Mitchell Institute for Fundamental Physics & Astronomy
College Station, Texas 77843
I discuss the relationship between extremal black holes, BPS states, and the Weak Gravity Conjecture (WGC) along with their connections to Calabi-Yau geometry. In particular, the tower/sublattice WGC requires an infinite tower of BPS particles in every charge direction in which extremal BPS black holes exist. In 5d theories arising from M-theory on a Calabi-Yau threefold, I show that these directions include the dual of the cone of effective divisors of the Calabi-Yau threefold. This implies the purely geometric conjecture that infinite towers of holomorphic curves exist in every direction within this cone, which I verify in several examples using the genus 0 Gopakumar-Vafa invariants. Along the way I uncover new facts about Calabi-Yau geometry as well as interconnections between swampland conjectures and novel mathematical ``cone conjectures’’.