The Weak Gravity Conjecture and BPS Particles

TL;DR

This paper explores the relationship between the Weak Gravity Conjecture, BPS black holes, and Calabi-Yau geometry, proposing geometric conjectures about infinite towers of holomorphic curves based on BPS particle counting.

Contribution

It establishes a connection between the Weak Gravity Conjecture and the geometry of Calabi-Yau threefolds, introducing new geometric conjectures supported by explicit examples.

Findings

Existence of extremal BPS black holes only in certain charge directions

Duality between charge lattice directions and cone of effective divisors

Verification of geometric conjectures via Gopakumar-Vafa invariants

Abstract

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 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.