Generic construction of the Standard Model gauge group and matter representations in F-theory

We describe general classes of 6D and 4D F-theory models with gauge group $(\operatorname{SU}(3) \times \operatorname{SU}(2) \times \operatorname{U}(1)) / \mathbb{Z}_6$. We prove that this set of constructions gives all possible consistent 6D supergravity theories with no tensor multiplets having this gauge group and the corresponding generic matter representations, which include those of the MSSM. We expect, though do not prove, that these models are similarly generic for 6D theories with tensor multiplets and for 4D $\mathcal{N} = 1$ supergravity theories. The largest class of these constructions comes from deforming an underlying geometry with gauge symmetry $\operatorname{SU}(4) \times \operatorname{SU}(3) \times \operatorname{SU}(2)$.