The Weak Gravity Conjecture in three dimensions

TL;DR

This paper investigates the Weak Gravity Conjecture in three-dimensional AdS space, showing that modular invariance and gauge group properties imply the existence of light charged operators, with implications for holography and discrete symmetries.

Contribution

It demonstrates that in AdS3, modular invariance and gauge group compactness imply the existence of light charged operators, extending the WGC to three dimensions.

Findings

Modular invariance implies light charged operators in AdS3.

Compact gauge groups lead to charged spectra below black hole threshold.

Discrete symmetries like b_N do not always guarantee light charged states.

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.