[Paper Review] Twistor spaces for hyperkaehler implosions

Andrew Dancer, Frances Kirwan+1 more|arXiv (Cornell University)|Aug 14, 2013

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper constructs a $K\times T\times\mathrm{SU}(2)$-equivariant embedding of the universal hyperkähler implosion $Q$ for $\mathrm{SU}(n)$ into a complex affine space $\mathcal{R}$, using moment maps and fundamental representations. It further defines a generically injective holomorphic map from the twistor space $\mathcal{Z}_Q$ to a vector bundle over $\mathbb{P}^1$, revealing the implosion's structure via hypertoric varieties and root plane arrangements, with a path toward generalizing the construction to other compact groups.

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.

Motivation & Objective

• To generalize the hyperkähler implosion construction beyond $\mathrm{SU}(n)$ to other compact Lie groups.
• To provide a geometric description of the twistor space $\mathcal{Z}_Q$ of the universal hyperkähler implosion $Q$ for $\mathrm{SU}(n)$.
• To establish a $K\times T\times\mathrm{SU}(2)$-equivariant embedding of $Q$ into a complex affine space $\mathcal{R}$ using fundamental representations and moment maps.
• To show that the image of $Q$ in $\mathcal{R}$ is the closure of the $K_{\mathbb{C}}$-sweep of a hypertoric variety associated to root planes in the maximal torus.
• To lay the foundation for extending the hyperkähler implosion construction to arbitrary compact groups via twistor space embeddings.

Proposed method

• Constructs a holomorphic map from the twistor space $\mathcal{Z}_Q$ of the hyperkähler implosion $Q$ to a complex vector bundle over $\mathbb{P}^1$ using the hyperkähler structure and complex structures parametrized by $\mathbb{P}^1$.
• Uses the hyperkähler quiver variety description of $Q$ as $ (T^*K_{\mathbb{C}})_{\mathrm{hkimpl}} $ to define the embedding $\sigma: Q \to \mathcal{R}$.
• Defines $\mathcal{R}$ as the space of holomorphic sections of a bundle over $\mathbb{P}^1$, specifically $ H^0(\mathbb{P}^1, \mathcal{O}(2) \otimes (\mathfrak{k}_{\mathbb{C}} \oplus \mathfrak{t}_{\mathbb{C}}) \oplus \bigoplus_{j=1}^{\dim V_{\varpi}-1} \mathcal{O}(\ell_j) \otimes \wedge^j V_{\varpi}) $.
• Establishes that the map $\sigma: Q \to \mathcal{R}$ is injective and $K\times T\times\mathrm{SU}(2)$-equivariant, using moment maps and the action of fundamental irreducible representations of $\mathrm{SU}(n)$.
• Describes the image of the hypertoric variety associated to root planes in $\mathfrak{t}$ under $\sigma$, showing it maps into $\mathcal{R}$ and that its $K_{\mathbb{C}}$-sweep is dense in $\mathcal{R}$.
• Extends the construction to $n=2$ and $n=3$ explicitly, verifying the maps and symmetries in low-rank cases.

Experimental results

Research questions

• RQ1 Can the twistor space of the universal hyperkähler implosion for $\mathrm{SU}(n)$ be described as a holomorphic subvariety of a vector bundle over $\mathbb{P}^1$?
• RQ2 How can the hyperkähler implosion $Q$ be embedded equivariantly into a complex affine space $\mathcal{R}$ using representation theory and moment maps?
• RQ3 What is the role of the hypertoric variety associated to root planes in the maximal torus in reconstructing the implosion $Q$?
• RQ4 Can the construction of $Q$ as the closure of the $K_{\mathbb{C}}$-sweep of a hypertoric variety in $\mathcal{R}$ be generalized to other compact Lie groups?
• RQ5 How does the $\mathrm{SU}(2)$-action rotating complex structures on $Q$ interact with the embedding into $\mathcal{R}$?

Key findings

• The map $\sigma: Q \to \mathcal{R}$ is injective and $K\times T\times\mathrm{SU}(2)$-equivariant, providing a concrete realization of the universal hyperkähler implosion in a complex affine space.
• The twistor space $\mathcal{Z}_Q$ admits a generically injective holomorphic map to a vector bundle over $\mathbb{P}^1$, constructed via the embedding $\sigma$ and the $\mathrm{SU}(2)$-rotation of complex structures.
• For $n=2$ and $n=3$, the maps from $Q$ and $\mathcal{Z}_Q$ are explicitly described, confirming the general construction in low-rank cases.
• The image of $Q$ in $\mathcal{R}$ is the closure of the $K_{\mathbb{C}}$-sweep of the image of the hypertoric variety associated to the root planes in $\mathfrak{t}$, establishing a geometric reconstruction of $Q$.
• The construction relies on the fundamental irreducible representations of $\mathrm{SU}(n)$ and the hyperplane arrangement of root planes in the Lie algebra of the maximal torus, suggesting a generalization to other compact groups.
• The paper provides a framework for extending hyperkähler implosion to arbitrary compact Lie groups by embedding the universal implosion into a $K_{\mathbb{C}}$-representation space $\mathcal{R}$ and analyzing its twistor space.