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We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theorycompactifications with N=1 supersymmetry. Extending our previous work on nearlytensionless heterotic strings in six dimensions, we show that under certainassumptions a tower of asymptotically massless states arises in the limit ofvanishing coupling of a U(1) gauge symmetry coupled to gravity. This towercontains super-extremal states whose charge-to-mass ratios are larger thanthose of certain extremal dilatonic Reissner-Nordstrom black holes, preciselyas required by the Weak Gravity Conjecture. Unlike in six dimensions, the towerof super-extremal states does not always populate a charge sub-lattice. Themain tool for our analysis is the elliptic genus of the emergent heteroticstring in the chiral N=1 supersymmetric effective theories. This also governssituations where the heterotic string is non-perturbative. We show how it canbe computed in terms of BPS invariants on elliptic four-folds, by making use ofvarious dualities and mirror symmetry. Compared to six dimensions, the geometryof the relevant elliptically fibered four-folds is substantially richer thanthat of the three-folds, and we classify the possibilities for obtainingcritical, nearly tensionless heterotic strings. We find that the(quasi-)modular properties of the elliptic genus crucially depend on the choiceof flux background. Our general results are illustrated in a detailed example.

Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions