We present a semiclassical proof of the weak gravity conjecture in @math spacetime dimensions for scalar matter gauged under a @math 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.
Qualitatively similar results to our entropy calculation appear in @cite_17 . However, not all of their quantitative results match ours exactly. We believe that this results from the formalism they use to calculate the entropy of the black hole, which is not exactly equivalent to ours. We also believe that their conclusions and interpretation of results differ significantly enough from our own. Moreover, they do not attempt to prove the weak gravity conjecture using entropy inequalities, although they allude to this possibility.
@cite_17
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Motivated by the desire to improve our understanding of the Weak Gravity Conjecture, we compute the one-loop correction of charged particles to the geometry and entropy of extremal black holes in 4d. We find that fermion loops provide evidence for the necessity of the magnetic' WGC cutoff. Moreover, for a certain regime of black holes, we find entropy corrections with unusual area scaling. The corrections are reduced when supersymmetry is present, and disappear in @math supergravity. We further provide some speculative arguments that in a theory with only sub-extremal particles, classical Reissner-Nordstrom black holes actually possess an infinite microcanonical entropy, though only a finite amount is visible to an external observer.