A Stringy Test of the Scalar Weak Gravity Conjecture

TL;DR

The paper tests the Scalar Weak Gravity Conjecture in 6d string compactifications with 8 supercharges by incorporating massless scalars and demonstrating that, near a weak coupling point, the dilatonic extremality bound and the scalar Yukawa force bound coincide for a tower of asymptotically massless states predicted by the Swampland Distance Conjecture. It provides a detailed F-theory realization, showing that the weak coupling limit yields a dilaton-like coupling with total extremality parameter $\mu = 1$, and that the spectrum from the elliptic genus lies on a sublattice whose charges satisfy the Sublattice WGC. The authors extend the analysis to multiple U(1) factors, where the elliptic genus becomes a higher-rank lattice Jacobi form, and confirm the sublattice structure in an explicit $n_V=2$ example on $\mathbb{F}_1$, connecting black hole extremality to arithmetic properties of Jacobi forms. Overall, the work links the WGC with stringy spectra and higher-rank modular objects, providing a concrete, testable framework for the WGC in theories with massless scalars and multiple abelian gauge factors.

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.

Figures (2)

• Figure 1: Charge-mass spectrum of a 6d F-theory, or heterotic, string compactification as determined by the elliptic genus, for a particular example of ref Lee:2018urn. Note how narrowly the bound of the Sublattice Weak Gravity Conjecture (solid blue line) is satisfied by the super-extremal, non-BPS string states (red dots). The new feature we show is how the blue line arises as the net sum of gravitational (dashed red line) and scalar (dotted green line) contributions. It is intriguing that this sum, which reflects a zero force property of the physics of extremal (non-BPS) black holes, conspires with the spectrum of super-extremal string states, which reflects certain mathematical properties of Jacobi forms.
• Figure 2: Shown on the left is the lowest lying charge-mass spectrum as determined by the elliptic genus given in eq. (\ref{['fullgen']}). The red dots denote the subset of super-extremal string states. The top picture on the right shows the charges of the maximally super-extremal states as determined in (\ref{['mostextremal']}), as viewed from the top of the left picture; obviously they lie on a sublattice of the full charge lattice. On the right bottom, the charges of all super-extremal states are shown. These do not form a lattice, rather are given as the union of shifted copies of the sublattice.