The Complete Search for the Supersymmetric Pati-Salam Models from Intersecting D6-Branes

We construct a systematic method to build all the possible three-family ${\cal N}=1$ supersymmetric Pati-Salam models from Type IIA orientifolds on $\mathbb{T}^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ with intersecting D6-branes, in which the $SU(4)_C\times SU(2)_L \times SU(2)_R $ gauge symmetry can be broken down to the $SU(3)_C \times SU(2)_L \times U(1)_Y$ Standard Model gauge symmetry by the D-brane splitting and supersymmetry preserving Higgs mechanism. This is essentially achieved by solving all the common solutions for the RR tadpole cancellation conditions, ${\cal N}=1$ supersymmetry conditions, and three generation conditions with deterministic algorithm. We find that there are $202752$ possible supersymmetric Pati-Salam models in total, and show that there are only $33$ independent models with different gauge coupling relations at string scale after modding out equivalent relations, such as T-dualities, etc. In particular, there is one and only one independent model which has gauge coupling unification. Furthermore, one can construct other types of intersecting D-brane models utilizing such deterministic algorithm, and therefore we suggest a brand new method for D-brane model building.