A Semiclassical, Entropic Proof of a Weak Gravity Conjecture

TL;DR

This work presents a semiclassical framework to test the weak gravity conjecture in $D=4$ for scalar matter gauged under $U(1)^N$ by computing the macroscopic entropy of an extremal black hole with scalar fluctuations. Using a heat-kernel analysis on the near-horizon $AdS_2\times S^2$ geometry, the authors obtain exact, non-perturbative quantum corrections to the entropy from neutral and gauged scalars, including a renormalization scheme that isolates logarithmic contributions. They demonstrate that, for scalars that violate or saturate the WGC, the gauged-scalar logarithmic entropy dwarfs the classical term at large charge and can violate the generalized second law, implying that such EFTs are thermodynamically sick and thus reside in the swampland. The results suggest that entropy inequalities provide a sharp IR diagnostic that can distinguish landscape from swampland theories and motivate extensions to arbitrary dimensions and higher-form gauge fields in future work.

Abstract

We present a semiclassical proof of the weak gravity conjecture in $D = 4$ spacetime dimensions for scalar matter gauged under a $U(1)^N$ gauge group. We compute the non-perturbative macroscopic entropy of a scalar field in an extremal black hole background at the level of linearized backreaction on the metric. The scalar field is assumed to violate or saturate the weak gravity conjecture. The scalar contributes a logarithmic correction to the entropy in the black hole geometry that outgrows the classical contribution. We demonstrate that the entropy of the gauged scalar violates the generalized second law in the limit of large black hole charge. Our result suggests that entropy inequalities may directly discriminate between effective field theories that live in the landscape versus the swampland.