## Journal of Differential Geometry

### 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.

#### Article information

Source
J. Differential Geom., Volume 97, Number 1 (2014), 37-77.

Dates
First available in Project Euclid: 9 July 2014

https://projecteuclid.org/euclid.jdg/1404912102

Digital Object Identifier
doi:10.4310/jdg/1404912102

Mathematical Reviews number (MathSciNet)
MR3229049

Zentralblatt MATH identifier
1296.32004

#### Citation

Dancer, Andrew; Kirwan, Frances; Swann, Andrew. Twistor spaces for hyperkähler implosions. J. Differential Geom. 97 (2014), no. 1, 37--77. doi:10.4310/jdg/1404912102. https://projecteuclid.org/euclid.jdg/1404912102