Integrability and Monopoles
Abstracts
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.