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

## TL;DR

This paper investigates the Weak Gravity Conjecture in four-dimensional F-theory compactifications with N=1 supersymmetry, showing the emergence of a tower of super-extremal states linked to elliptic genera and BPS invariants, with implications for quantum gravity.

## Contribution

It extends previous six-dimensional analyses to four dimensions, classifies elliptic four-folds for tensionless heterotic strings, and relates elliptic genera to BPS invariants using dualities.

## Key findings

- A tower of super-extremal states arises in the weak coupling limit.
- The elliptic genus can be computed via BPS invariants on elliptic four-folds.
- The modular properties of the elliptic genus depend on flux backgrounds.

## 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-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.