The geometry of the space of vortices on a two-sphere in the Bradlow limit

TL;DR

This paper proves that the normalized L^2 metric on the moduli space of n-vortices on a two-sphere converges to the Fubini-Study metric in the Bradlow limit, confirming a longstanding conjecture.

Contribution

It rigorously establishes the convergence of the vortex moduli space metric to the Fubini-Study metric in the Bradlow limit, confirming a conjecture by Baptista and Manton.

Findings

Normalized L^2 metric converges uniformly to Fubini-Study metric

Convergence holds for any Riemannian metric on the two-sphere

Provides a rigorous proof of a longstanding informal conjecture

Abstract

It is proved that the normalized $L^2$ metric on the moduli space of $n$-vortices on a two-sphere, endowed with any Riemannian metric, converges uniformly in the Bradlow limit to the Fubini-Study metric. This establishes, in a rigorous setting, a longstanding informal conjecture of Baptista and Manton.