Andrew Dancer

Nahm's equations and hyperkähler geometry

The geometry of certain moduli spaces of solutions to Nahms equations is studied, and a family of gravitational instationsj is shown to arise as a deformation of theh Atiyah-Hitchin maniford.

Hyperkähler metrics of cohomogeneity one

Abstract Irreducible hyperkahler manifolds of dimension greater than four admitting a cohomogeneity-one action of a compact simple Lie group are classified via coadjoint orbits. It is shown that the only complete example is the Calabi metric on T ∗ C P (n) .

Hyperkähler metrics associated to compact Lie groups

It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G , then this structure is preserved by the actions of G on T * G induced by left and right translation on G . We refer to these as the left and right actions of G on T * G .

Nahm data and SU(3) monopoles

A moduli space of Nahm data is shown to produce, via the ADHM-Nahm construction, SU(3) monopoles with minimal symmetry breaking. The author finds analytical and numerical expressions for the Higgs fields of such monopoles in the spherically and axially symmetric cases, and study their dynamics.

Dihedral singularities and gravitational instantons

Abstract We produce a family of gravitational instantons which arise as deformations of dihedral singularities.

QUATERNIONIC KAHLER MANIFOLDS OF COHOMOGENEITY ONE

Classification results are given for (i) compact quaternionic Kahler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperKahler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomogeneity one with respect to a group of 3-Sasakian symmetries. Information is also obtained about non-compact quaternionic Kahler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras.

Toric hypersymplectic quotients

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R 4d by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in R 3n . We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on R 4n and metrics on complements of hypersurfaces in compact manifolds.

Einstein metrics on tangent bundles of spheres

We give an elementary treatment of the existence of complete Kahler–Einstein metrics with nonpositive Einstein constant and underlying manifold diffeomorphic to the tangent bundle of the (n + 1)-sphere.

Implosion for hyperkähler manifolds

We introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of SU(n) actions. Our space is a stratified hyperkahler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

The Geometry of Singular Quaternionic Kähler Quotients

Two descriptions of quaternionic Kahler quotients by proper group actions are given: the first as a union of smooth manifolds, some of which come equipped with quaternionic Kahler or locally Kahler structures; the second as a union of quaternionic Kahler orbifolds. In particular the quotient always has an open set which is a smooth quaternionic Kahler manifold. When the original manifold and the group are compact, we describe a length space structure on the quotient. Similar descriptions of singular hyperKahler and 3-Sasakian quotients are given.