Monopoles and the Gibbons–Manton Metric

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 L ²-invariant hyperkähler 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. The construction of the Gibbons–Manton metric in terms of Nahm's equations yields a class of interesting (pseudo)-hyperkähler metrics. For example we show, for each semisimple Lie group G and a maximal torus T⋚G, the existence of a G×T-invariant (pseudo)-hyperkähler manifold whose hyperkähler quotients by T are precisely Kronheimer's hyperkähler metrics on G ^?/T ^ℂ. A similar result holds for Kronheimer's ALE-spaces.