N = 1 supersymmetric S U ( 12 ) C × S U ( 2 ) L × S U ( 2 ) R models, S U ( 4 ) C × S U ( 6 ) L × S U ( 2 ) R models, and S U ( 4 ) C × S U ( 2 ) L × S U ( 6 ) R models from intersecting D6-branes

For the first time we systematically discuss the $N=1$ supersymmetric $SU(12{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}$ models, $SU(4{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(6{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}$ models, and $SU(4{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(6{)}_{R}$ models from the type IIA orientifolds on ${T}^{6}/({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})$ with intersecting D6-branes. These gauge symmetries can be broken down to the Pati-Salam gauge symmetry $SU(4{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}$ 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 $SU(4{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(6{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}$ models, and the right-handed three-family SM fermion unification in the $SU(4{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(6{)}_{R}$ models. Utilizing mathematical analysis, we exclude the generalized $SU(12{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}$ models by requiring the conditions for constructing Minimal Supersymmetric Standard Model models. Moreover, the $SU(4{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(6{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{R}$ models and $SU(4{)}_{C}\ifmmode\times\else\texttimes\fi{}SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}SU(6{)}_{R}$ 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.