Monopoles and the Gibbons-Manton metric

TL;DR

This paper demonstrates that the L^2-metric on the moduli space of well-separated monopoles closely approximates the Gibbons-Manton hyperk"ahler metric, using twistor methods and Nahm's equations.

Contribution

It provides a rigorous proof that the monopole moduli space metric asymptotically matches the Gibbons-Manton metric, linking twistor theory and Nahm's equations.

Findings

L^2-metric is exponentially close to Gibbons-Manton metric for well-separated monopoles

Twistor methods determine the asymptotic form of the monopole metric

Description of Gibbons-Manton metric via solutions to Nahm's equations

Abstract

We show that, in the region where monopoles are well separated, the L^2-metric on the moduli space of n-monopoles is exponentially close to the T^n-invariant hyperk\"ahler metric proposed by Gibbons and Manton. The proof is based on a description of the Gibbons-Manton metric as a metric on a certain moduli space of solutions to Nahm's equations, and on twistor methods. In particular, we show how the twistor description of monopole metrics determines the asymptotic metric.