Abstract
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-Nordström 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.
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Lee, SJ., Lerche, W. & Weigand, T. Modular fluxes, elliptic genera, and weak gravity conjectures in four dimensions. J. High Energ. Phys. 2019, 104 (2019). https://doi.org/10.1007/JHEP08(2019)104
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DOI: https://doi.org/10.1007/JHEP08(2019)104