# Discrete Gauge Symmetries and the Weak Gravity Conjecture

## TL;DR

This paper investigates how the Weak Gravity Conjecture extends to discrete gauge symmetries, using black hole arguments and spontaneous symmetry breaking to derive constraints and explore implications for fundamental scales.

## Contribution

It provides a detailed analysis of applying the WGC to discrete symmetries via dual descriptions and spontaneous symmetry breaking, establishing bounds consistent with black hole arguments.

## Key findings

- Constraints saturate but do not violate existing bounds
- Discrete hair can be lost without altering gravity
- The cutoff scale relates to the symmetry breaking scale

## Abstract

In theories with discrete Abelian gauge groups, requiring that black holes be able to lose their charge as they evaporate 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. This suggests that a non-trivial version of the Weak Gravity Conjecture (WGC) may also apply to gauge symmetries that are discrete, despite there being no associated massless field, therefore pushing the conjecture beyond the slogan that `gravity is the weakest force'. 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 theories of spontaneous symmetry breaking. We show that applying the WGC to a dual description of an Abelian Higgs model leads to constraints that allow us to saturate but not violate existing bounds on discrete symmetries based on black hole arguments. In this setting, considering the effect of discrete hair on black holes naturally identifies the cutoff of the effective theory with the scale of spontaneous symmetry breaking, and provides a mechanism through which discrete hair can be lost without modifying the gravitational sector. We explore the possible implications of these arguments for understanding the smallness of the weak scale compared to $M_{Pl}$.