Discrete Gauge Symmetries and the Weak Gravity Conjecture

In theories with discrete Abelian gauge groups, avoiding stable remnants leads to an upper bound on the product of a charged particle's mass and the cutoff scale above which the effective description of the theory breaks down. To the extent that discrete gauge symmetries can arise at low energies from the spontaneous breaking of continuous ones, this suggests that a residual of the Weak Gravity Conjecture may persist in the Higgs phase. Here, we take a step towards making this expectation more precise by studying $\mathbb{Z}_N$ and $\mathbb{Z}_2^N$ gauge symmetries realised via Abelian Higgs models. In this setting, considering the effects of discrete hair on black holes reproduces existing bounds when the cutoff scale is identified with the scale of spontaneous symmetry breaking, and provides a mechanism through which discrete hair can be lost without modifying the gravitational sector. Moreover, applying the electric form of the conjecture to a dual description of the Abelian Higgs model leads to constraints consistent with existing bounds on discrete gauge groups. We explore the possible implications of these arguments for understanding the smallness of the weak scale compared to $M_{Pl}$.