Motivated by the Weak Gravity Conjecture, we uncover an intricate interplay between black holes, BPS particle counting, and Calabi‐Yau geometry in five dimensions. In particular, we point out that extremal BPS black holes exist only in certain directions in the charge lattice, and we argue that these directions fill out a cone that is dual to the cone of effective divisors of the Calabi‐Yau threefold. The tower and sublattice versions of the Weak Gravity Conjecture require an infinite tower of BPS particles in these directions, and therefore imply purely geometric conjectures requiring the existence of infinite towers of holomorphic curves in every direction within the dual of the cone of effective divisors. We verify these geometric conjectures in a number of examples by computing Gopakumar‐Vafa invariants.
This content will become publicly available on December 1, 2023
 Award ID(s):
 1914934
 Publication Date:
 NSFPAR ID:
 10388303
 Journal Name:
 Journal of High Energy Physics
 Volume:
 2022
 Issue:
 12
 ISSN:
 10298479
 Sponsoring Org:
 National Science Foundation
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