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Hyperkaehler Marriage of the two sphere with the hyperbolic space

Alice Barbora Tumpach

TL;DR

This work analyzes the hyperkähler extension of the 2-sphere, identifying $T^*\mathbb{CP}(1)$ with a complex coadjoint orbit $\mathcal{O}_x^{\mathbb{C}}$ of $SL(2,\mathbb{C})$ and exhibiting a fibration over $\mathbb{CP}(1)\simeq\mathbb{S}^2$ with fiber $\mathbb{H}^2$. By employing Mostow decomposition, the authors construct a $G$-equivariant diffeomorphism $\Phi: T\mathbb{CP}^1\to\mathcal{O}_x^{\mathbb{C}}$ and show that the hyperkähler complex structure $I_2$, extending the compact-type complex structure, preserves the hyperbolic fiber but does not coincide with the natural complex structure on $\mathbb{H}^2\subset\mathbb{C}$. The paper provides explicit parameterizations and computational formulas for the non-compact orbit, details the fibration, and demonstrates that the induced complex structure on $\mathbb{H}^2$ is not holomorphic with respect to its standard embedding, clarifying subtle aspects of hyperkähler geometry in coadjoint-orbit settings. These results advance understanding of how hyperkähler structures interact with complexifications of Hermitian-symmetric spaces and their real-analytic embeddings, with explicit geometric and representation-theoretic consequences.

Abstract

The Eguchi-Hanson metric is a natural metric on the total space of the cotangent bundle $T^*\mathbb{CP}(1)$ of the complex projective line $\mathbb{CP}(1) \simeq \mathbb{S}^2$, which extends the Fubini-Study metric of $\mathbb{CP}(1)$. By virtue of the Mostow decomposition theorem, $T^*\mathbb{CP}(1)$ is isomorphic, as $SU(2)$-equivariant fiber bundle over $\mathbb{CP}(1)$, to a complex (co-)adjoint orbit of $SL(2, \mathbb{C})$. In fact, this complex (co-)adjoint orbit is fibered over $\mathbb{CP}(1)\simeq \mathbb{S}^2$ with each fiber isomorphic to the hyperbolic disc $\mathbb{H}^2$. In this paper, we are interested in the complex structure inherited on the hyperbolic disc $\mathbb{H}^2$ by the hyperkähler extension of the $2$-sphere. Contrary to what is generally believed, we show that it differs from the natural complex structure of $\mathbb{H}^2\subset \mathbb{C}$ inherited from its embedding in $\mathbb{C}$. In other words, the embedding of $\mathbb{H}^2$ with its Hermitian-symmetric structure into the hyperkähler manifold $T^*\mathbb{CP}(1)$ is not holomorphic.

Hyperkaehler Marriage of the two sphere with the hyperbolic space

TL;DR

This work analyzes the hyperkähler extension of the 2-sphere, identifying with a complex coadjoint orbit of and exhibiting a fibration over with fiber . By employing Mostow decomposition, the authors construct a -equivariant diffeomorphism and show that the hyperkähler complex structure , extending the compact-type complex structure, preserves the hyperbolic fiber but does not coincide with the natural complex structure on . The paper provides explicit parameterizations and computational formulas for the non-compact orbit, details the fibration, and demonstrates that the induced complex structure on is not holomorphic with respect to its standard embedding, clarifying subtle aspects of hyperkähler geometry in coadjoint-orbit settings. These results advance understanding of how hyperkähler structures interact with complexifications of Hermitian-symmetric spaces and their real-analytic embeddings, with explicit geometric and representation-theoretic consequences.

Abstract

The Eguchi-Hanson metric is a natural metric on the total space of the cotangent bundle of the complex projective line , which extends the Fubini-Study metric of . By virtue of the Mostow decomposition theorem, is isomorphic, as -equivariant fiber bundle over , to a complex (co-)adjoint orbit of . In fact, this complex (co-)adjoint orbit is fibered over with each fiber isomorphic to the hyperbolic disc . In this paper, we are interested in the complex structure inherited on the hyperbolic disc by the hyperkähler extension of the -sphere. Contrary to what is generally believed, we show that it differs from the natural complex structure of inherited from its embedding in . In other words, the embedding of with its Hermitian-symmetric structure into the hyperkähler manifold is not holomorphic.
Paper Structure (18 sections, 6 theorems, 30 equations, 1 figure)

This paper contains 18 sections, 6 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definition of hyperkähler metrics
  3. Hyperkähler metrics on cotangent spaces
  4. Isomorphism between cotangent spaces and complex coadjoint orbits
  5. Fibre bundle structure of complex coadjoint orbits
  6. Hyperkähler structures on complexifications of Hermitian-symmetric orbits of compact type
  7. Contributions
  8. Summary:
  9. Key points:
  10. The $2$-sphere and the hyperbolic disc are living in the same coadjoint orbit of $SL(2, \mathbb{C})$
  11. Presentation of the protagonists and their relatives
  12. Compact (co-)adjoint orbit $\mathcal{O}_x$, complex (co-)adjoint orbit $\mathcal{O}^\mathbb{C}_x$ and non-compact (co-)adjoint orbit $\mathcal{O}^{n.c.}_x$
  13. Explicit Mostow Decomposition of $SL(2, \mathbb{C})$
  14. Diffeomorphism between $T\mathbb{CP}^1$ and $\mathcal{O}^\mathbb{C}_x$
  15. The hyperbolic disc $\mathbb{H}^2$ inside $T\mathbb{CP}^1 \simeq \mathcal{O}^\mathbb{C}_x$
  16. ...and 3 more sections

Key Result

theorem thmcountertheorem

There exists a $G$-equivariant projection $\pi: \mathcal{O}^\mathbb{C}_x \twoheadrightarrow \mathcal{O}_x$ and a $G$-equivariant diffeomorphism $\Phi: T\mathcal{O}_x \rightarrow \mathcal{O}_x^{\mathbb{C}}$ which commutes with the projections $p$ and $\pi$.

Figures (1)

  • Figure 1: Representation of the complex structure induced on $\mathbb{H}^2$ by the complex structure $I_2$ of the hyperkähler coadjoint orbit $\mathcal{O}_x^\mathbb{C}\simeq T^*\mathbb{CP}(1)$. Each ellipse represents the complex structure $I_2$ at the center of the ellipse. The complex structure maps the tangent vector given at the center of each ellipse by half of one axis to the tangent vector given by half of the other axis.

Theorems & Definitions (10)

  • theorem thmcountertheorem: Tum1
  • theorem thmcountertheorem: BG1BG2BG3
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof