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Hyperkähler structures on leaves of hyper-Lie Poisson manifolds

Dadi Ni, Kaichuan Qi

TL;DR

This work investigates hyperkähler structures arising on leaves of the hyper-Lie Poisson manifold built from $\mathfrak{su}(2)^3_{\star}$. By classifying leaves as $N_{q,r}$ and analyzing their curvature, the authors establish a tripartite type taxonomy: type-0 leaves are flat, type-1 leaves have bounded curvature, and type-2 leaves have unbounded curvature. They demonstrate that type-0 and type-1 leaves reproduce Kronheimer’s hyperkähler structures on nilpotent and regular adjoint orbits of $\mathfrak{sl}(2,\mathbb{C})$, while type-2 leaves yield genuinely new hyperkähler manifolds not captured by Kronheimer’s moduli spaces. The results deepen the link between hyper-Lie Poisson geometry and gauge-theoretic moduli, and reveal a new family of hyperkähler geometries associated with $\mathfrak{su}(2)$-cubic setups. The approach combines explicit leaf descriptions, curvature computations, and a comparison with Nahm-equation moduli to map out where Kronheimer’s construction fits within Xu’s framework and where new structures arise.

Abstract

Due to its rich structure and close connection with gauge theory, hyperkähler manifolds have attracted increasing interest. Using infinite dimensional hyperkähler reduction, Kronheimer proved that certain adjoint orbits of complexified semisimple Lie algebras admits hyperkähler structures. Later on, Xu obtained a proof for the existence of hyperkähler structures on adjoint orbits of $\mathfrak{sl}(2,\mathbb{C})$ from the viewpoint of symplectic geometry. This paper aims to thoroughly investigate and elucidate the key differences as well as the underlying connections between two distinct construction methods.

Hyperkähler structures on leaves of hyper-Lie Poisson manifolds

TL;DR

This work investigates hyperkähler structures arising on leaves of the hyper-Lie Poisson manifold built from . By classifying leaves as and analyzing their curvature, the authors establish a tripartite type taxonomy: type-0 leaves are flat, type-1 leaves have bounded curvature, and type-2 leaves have unbounded curvature. They demonstrate that type-0 and type-1 leaves reproduce Kronheimer’s hyperkähler structures on nilpotent and regular adjoint orbits of , while type-2 leaves yield genuinely new hyperkähler manifolds not captured by Kronheimer’s moduli spaces. The results deepen the link between hyper-Lie Poisson geometry and gauge-theoretic moduli, and reveal a new family of hyperkähler geometries associated with -cubic setups. The approach combines explicit leaf descriptions, curvature computations, and a comparison with Nahm-equation moduli to map out where Kronheimer’s construction fits within Xu’s framework and where new structures arise.

Abstract

Due to its rich structure and close connection with gauge theory, hyperkähler manifolds have attracted increasing interest. Using infinite dimensional hyperkähler reduction, Kronheimer proved that certain adjoint orbits of complexified semisimple Lie algebras admits hyperkähler structures. Later on, Xu obtained a proof for the existence of hyperkähler structures on adjoint orbits of from the viewpoint of symplectic geometry. This paper aims to thoroughly investigate and elucidate the key differences as well as the underlying connections between two distinct construction methods.

Paper Structure

This paper contains 15 sections, 40 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Hyper-Lie Poisson structures
  3. Hyperkähler manifolds
  4. General theory of hyper-Lie Poisson structures
  5. The $\mathfrak{su}(2)$-case
  6. Leaves of the Hyper-Lie Poisson structure on $\mathfrak{su}(2)^3_{\star}$
  7. Description of leaves
  8. Hypersymplectic structures on leaves
  9. Sectional curvature of leaves, $N_{q,r}$
  10. Riemann curvature tensor on $N_{q,r}$
  11. Proof of Theorem \ref{['thm:firstmainresult']}
  12. Comparison with Kronheimer's constructions
  13. Identification of leaves of $\mathfrak{su}(2)^3_{\star}$ and adjoint orbits
  14. Kronheimer's work
  15. The main result

Key Result

Theorem 1.1

Hyperkähler metric on symplectic leaves of the hyper-Lie Poisson manifold $(\mathfrak{su}(2)^3_{\star}; \pi_1,\pi_2,\pi_3)$ of different types cannot be isomorphic as hyperkähler manifolds. Moreover, for any $t>0$, the diffeomorphism $\rho_{t}\colon N_{q,r}\rightarrow N_{qt,rt}, (a_1,a_2,a_3)\mapsto

Theorems & Definitions (68)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Theorem 2.7
  • proof
  • Corollary 2.8
  • ...and 58 more