Hyperkähler Nahm transform

Holomorphic Lagrangian fibrations of toric hyperkahler manifolds

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk{ä}hler moment map. We use the complex part of this moment map to construct a holomorphic Lagrangian fibration with generic fiber diffeomorphic to $(\mathbb{C}^*)^n$, and study the singular fibers.

Rozansky-Witten invariants of hyperkähler manifolds

We investigate invariants of compact hyperk{ä}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifold, arriving at a number. For certain graph homology classes, the invariants we get are Chern numbers, and in fact all characteristic numbers arise in this way. We use relations in graph homology to study and compare these hyperk{ä}hler manifold invariants. For example, we show t…

Lie groups, Nahm's equations and hyperkaehler manifolds

Lectures given at the summer school on Algebraic Groups, Goettingen, June 27 - July 15 2005

Cotangent bundles of toric varieties and coverings of toric hyperkähler manifolds

Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.

A hyperholomorphic line bundle on certain hyperkähler manifolds not admitting an $S^1$-symmetry

Generalizing work of Haydys and Hitchin, we prove the existence of a hyperholomorphic line bundle on certain hyperkähler manifolds that do not necessarily admit an $S^1$ action. As examples, we consider the moduli space of (non-strongly) parabolic Higgs bundles, the moduli space of solutions to Nahm's equations, and Nakajima quiver varieties.

Nahm-Kirchhoff 模空间、超凯勒商与二维场论

Nahm-Kirchhoff Moduli Spaces, Hyperkähler Quotient and 2D Field theories

本文为带边界箭图构造 Nahm-Kirchhoff 模空间，证明其超凯勒结构并等同于复辛商，进而建立二维场论与度量场论，揭示超凯勒细化对边长的依赖及退化行为。