Monopoles and the Gibbons-manton Metric

We show that, in the region where monopoles are well separated, the L-metric on the moduli space of n-monopoles is exponentially close to the Tn-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 moduli space Mn of (framed) static SU(2)-monopoles of charge n, i.e. solutions to Bogomolny equations dAΦ = ∗F , carries a natural hyperkähler metric [1]. The geodesic motion in this metric is a good approximation to the dynamics of low energy monopoles [26, 33]. For the charge n = 2 the metric has been determined explicitly by Atiyah and Hitchin [1], and it follows from their explicit formula that when the two monopoles are well separated, the metric becomes (exponentially fast) the Euclidean Taub-NUT metric with a negative mass parameter. It was also shown by N. Manton [27] that this asymptotic metric can be determined by treating well-separated monopoles as dyons. The equations of motion for a pair of dyons in R are found to be equivalent to the equations for geodesic motion on Taub-NUT space. For an arbitrary charge n, it was shown in [3] that, when the individual monopoles are well-separated, the L-metric is close (as inverse of the separation distance) to the flat Euclidean metric. Gibbons and Manton [14] have then calculated the Lagrangian for the motion of n dyons in R and shown that it is equivalent to the Lagrangian for geodesic motion in a hyperkähler metric on a torus bundle over the configuration space C̃n(R ). This metric is T -invariant and has a simple algebraic form. Gibbons and Manton have conjectured, by analogy with the n = 2 case, that the exact n-monopole metric differs from their metric by an exponentially small amount as the separation gets large. We shall prove this conjecture here. Our strategy is as follows. We construct certain moduli space M̃n of solutions to Nahm’s equations which carries a T -invariant hyperkähler metric. Using twistor methods we identify this metric as the Gibbons-Manton metric. Finally, we show that the metrics on M̃n and Mn are exponentially close. This proof adapts equally well to the asymptotic behaviour of SU(N)-monopole metrics with maximal symmetry breaking, as will be shown elsewhere. The asymptotic picture can be explained in the twistor setting. We recall that a monopole is determined (up to framing) by a curve S the spectral curve in TCP , which satisfies certain conditions [16]. One of these is triviality of the line bundle L over S, and a nonzero section of this bundle is the other ingredient needed to determine the metric [19, 1]. Asymptotically we have now the following situation. When the individual monopoles become well separated the spectral curve of the 1991 Mathematics Subject Classification. 53C25, 81T13. 1