Phenomenological implications of a minimal F-theory GUT with discrete symmetry

We discuss the origin of both non-Abelian discrete family symmetry and Abelian continuous family symmetry, as well as matter parity, from F-theory SUSY GUTs. We propose a minimal model based on the smallest GUT group $SU(5)$, together with the non-Abelian family symmetry $D_4$ plus an Abelian family symmetry, where fluxes are responsible for doublet-triplet splitting, leading to a realistic low energy spectrum with phenomenologically acceptable quark and lepton masses and mixing. We show how a $Z_2$ matter parity emerging from F-theory can suppress proton decay while allowing neutron-antineutron oscillations, providing a distinctive signature of the set-up.