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We propose magnetic quivers for partial implosion spaces. Such partial implosions involve a choice of parabolic subgroup, with the Borel subgroup corresponding to the standard implosion. In the subregular case we test the conjecture by verifying that reduction by the Levi group gives the appropriate nilpotent orbit closure. In the case of a parabolic corresponding to a hook diagram we are also able to carry out this verification provided we work at nonzero Fayet-Iliopoulos parameters.

We discuss symplectic and hyperk\"ahler implosion and present candidates for the symplectic duals of the universal hyperk\"ahler implosion for various groups., Comment: 18 pages, 2 figures

We review the quiver descriptions of symplectic and hyperk\"ahler implosion in the case of SU(n) actions. We give quiver descriptions of symplectic implosion for other classical groups, and discuss some of the issues involved in obtaining a similar description for hyperk\"ahler implosion., Comment: To appear in the Proceedings of the 2013 Arbeitstagung in memory of Friedrich Hirzebruch

We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. The underlying manifolds are of the form ℝ2 × M2 × ⋯ × Mr where Mi are arbitrary Einstein manifolds with positive scalar curvature. On the same spaces we also obtain a family of complete non-Kähler Ricci-flat metrics with asymptotically locally conical asymptotics. Among these new Ricci-flat and soliton examples are pairs with dimension 4m + 3 which are homeomorphic but not diffeomorphic.

We consider complete multiple warped product type Riemannian metrics on manifolds of the form R2 × M2 × · · · × Mr, where r ≥ 2 and Mi are arbitrary closed Einstein spaces with positive scalar curvature. We construct on these spaces a family of non-Kähler, non-Einstein, expanding gradient Ricci solitons with conical asymptotics as well as a family of Einstein metrics with negative scalar curvature. The 2-dimensional Euclidean space factor allows us to obtain homeomorphic but not diffeomorphic examples which have analogous cone structure behaviour at infinity. We also produce numerical evidence for complete expanding solitons on the vector bundles whose sphere bundles are the twistor or Sp(1) bundles over quaternionic projective space.

We show how to view the equations for a cohomogeneity one Ricci soliton as a Hamiltonian system with a constraint. We investigate conserved quantities and superpotentials, and use this to find some explicit formulae for Ricci solitons not of Kahler type in five dimensions.

We discuss various aspects of moment map geometry in symplectic and hyperKahler geometry. In particular, we classify complete hyperKahler manifolds of dimension \(4n\) with a tri-Hamiltonian action of a torus of dimension \(n\), without any assumption on the finiteness of the Betti numbers. As a result we find that the hyperKahler moment in these cases has connected fibres, a property that is true for symplectic moment maps, and is surjective. New examples of hypertoric manifolds of infinite topological type are produced. We provide examples of non-Abelian tri-Hamiltonian group actions of connected groups on complete hyperKahler manifolds such that the hyperKahler moment map is not surjective and has some fibres that are not connected. We also discuss relationships to symplectic cuts, hyperKahler modifications and implosion constructions.

We give a classification of continuously differentiable on-shell integrals which are linear or quadratic in momenta for a Hamiltonian system with constraint associated to the Ricci-flat cohomogeneity one Einstein equations for two special classes of principal orbits/hypersurfaces. Relations to known on-shell integrals and to Painleve analysis are discussed.

We introduce a general formalism for studying cohomogeneity on Ricci solitons, and illustrate this by producing new families of examples. © 2011 American Institute of Physics.

We study the geometry of the twistor space of the universal hyperkahler implosion $Q$ for $SU(n)$. Using the description of $Q$ as a hyperkahler quiver variety, we construct a holomorphic map from the twistor space $\mathcal{Z}_Q$ of $Q$ to a complex vector bundle over $\mathbb{P}^1$, and an associated map of $Q$ to the affine space $\mathcal{R}$ of the bundle’s holomorphic sections. The map from $Q$ to $\mathcal{R}$ is shown to be injective and equivariant for the action of $SU(n) \times T^{n-1} \times SU(2)$. Both maps, from $Q$ and from $\mathcal{Z}_Q$, are described in detail for $n = 2$ and $n = 3$. We explain how the maps are built from the fundamental irreducible representations of $SU(n)$ and the hypertoric variety associated to the hyperplane arrangement given by the root planes in the Lie algebra of the maximal torus. This indicates that the constructions might extend to universal hyperkahler implosions for other compact groups.

