A Stringy Test of the Scalar Weak Gravity Conjecture

TL;DR

This paper proves a version of the Scalar Weak Gravity Conjecture in 6d string compactifications, incorporating massless scalars and demonstrating the bound's validity at weak coupling through detailed mathematical analysis.

Contribution

It extends previous work by including massless scalar fields in the Weak Gravity Conjecture proof for 6d F-theory and heterotic string compactifications, establishing the bound's precise numerical form.

Findings

The scalar-modified extremality condition aligns with the charge-to-mass ratio bound.

The bound is satisfied at weak coupling in F-theory compactifications.

The elliptic genus in models with multiple gauge factors forms a lattice quasi-Jacobi form.

Abstract

We prove a version of the Weak Gravity Conjecture for 6d F-theory or heterotic string compactifications with 8 supercharges. This sharpens our previous analysis by including massless scalar fields. The latter are known to modify the Weak Gravity Conjecture bound in two a priori independent ways: First, the extremality condition of a charged black hole is modified, and second, the test particles required to satisfy the Weak Gravity Conjecture are subject to additional Yukawa type interactions. We argue on general grounds that at weak coupling, the two types of effects are equivalent for a tower of asymptotically massless charged test particles predicted by the Swampland Distance Conjecture. We then specialise to F-theory compactified on elliptic Calabi-Yau three-folds and prove that the precise numerical bound on the charge-to-mass ratio is satisfied at weak coupling. This amounts to an intriguing coincidence of two a priori different notions of extremality, namely one based on the balance of gauge, gravitational and scalar forces for extremal (non-BPS) black holes, and the other encoded in the modular properties of certain Jacobi forms. In the presence of multiple abelian gauge group factors, the elliptic genus counting these states is a lattice quasi-Jacobi form of higher rank, and we exemplify this in a model with two abelian gauge group factors.