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Weak Gravity Conjecture and Extremal Black Holes

William Cottrell, Gary Shiu, Pablo Soler

TL;DR

This work investigates the Weak Gravity Conjecture by computing one-loop corrections to extremal black hole geometry and entropy from charged particles, using the quantum entropy function on an $\mathrm{AdS}_{2}\times S^{2}$ near-horizon background. It introduces the key parameter $\Delta m^{2}= m^{2}-2 q^{2} M_{P}^{2}$ to classify charged-particle effects as subextremal, extremal, or superextremal, and finds logarithmic entropy corrections for massless neutral and extremal cases, with distinct but suppressed corrections for subextremal states; superextremal states lead to instabilities that signal discharge of the black hole. Supersymmetry, especially ${\cal N}=4$, suppresses or cancels loop corrections, while generalizations to dyonic and multi-U(1) setups reveal how electric and magnetic charges modify the near-horizon data and spectral densities. The results underscore a tension between microscopic bound-state entropy expectations in WGC-violating theories and semiclassical EFT descriptions, and highlight the potential necessity of a magnetic WGC cutoff $\Lambda \lesssim q M_{P}$ for a consistent quantum gravity embedding. Overall, the paper argues that the presence of at least one sub- or extremal-allowed particle is crucial to avoid entropy-pathologies and to maintain consistency with holographic and cosmic censorship principles.

Abstract

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 ${\cal N}=4$ 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.

Weak Gravity Conjecture and Extremal Black Holes

TL;DR

This work investigates the Weak Gravity Conjecture by computing one-loop corrections to extremal black hole geometry and entropy from charged particles, using the quantum entropy function on an near-horizon background. It introduces the key parameter to classify charged-particle effects as subextremal, extremal, or superextremal, and finds logarithmic entropy corrections for massless neutral and extremal cases, with distinct but suppressed corrections for subextremal states; superextremal states lead to instabilities that signal discharge of the black hole. Supersymmetry, especially , suppresses or cancels loop corrections, while generalizations to dyonic and multi-U(1) setups reveal how electric and magnetic charges modify the near-horizon data and spectral densities. The results underscore a tension between microscopic bound-state entropy expectations in WGC-violating theories and semiclassical EFT descriptions, and highlight the potential necessity of a magnetic WGC cutoff for a consistent quantum gravity embedding. Overall, the paper argues that the presence of at least one sub- or extremal-allowed particle is crucial to avoid entropy-pathologies and to maintain consistency with holographic and cosmic censorship principles.

Abstract

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 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.

Paper Structure

This paper contains 18 sections, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Microphysical Intuition
  3. Macroscopic Description
  4. Classical entropy function
  5. One-Loop Correction
  6. Neutral particles: $\Delta m^2=m^2$
  7. Superextremal Particles $(m^2<2q^2 M_P^2)$: Unstable black holes
  8. The heat kernel for charged particles
  9. Subextremal Particles: $m^2>2q^2M_P^2$
  10. Extremal Particles: $m^2=2q^2 M_P^2$
  11. Comments On Cutoffs
  12. Generalizations
  13. Dyonic Black Holes
  14. Multi-centered black holes and primitive charges
  15. String theory black holes
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Spectral densities for large and small $q E$. One can see that the spectral density for $q E \gg 1$ resembles a step function.
  • Figure 2: Maximal extension of the Reissner-Nordstrom metric reproduced from hawking1975large. The diagram repeats an infinite number of times in the vertical direction.