Twistor spaces for hyperkaehler implosions

Abstract

We study the geometry of the twistor space of the universal hyperkaehler implosion Q for SU(n). Using the description of Q as a hyperkaehler quiver variety, we construct a holomorphic map from the twistor space Z_Q of Q to a complex vector bundle over P^1, 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) x T^{n-1} x SU(2). Both maps, from Q and from 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 hyperkaehler implosions for other compact groups.