6{)}_R$ models from the type IIA orientifolds on $T^6/({\mathbb{Z}}_2\times {\mathbb{Z}}_2)$ with intersecting D6-branes. These gauge symmetries can be broken down to the Pati-Salam gauge symmetry $SU(4{)}_C\times SU(2{)}_L\times SU(2{)}_R$ via three $SU(12{)}_C/SU(6{)}_L/SU(6{)}_R$ adjoint representation Higgs fields, and further down to the Standard Model (SM) via the D-brane splitting and Higgs mechanism. We obtain three families of the SM fermions, and have the left-handed three-family SM fermion unification in the $SU(4{)}_C\times SU(6{)}_L\times SU(2{)}_R$ models, and the right-handed three-family SM fermion unification in the $SU(4{)}_C\times SU(2{)}_L\times SU(6{)}_R$ models. Utilizing mathematical analysis, we exclude the generalized $SU(12{)}_C\times SU(2{)}_L\times SU(2{)}_R$ models by requiring the conditions for constructing Minimal Supersymmetric Standard Model models. Moreover, the $SU(4{)}_C\times SU(6{)}_L\times SU(2{)}_R$ models and $SU(4{)}_C\times SU(2{)}_L\times SU(6{)}_R$ models are related by the left and right gauge symmetry exchanging, as well as a variation of type II T-duality. The hidden sector contains $USp(n)$ branes, which are parallel with the orientifold planes or their ${\mathbb{Z}}_2$ images and might break the supersymmetry via gaugino condensations.

• Received 15 August 2020
• Accepted 8 June 2021

DOI:https://doi.org/10.1103/PhysRevD.104.046018