The Asymptotic Weak Gravity Conjecture in M-theory

The tower Weak Gravity Conjecture predicts infinitely many super-extremal states along every ray in the charge lattice of a consistent quantum gravity theory. We show this far-reaching claim in five-dimensional compactifications of M-theory on Calabi--Yau 3-folds for gauge groups with a weak coupling limit. We first characterize the possible weak coupling limits, building on an earlier classification of infinite distance limits in the K\"ahler moduli space of M-theory compactifications. We find that weakly coupled gauge groups are associated to curves on the compactification space contained in generic fibers or in fibers degenerating at finite distance in their moduli space. These always admit an interpretation as a Kaluza--Klein or winding U$(1)$ in a dual frame or as part of a dual perturbative heterotic gauge group, in agreement with the Emergent String Conjecture. Using the connection between Donaldson--Thomas invariants and Noether--Lefschetz theory, we then show that every ray in the associated charge lattice either supports a tower of BPS states or of non-BPS states, and prove that these satisfy the super-extremality condition, at least in the weak coupling regime.