Swampland bounds on the Abelian gauge sector

Abstract

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 $N\ensuremath{\le}20$ (or $N\ensuremath{\le}22$ in the absence of charged matter). For any geometric $F$-theory realization with at least one tensor multiplet, we establish the bound $N\ensuremath{\le}16$ 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.