The Weak Gravity Conjecture in three dimensions

TL;DR

This work analyzes the Weak Gravity Conjecture in AdS$_3$ with a weakly coupled U(1) gauge field, showing that modular invariance of the holographic CFT together with gauge-compactness implies the existence of light charged operators below the charged BTZ black hole threshold, realizing a 3D version of the WGC. The authors derive this from the Sugawara decomposition of the stress tensor and the spectral-flow automorphism, which together guarantee light, charged primaries in appropriate sectors. They also study discrete $ ext{Z}_N$ charges, demonstrating that modular invariance alone does not guarantee light $ ext{Z}_N$-charged states and providing a concrete lattice example where the lowest charged state can be arbitrarily heavy. Additionally, a modular bootstrap approach is developed to bound charged operator dimensions in $ ext{Z}_N$ sectors, revealing nontrivial constraints for small $N$ but no universal bound for larger $N$. Overall, the paper argues that modular invariance and current-algebra structure imply a form of the WGC in three dimensions, while highlighting important distinctions from higher-dimensional remnants arguments and from discrete-charge sectors.

Abstract

We study weakly coupled $U(1)$ theories in $AdS_3$, their associated charged BTZ solutions, and their charged spectra. We find that modular invariance of the holographic dual two-dimensional CFT and compactness of the gauge group together imply the existence of charged operators with conformal dimension significantly below the black hole threshold. We regard this as a form of the Weak Gravity Conjecture (WGC) in three dimensions. We also explore the constraints posed by modular invariance on a particular discrete $\mathbb{Z}_N$ symmetry which arises in our discussion. In this case, modular invariance does not guarantee the existence of light $\mathbb{Z}_N$-charged states. We also highlight the differences between our discussion and the usual heuristic arguments for the WGC based on black hole remnants.