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A bstract We propose magnetic quivers for the complex-symplectic contraction spaces, which are related to implosions and have a natural interpretation in terms of the Moore-Tachikawa category. We use 3-d mirrors to provide computational checks.

A bstract 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.

A bstract We propose quivers for Coulomb branch constructions of universal implosions for orthogonal and symplectic groups, extending the work on special unitary groups in [1]. The quivers are unitary-orthosymplectic as opposed to the purely unitary quivers in the A-type case. Where possible we check our proposals using Hilbert series techniques.

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

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

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 Painlevé analysis are discussed.

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

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R4d\\mathbb {R}^{4d} by the action of a compact Abelian group. These 4n4n-dimensional quotients carry a multi-Hamilitonian action of an nn-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in R3n\\mathbb {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 R4n\\mathbb {R}^{4n} and metrics on complements of hypersurfaces in compact manifolds.

We analyse properties of hypertoric manifolds of infinite topological type, including their topology and complex structures. We show that our manifolds have the homotopy type of an infinite union of compact toric varieties. We also discuss hypertoric analogues of the periodic Ooguri-Vafa spaces.

We give a survey of the implosion construction, extending some of its aspects relating to hypertoric geometry from type \\(A\\) to a general reductive group, and interpret it in the context of the Moore-Tachikawa category. We use these ideas to discuss how the contraction construction in symplectic geometry can be generalised to the hyperk\"ahler or complex symplectic situation.