We give an overview of our earlier classification results in [DW4] and [DW6] for superpotentials of scalar curvature type of the cohomogeneity one Ricci-flat equations. We then give an account of the classification in the case where the isotropy representation of the principal orbit consists of exactly three distinct irreducible real summands--the leftover case from [DW6]., Comment: final version

We use Nahm data to give a description of a candidate for the universal hyperkahler implosion with respect to a compact Lie group, Version to appear in Communications in Mathematical Physics

We introduce a multiplicative version of complex-symplectic implosion in the case of $SL(n, \C)$. The universal multiplicative implosion for $SL(n, \C)$ is an affine variety and can be viewed as a nonreductive geometric invariant theory quotient. It carries a torus action. and reductions by this action give the Steinberg fibres of $SL(n, \C)$. We also explain how the real symplectic group-valued universal implosion introduced by Hurtubise, Jeffrey and Sjamaar may be identified inside this space., Comment: To appear in European Journal of Mathematics

A construction is introduced for modifying hyperkähler manifolds with tri-Hamiltonian circle action, that in favourable situations increases the second Betti number by one. This is based on the symplectic cut construction of Lerman. In 4 or 8 dimensions the construction may be interpreted as adding a D6-brane. A number of examples are given and a generalisation to three-Sasaki geometry discussed.

Using ideas from convex geometry, we prove a classification theorem, under suitable hypotheses, for superpotentials of the Hamiltonian form of the cohomogeneity one Ricci-flat equations. Some new superpotentials are also constructed and their associated first order systems analysed. The situation with a cosmological constant is briefly examined. © Springer-Verlag 2005.

We apply techniques of Painlevé-Kowalewski analysis to a Hamiltonian system arising from symmetry reduction of the Ricci-flat Einstein equations. In the case of doubly warped product metrics on a product of two Einstein manifolds over an interval, we show that the cases when the total dimension is 10 or 11 are singled out by our analysis. © 2001 Elsevier Science B.V.

Classification results are given for (i) compact quaternionic Kähler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperKähler 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 Kähler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras.

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) .

Charge-two SU(3) monopoles with minimal symmetry breaking can be generated via Nahm's equations. This paper investigates the detailed structure of such monopoles, through explicit calculation of the energy density, and the norm and discriminant of the Higgs field. Monopoles may be classified according to the maxima and minima of these quantities.

We use analytical and numerical methods to investigate the equations for cohomogeneity one shrinking gradient Ricci solitons. We show the existence of a winding number for this system around the subvariety of phase space corresponding to Einstein solutions and obtain some estimates for it. We prove a non-existence result for certain orbit types, analogous to that of Bohm in the Einstein case. We also carry out numerical investigations for selected orbit types., Comment: 7 figures : references added

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., 48 pages

The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined byis closed. It is well-known that if this condition holds then Ω is in fact covariant constant.

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

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

We study geodesic flow on a totally geodesic submanifold of a moduli space of solutions to Nahm’s equations, and draw conclusions about the dynamics of SU (3) monopoles with minimal symmetry breaking.

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.

We produce non-Kähler complete steady gradient Ricci solitons generalisin those constructed by Bryant and Ivey. © International Press 2009.

We produce new examples of non-Kähler complete expanding gradient Ricci solitons on trivial vector bundles over a product of Einsteinmanifolds with positive scalar curvature. © The Author 2009. Published by Oxford University Press. All rights reserved.

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansatze of cohomogeneity one type to produce new explicit examples of complete Kahler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kahler-Einstein manifolds or over coadjoint orbits of a compact connected semisimple Lie group., 29 pages

We extend our previous classification [DW4] of superpotentials of "scalar curvature type" for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in [DW4], i.e., when some weight vector of the superpotential lies outside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano Kähler-Einstein product. © 2008 Springer-Verlag.

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., Comment: 26 pages, 6 figures, small linguistic corrections

We investigate the existence of Painlevé-Kovalevskaya expansions for various reductions to ordinary differential equations of the Ricci-flat equations. We investigate links between such expansions and metrics of exceptional holonomy. © 2003 Elsevier Science B.V. All rights reserved.

We apply techniques of Painlevé-Kowalewski analysis to certain ODE reductions of the Ricci-flat equations. We particularly focus on two examples when the hypersurface is an Aloff-Wallach space or a circle bundle over a Fano product. © 2003 American Institute of Physics.