On the Geometry of the Space of Monopole-Clusters

We review the results pertaining to the space of monopole-clusters, Mk,l, which was first proposed by Roger Bielawski. In particular, it has a pseudo-hyperk¨ahler metric which approximates the metric of the moduli space of SU(2)-monopoles on R 3 with exponential accuracy. We define actions of the groups R 3 , T 2 and SO(3) on Mk,l, and show that they are all isometry groups. In the case (k, l) = (1, 2), we express the monopole-clusters in terms of elliptic functions, and verify that they approach the true monopoles with rate inversely proportional to the separation distance between the clusters. For some SO^(2) ⊂ SO(3), the subgroups of SO^(2) × T 2 that admit a fixed point in the asymptotic region of M1,2 will be classified; their fixed point sets will be parametrized in terms of real coordinates and hence are manifolds. Finally, we compute the induced metric on an axially symmetric manifold in such family of manifolds, and show that it is asymptotically flat.