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I think I've understood the general idea of the WGC (proposed in https://arxiv.org/abs/hep-th/0601001). From a more recent review, I quote:

In a consistent effective field theory (EFT) coupled to gravity, gravity must always be the weakest force.

(The reference is https://arxiv.org/abs/1711.00864). This implies that in the theory there should be at least one charged particle for which we have $m\le M_p q$, where $m$ is the mass of the particle, $q$ is the charge and $M_p$ is the Planck mass. This is connected to a hidden ultraviolet scale beneath the Planck mass.

What I don't understand is how can we practically use this conjecture in the framework of the Swampland program. What should we do in the EFT? Should we search for a charged particle satisfying the mass-charge relation or do something with the new scale? Could you give me a simple example of application?

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Here is a recent comprehensive (200 page) review:

Abstract:

The Swampland program aims to distinguish effective theories which can be completed into quantum gravity in the ultraviolet from those which cannot. This article forms an introduction to the field, assuming only a knowledge of quantum field theory and general relativity. It also forms a comprehensive review, covering the range of ideas that are part of the field, from the Weak Gravity Conjecture, through compactifications of String Theory, to the de Sitter conjecture.

Some of the section titles that reference the WGC:

  • First encounter with the Weak Gravity Conjecture

  • The Weak Gravity Conjecture and charged black holes

  • Refinements of the Weak Gravity Conjecture

  • The Weak Gravity Conjecture and Heterotic Strings

  • The Weak Gravity Conjecture in type II string theory

Hope this helps.

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