The global gauge group structure of F-theory compactification with U(1)s

We show that F-theory compactifications with abelian gauge factors generally exhibit a non-trivial global gauge group structure. The geometric origin of this structure lies with the Shioda map of the Mordell-Weil generators. This results in constraints on the u 1 u(1) charges of non-abelian matter consistent with observations made throughout the literature. In particular, we find that F-theory models featuring the Standard Model algebra actually realise the precise gauge group [SU(3) × SU(2) × U(1)]/ℤ6. Furthermore, we explore the relationship between the gauge group structure and geometric (un-)higgsing. In an explicit class of models, we show that, depending on the global group structure, an s u 2 ⊕ u 1 su(2)⊕ u(1) gauge theory can either unhiggs into an SU(2) × SU(2) or an SU(3) × SU(2) theory. We also study implications of the charge constraints as a criterion for the F-theory ‘swampland’.