# The Weak Gravity Conjecture and Scalar Fields

## TL;DR

This paper generalizes the Weak Gravity Conjecture to include scalar fields, suggesting that particles exist with forces stronger than gravity, and explores implications for string theory and scalar field behavior.

## Contribution

It introduces a new formulation of the Weak Gravity Conjecture involving scalar forces and provides a proof relating scalar field distances to moduli in string compactifications.

## Key findings

- Scalar forces can dominate over gravity for certain particles.
- Mass of particles decreases exponentially with scalar field variation.
- Field distance in string compactifications grows at most logarithmically.

## Abstract

We propose a generalisation of the Weak Gravity Conjecture in the presence of scalar fields. The proposal is guided by properties of extremal black holes in ${\cal N}=2$ supergravity, but can be understood more generally in terms of forbidding towers of stable gravitationally bound states. It amounts to the statement that there must exist a particle on which the gauge force acts more strongly than gravity and the scalar forces combined. We also propose that the scalar force itself should act on this particle stronger than gravity. This implies that generically the mass of this particle decreases exponentially as a function of the scalar field expectation value for super-Planckian variations, which is behaviour predicted by the Refined Swampland Conjecture. In the context of ${\cal N}=2$ supergravity the Weak Gravity Conjecture bound can be tied to bounds on scalar field distances in field space. Guided by this, we present a general proof that for any linear combination of moduli in any Calabi-Yau compactification of string theory the proper field distance grows at best logarithmically with the moduli values for super-Planckian distances.