The Weak Gravity Conjecture in three dimensions

We study weakly coupled U(1) theories in AdS3, 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 ℤ N symmetry which arises in our discussion. In this case, modular invariance does not guarantee the existence of light 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.