Integrability and Monopoles


Oliver Nash:   A new approach to monopole moduli spaces
Euclidean monopoles have hyperkahler moduli spaces. The natural geometry on hyperbolic monopole moduli spaces is unknown. The hyperkahler geometry of the Euclidean monopole moduli spaces can be understood from a number of points of view but none of these carry over in a straightforward way to the hyperbolic case. I will introduce a new way to study both Euclidean and hyperbolic monopole moduli spaces by considering an appropriate family of deformations of the spectral curves. Furthermore I will show how to recover the hyperkahler geometry in the Euclidean case from this point of view and outline the sort of geometry the hyperbolic monopole moduli space carries.

Harry Braden:   Explicit Monopoles
Finite gap integration is applied to the explicit construction of monopoles. The Ercolani-Sinha construction will be reviewed and extended.  Some open geometric questions will be discussed for a class of charge 3 su(2) monopoles.

Michael Singer:   Gluing constructions for monopoles revisited
I will report on a new approach to the construction and compactification of (euclidean) monopole moduli spaces which is being developed in joint work with Richard Melrose.  The novel feature is that the compactification of the moduli space will appear as a compact manifold with corners.  The long-term goal is to understand the moduli space and its metric in sufficient detail to study Sen's conjectures, but so far we only have preliminary results.

Nigel Hitchin:  Monopole metrics
Kronheimer described the constraints on the generic spectral curve for a Euclidean monopole with singularities and this was used by the speaker and S Cherkis to give formulae for the hyperkaehler metric on the four-dimensional ALF moduli space of charge two solutions. In general this has no Killing fields but for special values of the parameters there is a circle action, and in particular a 2-sphere fixed by it. We address the question of finding this sphere and the metric on it, and its relationship to a similar problem in the ALE case.

Roger Bielawski:  Monopoles and clusters
It is known from the work of Taubes and Uhlenbeck that an infinite sequence of monopoles of charge n has a subsequence which looks like a bunch of clusters (of lower charges) receding from one another (with the centers of clusters receding in definite directions). The aim of this work is to capture this asymptotic picture in metric terms (thus generalising the Gibbons-Manton picture for particles). The motion of the clusters turns out to be governed by a (pseudo)-hyperkaehler metric (not a product metric as it captures the interaction of the clusters). Just as in the case of particles, the cluster metric is an exponentially good approximation to the monopole metric.