Dense Geodesics, Tower Alignment, and the Sharpened Distance Conjecture

Abstract: The Sharpened Distance Conjecture and Tower Scalar Weak Gravity Conjecture are closely related but distinct conjectures, neither one implying the other. Motivated by examples, I propose that both are consequences of two new conjectures: 1. The infinite distance geodesics passing through an arbitrary point $\phi$ in the moduli space populate a dense set of directions in the tangent space at $\phi$. 2. Along any infinite distance geodesic, there exists a tower of particles whose scalar-charge-to-mass ratio ($-\nabla \log m$) projection everywhere along the geodesic is greater than or equal to $1/\sqrt{d-2}$. I perform several nontrivial tests of these new conjectures in maximal and half-maximal supergravity examples. I also use the Tower Scalar Weak Gravity Conjecture to conjecture a sharp bound on exponentially heavy towers that accompany infinite distance limits.