Abstract
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 mathfrak{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 mathfrak{s}mathfrak{u}(2)oplus mathfrak{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’.
Highlights
Inherently global objects that can only be fully described within a globally defined geometry
We show that F-theory compactifications with abelian gauge factors generally exhibit a non-trivial global gauge group structure
Only non-local operators such as line operators are sensitive to this quotient structure [38], one often refers to G/Z as the structure of the global gauge group, in order to distinguish it from the gauge algebra that is seen by local operators
Summary
Because our main argument is based on the Shioda map, we will first present a brief review of its prominent role in F-theory, which will help to set up the notation. The first two conditions ensure that the intersection product of φ(σ) with the generic fibre f and any curve CB of the base (lifted by the zero-section) vanishes This is related to the requirement that the u(1) gauge field lifts properly from d − 1 to d dimensions in the M-/F-theory duality. For the purpose of these notes, let us fix the factor λ in the Shioda map to 1: φ(σ) := S − Z + π−1(DB) + li Ei. Since the following discussion revolves around the fractional coefficients li, let us recall that they arise from requiring the intersection numbers of φ(σ) with the fibre P1i s of the exceptional divisors Ei to vanish, see (2.4). All arguments and conclusions hold for F-theory compactifications to six, four and two dimensions
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