The Weak Gravity Conjecture and Scalar Fields

TL;DR

This work extends the Weak Gravity Conjecture to include scalar fields, motivated by extremal black holes in ${\cal N}=2$ supergravity and the principle of forbidding stable gravitationally bound towers. It introduces a field-space invariant bound that ensures gauge forces dominate over gravity and scalar interactions for some WGC state, and connects this to a scalar version of the no-force condition in N=2 theories. The analysis yields a robust logarithmic bound on the growth of proper field distances for linear moduli in Calabi–Yau compactifications and identifies towers of states whose masses decay exponentially, providing evidence for the Refined Swampland Conjecture. Additionally, the work argues that scalar forces can act as the decisive component ensuring gravity remains the weakest force, with implications for axions, Lattice-WGC, and potential links to high-scale supersymmetry.

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.