Vortex motion on surfaces of small curvature

• We study an Abelian Higgs vortex on a surface with small curvature.
• A universal expansion for the moduli space metric is proposed.
• We numerically check the universality at low orders.
• Vortex motion differs from point particle motion because a vortex has a finite size.
• Moduli space geometry has similarities with the geometry arising from Ricci flow.

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