Proof of the Weak Gravity Conjecture from Black Hole Entropy

TL;DR

The paper demonstrates that higher-dimension operators arising at tree level from heavy fields increase black hole entropy at fixed mass and charge, and translates this into a free-energy inequality that yields new positivity bounds on EFT coefficients. These bounds imply that extremal black holes shift to higher charge-to-mass ratios, ensuring that large extremal black holes are unstable to decay and thereby automatically satisfying the Weak Gravity Conjecture. The analysis is carried out in a model-independent EFT framework, extended to arbitrary spacetime dimensions and multiple gauge fields, with explicit connections to unitarity, RG flows, and entanglement entropy interpretations. The results offer a concrete, entropy-based route to WGC validity and invite exploration of broader swampland implications and generalizations to other BH backgrounds.

Abstract

We prove that higher-dimension operators contribute positively to the entropy of a thermodynamically stable black hole at fixed mass and charge. Our results apply whenever the dominant corrections originate at tree level from quantum field theoretic dynamics. More generally, positivity of the entropy shift is equivalent to a certain inequality relating the free energies of black holes. These entropy inequalities mandate new positivity bounds on the coefficients of higher-dimension operators. One of these conditions implies that the charge-to-mass ratio of an extremal black hole asymptotes to unity from above for increasing mass. Consequently, large extremal black holes are unstable to decay to smaller extremal black holes and the weak gravity conjecture is automatically satisfied. Our findings generalize to arbitrary spacetime dimension and to the case of multiple gauge fields. The assumptions of this proof are valid across a range of scenarios, including string theory constructions with a dilaton stabilized below the string scale.

Figures (2)

• Figure 1: Black holes of maximal charge shown as a function of mass $m$ and charge-to-mass ratio $q/m$. Higher-dimension operators induce corrections to the extremality condition. If these corrections are positive, then the WGC is automatically satisfied (upper solid curve) since large black holes are unstable to decay to smaller ones. If these corrections are negative (lower solid curve), then the WGC mandates additional light, superextremal particles to avoid an infinite number of stable extremal black hole remnants.
• Figure 2: Constraints on higher-dimension operator coefficients derived from black hole entropy. The shaded regions are excluded, with the gradations corresponding to incremental values of $\xi \in (0,1/2)$. The left and right panels correspond to $d_3>0$ and $d_3 <0$, respectively. In either case, $d_0<0$ is forbidden so $d_0 >0$ and the WGC is automatically satisfied.