# The Weak Gravity Conjecture and the Axionic Black Hole Paradox

## TL;DR

This paper explores a paradox in axionic black holes related to Wilson lines and proposes a Weak Gravity Conjecture-like resolution involving light strings, linking small axion decay constants to the necessity of light charged objects.

## Contribution

It introduces a new Weak Gravity Conjecture analogue for 2-form fields and black holes, connecting small decay constants to the presence of light strings or a low cutoff.

## Key findings

- Light strings resolve the axionic black hole paradox.
- Small axion decay-constant implies the existence of light strings.
- The argument extends to models with gauged vector fields and multiple 2-forms.

## Abstract

In theories with a perturbatively massless 2-form (dual to an axion), a paradox may arise in the process of black hole evaporation. Schwarzschild black holes can support a non-trivial Wilson-line-type field, the integral of the 2-form around their horizon. After such an 'axionic black hole' evaporates, the Wilson line must be supported by the corresponding 3-form field strength in the region formerly occupied by the black hole. In the limit of small axion decay-constant f, the energy required for this field configuration is too large. The natural resolution is through the presence of light strings, which allow the black hole to "shed" its axionic hair sufficiently early. This gives rise to a new Weak-Gravity-type argument in the 2-form context: Small coupling, in this case f, enforces the presence of light strings or a low cutoff. We also discuss how this argument may be modified in situations where the weak coupling regime is achieved in the low-energy effective theory through an appropriate gauging of a model with a vector field and two 2-forms.