Three family supersymmetric Pati-Salam model from rigid intersecting D6-branes

We construct, for the first time, a three-family $\mathcal{N}=1$ supersymmetric Pati-Salam model from rigid intersecting D6-branes on a factorizable ${\mathbb{T}}^6\mathrm{/}({\mathbb{Z}}_2\times {\mathbb{Z}}_2^{\mathrm{\prime }})$ orientifold with discrete torsion. The factorizable geometry allows for explicit control over rigid cycles and moduli stabilization. We can break the Pati-Salam gauge symmetry down to the Standard Model (SM) gauge symmetry via the supersymmetry preserving Higgs mechanism, generate the SM fermion masses and mixings, and break the supersymmetry via gaugino condensations in the hidden sector.