The dynamics of vortices on S^2 near the Bradlow limit

TL;DR

This paper develops an approximate solution for vortex dynamics on a sphere near the Bradlow limit, enabling the use of the geodesic approximation to describe vortex interactions with a moduli space metric proportional to the Fubini-Study metric.

Contribution

It introduces an approximate solution to the Bogomolny equations for vortices on a sphere near the Bradlow limit, facilitating the analysis of vortex dynamics via the geodesic approximation.

Findings

Derived an approximate solution valid near the Bradlow limit.

Computed the moduli space metric as proportional to the Fubini-Study metric.

Provided a complete description of vortex particle dynamics on the sphere.

Abstract

The explicit solutions of the Bogomolny equations for N vortices on a sphere of radius R^2 > N are not known. In particular, this has prevented the use of the geodesic approximation to describe the low energy vortex dynamics. In this paper we introduce an approximate general solution of the equations, valid for R^2 close to N, which has many properties of the true solutions, including the same moduli space CP^N. Within the framework of the geodesic approximation, the metric on the moduli space is then computed to be proportional to the Fubini- Study metric, which leads to a complete description of the particle dynamics.