Twistor spaces for hyperkahler implosions

Abstract

We study the geometry of the twistor space of the universal hyperk ähler implosion Q for SU(n). Using the description of Q as a hyperkähler quiver variety, we construct a holomorphic map from the twistor space ZQ of Q to a complex vector bundle over P1, and an associated map of Q to the affine space R of the bundle's holomorphic sections. The map from Q to R is shown to be injective and equivariant for the action of SU(n)×T n?1×SU(2). Both maps, from Q and from ZQ, 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 hyperkähler implosions for other compact groups.