nLab
Calabi-Penrose fibration

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Complex geometry

Contents

Idea

The twistor fibration P 3S 4\mathbb{C}P^3 \to S^4 (Atiyah 79, Sec. III.1, see also Bryant 82, ArmstrongSalamon14, ABS 19), also called, in its coset space-version SO(5)/U(2)SO(5)/SO(4)SO(5)/U(2) \to SO(5)/SO(4), the Calabi-Penrose fibration (apparently starting with Lawson 85, Sec. 3, see also, e.g., Loo 89, Seade-Verjovsky 03, 3 for this usage, see Nordstrom 08, Lemma 2.31 for review of Calabi’s construction Calabi 67, Calabi 68) is a fiber bundle-structure on complex projective 3-space over the 4-sphere with 2-sphere (Riemann sphere) fibers:

P 1 P 3 p S 4 \array{ \mathbb{C}P^1 &\longrightarrow& \mathbb{C}P^3 \\ && \big\downarrow^{\mathrlap{p}} \\ && S^4 }

If one identifies the 4-sphere as the quaternionic projective line S 4P 1S^4 \simeq \mathbb{H}P^1, then the fibration pp here is given by sending complex lines to the quaternionic lines which they span (Atiyah 79, III (1.1), see also Seade-Verjovsky 03, p. 198):

P 3 p P 1 {xz|z} {xq|q}, \array{ \mathbb{C}P^3 &\overset{p}{\longrightarrow}& \mathbb{H}P^1 \\ \{x \cdot z \vert z \in \mathbb{C}\} &\mapsto& \{x \cdot q \vert q \in \mathbb{H}\} } \,,

for any x 4 2x \in \mathbb{C}^4 \simeq_{\mathbb{R}} \mathbb{H}^2.

Generalizations

It is possible to define a twistor fibration over each S 2nS^{2n}, where the resulting manifold is a complex manifold endowed with a holomorphic? nn-plane field transverse to the fibers of the mapping. Namely, writing S 2n=SO(2n+1)/SO(2n)S^{2n}=SO(2n+1)/SO(2n), then, using the inclusion U(n)SO(2n)U(n) \subset SO(2n), one has the coset fibration

Z n=SO(2n+1)/U(n)SO(2n+1)/SO(2n).Z_n=SO(2n+1)/U(n) \to SO(2n+1)/SO(2n).

The manifold Z nZ_n canonically has the structure of a complex manifold and is known as the twistor space of S 2nS^{2n}.

There are generalizations of this picture for each of the so-called ‘inner’ symmetric spaces G/KG/K where KK is the fixed subgroup of an involution that is an inner automorphism of GG. The twistor fibration is of the form G/UG/KG/U \to G/K where UKU \subset K is a subgroup such that K/UK/U (the typical fiber of the fibration) is an Hermitian symmetric space. There are also other kinds of twistor spaces over G/KG/K that are flag manifolds of the form G/TG/T where TKT \subset K is a maximal torus. (For these generalizations see Bryant 85.)

References

In noncommutative geometry:

  • Simon Brain, Giovanni Landi, Differential and Twistor Geometry of the Quantum Hopf Fibration, Commun. Math. Phys. 315 (2012):489-530 (arXiv:1103.0419)

In higher dimensions:

Over P 3\mathbb{H}P^3:

Last revised on November 21, 2020 at 14:32:09. See the history of this page for a list of all contributions to it.