An F-theory Realization of the Chiral MSSM with $\mathbb{Z}_2$-Parity

Using F-theory we construct 4D ${\cal N}=1$ SUGRA theories with the Standard Model gauge group, three chiral generations, and matter parity in order to forbid all dimension four baryon and lepton number violating operators. The underlying geometries are derived by constructing smooth genus-one fibered Calabi--Yau fourfolds using toric tops that have a Jacobian fibration with rank one Mordell--Weil group and $SU(3) \times SU(2) $ singularities. The necessary gauge backgrounds on the smooth fourfolds are shown to be fully compatible with the quantization condition, including positive integer D3-tadpoles. This construction realizes for the first time a consistent UV completion of an MSSM-like model with matter parity in F-theory. Moreover our construction is general enough to also exhibit other relevant $\mathbb{Z}_2$ charge extensions of the MSSM such as lepton and baryon parity. Such models however are rendered inconsistent by non-integer fluxes, which are necessary for producing the exact MSSM chiral spectrum. These inconsistencies turn out to be intimately related to field theory considerations regarding a UV-embedding of the $\mathbb{Z}_2$ into a $U(1)$ and the resulting discrete anomalies.