Sharpening the Weak Gravity Conjecture with Dimensional Reduction

TL;DR

The paper investigates the Weak Gravity Conjecture (WGC) in the presence of dimensional reduction and multiple gauge fields, showing that circle compactification can preserve or weaken the WGC depending on moduli dynamics, and that KK photons complicate simple single-U(1) bounds via the convex hull condition.It strengthens the WGC into the Lattice Weak Gravity Conjecture (LWGC), positing a superextremal state for every charge vector in the charge lattice, and provides string-theoretic evidence that the LWGC holds in the perturbative heterotic string, including via toroidal compactifications.Extending the WGC to axions through gravitational instantons, the authors show that, with a nonzero dilaton coupling, instantons obey an extremality bound analogous to black holes and that the instanton action matches the action of wrapped higher-dimensional objects in a UV completion, supporting an axion version of the WGC, which remains consistent under dimensional reduction.

Abstract

We investigate the behavior of the Weak Gravity Conjecture (WGC) under toroidal compactification and RG flows, finding evidence that WGC bounds for single photons become weaker in the infrared. By contrast, we find that a photon satisfying the WGC will not necessarily satisfy it after toroidal compactification when black holes charged under the Kaluza-Klein photons are considered. Doing so either requires an infinite number of states of different charges to satisfy the WGC in the original theory or a restriction on allowed compactification radii. These subtleties suggest that if the Weak Gravity Conjecture is true, we must seek a stronger form of the conjecture that is robust under compactification. We propose a "Lattice Weak Gravity Conjecture" that meets this requirement: a superextremal particle should exist for every charge in the charge lattice. The perturbative heterotic string satisfies this conjecture. We also use compactification to explore the extent to which the WGC applies to axions. We argue that gravitational instanton solutions in theories of axions coupled to dilaton-like fields are analogous to extremal black holes, motivating a WGC for axions. This is further supported by a match between the instanton action and that of wrapped black branes in a higher-dimensional UV completion.

Figures (4)

• Figure 1: Boosting a black string charged under a 1-form. Finite boosts of sub-extremal black strings remain sub-extremal black holes after dimensional reduction. Infinite boosts yield the extremal KK charged black hole, whereas finite boosts of extremal black strings map out the remaining extremal black holes with both charges nonzero.
• Figure 2: Boosting a black string charged under a 2-form. Here, the extremal black string is invariant under boosts, and instead extremal black holes in the dimensionally reduced theory come from maximally boosting while simultaneously taking $r_+ \rightarrow r_-$. Interestingly, the extremality condition takes the form of a linear relation $M \geq c_F |Q_F| + c_H |Q_H|$ rather than a quadratic one.
• Figure 3: Stabilizing the scalars shrinks the black hole region, which becomes ellipsoidal. In the case without a $D$-dimensional dilaton ($\alpha = 0$) the stabilized extremal boundary intersects the unstabilized extremal boundary at four points, whereas with a dilaton ($\alpha>0$), the stabilized extremal boundary lies strictly inside the unstabilized one.
• Figure 4: The CHC for a theory with a KK $U(1)$ plus another $U(1)$. It is possible for the charge-to-mass vector of every KK mode to obey $|\vec{Z}_n| \geq 1$ without satisfying the CHC, as shown at left. Instead, the charge-to-mass vectors must be sufficiently large that the line segments connecting them lie outside the unit disk, as shown at right.