Twistor spaces for hyperkähler implosions

Abstract

We study the geometry of the twistor space of the universal hyperkähler implosion $Q$ for $SU(n)$. Using the description of $Q$ as a hyperkähler 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 hyperkähler implosions for other compact groups.