  We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theory
compactifications with N=1 supersymmetry. Extending our previous work on nearly
tensionless heterotic strings in six dimensions, we show that under certain
assumptions a tower of asymptotically massless states arises in the limit of
vanishing coupling of a U(1) gauge symmetry coupled to gravity. This tower
contains super-extremal states whose charge-to-mass ratios are larger than
those of certain extremal dilatonic Reissner-Nordstrom black holes, precisely
as required by the Weak Gravity Conjecture. Unlike in six dimensions, the tower
of super-extremal states does not always populate a charge sub-lattice. The
main tool for our analysis is the elliptic genus of the emergent heterotic
string in the chiral N=1 supersymmetric effective theories. This also governs
situations where the heterotic string is non-perturbative. We show how it can
be computed in terms of BPS invariants on elliptic four-folds, by making use of
various dualities and mirror symmetry. Compared to six dimensions, the geometry
of the relevant elliptically fibered four-folds is substantially richer than
that of the three-folds, and we classify the possibilities for obtaining
critical, nearly tensionless heterotic strings. We find that the
(quasi-)modular properties of the elliptic genus crucially depend on the choice
of flux background. Our general results are illustrated in a detailed example.

High Energy Physics - Theory