Swampland bounds on the Abelian gauge sector

We derive bounds on the number of Abelian gauge group factors in six-dimensional gravitational theories with minimal supersymmetry and in their F-theoretic realizations. These bounds follow by requiring consistency of certain Bogomol'nyi-Prasad-Sommerfield strings in the spectrum of the theory, as recently proposed in the literature. Under certain assumptions, this approach constrains the number of Abelian gauge group factors in six-dimensional supergravity theories with at least one tensor multiplet to be N20 (or N22 in the absence of charged matter). For any geometric F-theory realization with at least one tensor multiplet, we establish the bound N16 by demanding unitarity of a heterotic solitonic string which exists even in the absence of a perturbative heterotic dual. This result extends to four-dimensional F-theory vacua on any blowup of a rational fibration. Our findings lead to universal bounds on the rank of the Mordell-Weil group of elliptically fibered Calabi-Yau 3-folds.