Vortex motion on surfaces of small curvature Original Research Article

We consider a single Abelian Higgs vortex on a surface image whose Gaussian curvature image 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 image with a modified metric, and we propose that this metric has a universal expansion, in terms of image and its derivatives, around the initial metric on image. Using an integral expression for the Kähler 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 image, but differs at higher order. We compare the vortex motion with the motion of a point particle along geodesics of image. Relative to a particle geodesic, the vortex experiences an additional force, which to leading order is proportional to the gradient of image. This force is analogous to the self-force on bodies of finite size that occurs in gravitational motion.