Sharpening the Weak Gravity Conjecture with 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.

• The paper introduces the Lattice Weak Gravity Conjecture (LWGC) to strengthen the original WGC under dimensional reduction.
• It demonstrates that Kaluza-Klein photons require stringent conditions after compactification to maintain the weak gravity bounds.
• The findings bridge gauge theory with gravitational instantons and axion dynamics, impacting string theory and quantum cosmology.

Analysis of "Sharpening the Weak Gravity Conjecture with Dimensional Reduction"

The paper by Heidenreich, Reece, and Rudelius addresses the Weak Gravity Conjecture (WGC) and its preservation across different dimensionalities through compactifications and renormalization group flows. The authors propose a rigorous examination of the WGC when subjected to toroidal compactification, particularly focusing on Kaluza-Klein (KK) photons and their properties post compactification. The central proposition is the introduction of a stronger form—the Lattice Weak Gravity Conjecture (LWGC)—which they argue is necessary for ensuring consistency of WGC under dimensional reduction.

Key Findings

1. WGC and Infrared Behavior:

The study highlights that WGC bounds for gauge couplings, specifically for single photons in four-dimensional scenarios, appear to become less stringent in an infrared limit. However, this does not hold uniformly; specific configurations, particularly those involving KK photons, exhibit contrasting behavior.

2. KK Photons and Dimensionality:

Upon compactifying a U(1) gauge theory, the authors discover that a photon already satisfying WGC in a higher-dimensional theory might not satisfy it post-compactification unless subject to stringent conditions. These conditions include either a necessity to allow infinite states of varying charges in the original theory or imposing constraints on the radius of compactification.

3. Proposal of the Lattice Weak Gravity Conjecture:

To address these subtleties, this work posits the "Lattice Weak Gravity Conjecture", which suggests that for every charge in the charge lattice, there should exist a corresponding superextremal particle. This conjecture proves robust across compactifications and is supported by perturbative heterotic string theory.

4. Axions and Gravitational Instantons:

The article extends the conjecture to axions, juxtaposing gravitational instanton solutions with extremal black holes. The authors show that gravitational instantons behave analogously to extremal black holes and that instanton solutions hold implications for axions, supporting a version of WGC for these fields. The analogous extremality conditions align with deviation actions of wrapped higher-dimensional objects.

Implications and Future Directions

The implications of this research are significant, suggesting that particle physics constraints derived from quantum gravity could apply universally to all compactified dimensions, thus offering a pathway to reconcile low-energy phenomena with high-energy theoretical frameworks.

• Practical Implications: The validation of the LWGC could impact various model-building approaches in string theory and quantum cosmology, supporting more robust predictions in scenarios like large-field inflation.
• Theoretical Implications: This research highlights intricate links between gravity, gauge theory, and quantum field theory, suggesting interdependencies not initially apparent. The adoption of a lattice framework over possibly restricted forms of WGC underscores the holistic nature required for a consistent theory of quantum gravity.
• Future Directions: Further exploration of the LWGC’s applicability across diverse topologies and gauge groups in string theory could yield novel insights. Additionally, rigorous analysis within frameworks such as AdS/CFT could provide grounded support or pose challenges to current conjectures.

This paper advances the discourse on Weak Gravity Conjecture, providing robust arguments for its adaptations when considering behaviors across compactified and extrapolated dimensions. The propositions laid out, particularly the LWGC, advocate for a foundational reevaluation of gravitational theories and their respective low-energy and high-energy interactions.