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) 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) x SU(2) x U(1)]/Z(6). 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 su(2) circle plus u(1) gauge theory can either unhiggs into an SU(2) x SU(2) or an SU(3) x SU(2) theory. We also study implications of the charge constraints as a criterion for the F-theory 'swampland'.