Vortex Motion on Surfaces of Small Curvature

TL;DR

This paper analyzes the motion of a vortex on a curved surface with small curvature, proposing a universal expansion for the moduli space metric and comparing vortex dynamics to point particle geodesics.

Contribution

It introduces a universal expansion of the vortex moduli space metric in terms of curvature and its derivatives, supported by numerical evidence and comparison with Ricci flow.

Findings

The moduli space metric expansion agrees with Ricci flow at first order.

Vortex experiences an additional force proportional to the gradient of curvature.

Numerical calculations suggest universality of the expansion coefficients.

Abstract

We consider a single Abelian Higgs vortex on a surface {\Sigma} whose Gaussian curvature K is small relative to the size of the vortex, and analyse vortex motion by using geodesics on the moduli space of static solutions. The moduli space is {\Sigma} with a modified metric, and we propose that this metric has a universal expansion, in terms of K and its derivatives, around the initial metric on {\Sigma}. Using an integral expression for the K\"ahler potential on the moduli space, we calculate the leading coefficients of this expansion numerically, and find some evidence for their universality. The expansion agrees to first order with the metric resulting from the Ricci flow starting from the initial metric on {\Sigma}, but differs at higher order. We compare the vortex motion with the motion of a point particle along geodesics of {\Sigma}. Relative to a particle geodesic, the vortex experiences an additional force, which to leading order is proportional to the gradient of K. This force is analogous to the self-force on bodies of finite size that occurs in gravitational motion.