Electrostatics and geodesics on $K3$ surfaces

TL;DR

This paper uses advanced geometric constructions to locate and analyze closed geodesics on K3 surfaces, linking mathematical physics, electrostatics, and string theory in a novel way.

Contribution

It applies Foscolo's Ricci-flat Kähler metrics to identify geodesics on K3 surfaces, connecting geometric analysis with classical electrostatics and physics.

Findings

Located several closed geodesics with high precision

Computed their indices and approximate lengths

Linked geodesic construction to Maxwell's electrostatics problem

Abstract

Motivated by some conjectures originating in the Physics literature, we use Foscolo's construction of Ricci-flat Kahler metrics on K3 surfaces to locate, with high precision, several closed geodesics and compute their index (their length is also approximately known). Interestingly, the construction of these geodesics is related to an open problem in electrostatics posed by Maxwell in 1873. Our construction is also of interest to modern Physicists working on (supersymmetric) non-linear sigma models with target space such a K3 surface.