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Merging the Weak Gravity and Distance Conjectures Using BPS Extremal Black Holes

Naomi Gendler, Irene Valenzuela

Abstract

We analyze the charge-to-mass structure of BPS states in general infinite-distance limits of $\mathcal{N}=2$ compactifications of Type IIB string theory on Calabi-Yau three-folds, and use the results to sharpen the formulation of the Swampland Conjectures in the presence of multiple gauge and scalar fields. We show that the BPS bound coincides with the black hole extremality bound in these infinite distance limits, and that the charge-to-mass vectors of the BPS states lie on degenerate ellipsoids with only two non-degenerate directions, regardless of the number of moduli or gauge fields. We provide the numerical value of the principal radii of the ellipsoid in terms of the classification of the singularity that is being approached. We use these findings to inform the Swampland Distance Conjecture, which states that a tower of states becomes exponentially light along geodesic trajectories towards infinite field distance. We place general bounds on the mass decay rate $λ$ of this tower in terms of the black hole extremality bound, which in our setup implies $λ\geq 1/\sqrt{6}$. We expect this framework to persist beyond $\mathcal{N}=2$ as long as a gauge coupling becomes small in the infinite field distance limit.

Merging the Weak Gravity and Distance Conjectures Using BPS Extremal Black Holes

Abstract

We analyze the charge-to-mass structure of BPS states in general infinite-distance limits of compactifications of Type IIB string theory on Calabi-Yau three-folds, and use the results to sharpen the formulation of the Swampland Conjectures in the presence of multiple gauge and scalar fields. We show that the BPS bound coincides with the black hole extremality bound in these infinite distance limits, and that the charge-to-mass vectors of the BPS states lie on degenerate ellipsoids with only two non-degenerate directions, regardless of the number of moduli or gauge fields. We provide the numerical value of the principal radii of the ellipsoid in terms of the classification of the singularity that is being approached. We use these findings to inform the Swampland Distance Conjecture, which states that a tower of states becomes exponentially light along geodesic trajectories towards infinite field distance. We place general bounds on the mass decay rate of this tower in terms of the black hole extremality bound, which in our setup implies . We expect this framework to persist beyond as long as a gauge coupling becomes small in the infinite field distance limit.

Paper Structure

This paper contains 19 sections, 151 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Swampland conjectures in the presence of scalar fields
  3. BPS states and extremal black holes in 4d $\mathcal{N}=2$ effective field theories
  4. Review: $\mathcal{N}=2$ EFTs and BPS states in Calabi-Yau compactifications
  5. BPS and extremality bounds in the infinite distance limit
  6. General structure of BPS charge-to-mass spectra
  7. Example 1: one modulus case
  8. Example 2: two moduli case
  9. Magnetic charge and dyons
  10. General bounds in the asymptotic limit
  11. Review: asymptotic Hodge theory techniques
  12. Asymptotic form of the charge-to-mass ratio
  13. General bounds on mass decay rate
  14. Examples
  15. Emergence and generalization to other setups
  16. ...and 4 more sections

Figures (10)

  • Figure 1: The charge-to-mass ratios of BPS states in a one-modulus compactification. On the left, the axion is set to zero, while on the right, the axion is non-zero. The black arrow a principle axis of the ellipse.
  • Figure 2: The charge-to-mass vectors for BPS states in a compactification with $h^{2,1}=2$. The black line indicates the ellipse whose principal radii are given by Eqs. \ref{['eigenvalue1ex2']} and \ref{['eigenvalue2ex2']}, the red ellipses indicate the outer boundaries of the BPS region, and the green ellipses indicate the boundaries $p^1 q_0 =0$ associated with the single-charge states.
  • Figure 3:
  • Figure 4: The charge-to-mass vectors for BPS states in a single field limit with $d=3$. The red ellipse indicates the extremality region, while the green points represent only those BPS states which are stable against crossing walls of marginal stability.
  • Figure 5:
  • ...and 5 more figures