Further Advances in Twistor Theory: Volume II: Integrable Systems, Conformal Geometry and Gravitation 0582004659, 9780582004658

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Titles in this series. A full list is available from the publisher 275 Green functions for second order

M G Garroni and J F Menaldi 276 Riemann waves and their applications M W Kalinowski and

Y A Abramovich, A K Kitover

278 Limit algebras: C*-algebras

an

E

operators

L

introduction

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and

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S C Power 279 and

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Emerging

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Free

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P

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Rasmussen

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Y Xu

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K-theory

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Multigrid

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316

vector

spaces,

algebras

and related

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317

porous media 319

methods

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319

Chipot

297 Nonstandard methods in the calculus of variations

C Tuckey 298 Barrelledness, Baire-like- and (LF)-spaces M Kunzinger Nonlinear 299 partial differential equations applications. Collège de France Seminar.

and

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321

their

322

Volume XI H Brezis and J L Lions

Topological areas

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J Bramble 295 Entropy and partial differential equations W A Day 296 Progress in partial differential equations: the M

Extending modules N V Dung, D van Huynh, P

R Wisbauer

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groups V Baldoni and M Picardello 312 Common zeros of polynomials in several variables and higher dimensional quadrature

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applications

V Kiryakova 302 Nonlinear partial differential equations and their applications. Collège de France Seminar Volume XII H Brezis and J L Lions

preserving

Arenson

request.

300 Introduction to operator theory T Yoshino 301 Generalized fractional calculus and

parabolic

integro-differential problems

277 Banach C(K)-modules disjointness

on

R Raghavan and E Ozkan Asymptotic theories for plates and shells R P Gilbert and K Hackl Nonlinear variational problems and partial differential equations A Marino and M K V Murthy Asymptotic theories for plates and shells R P Gilbert and K Hackl Nonlinear variational problems and partial differential equations A Marino and M K V Murthy Topics in abstract differential equations II S Zaidman Diffraction by wedges B Budaev

L J Mason St Peter's College and the Mathematical Institute, Oxford

LP Hughston

Merrill Lynch International, and King's College, London

P Z Kobak Scuola Internazionale Superiore di Studi Avanzati, Trieste (Editors)

Further advances in twistor theory Volume II: Integrable systems, conformal geometry and gravitation

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Contents

Chapter 1: Integrable and soluble systems II. 1.1 Introduction

by

II.1.2 Twistors and

SU(3) monopoles by

II.1.3

Monopoles

and

L.J.Mason A.Dancer 11

Yang-Baxter equations by M.F.Atiyah

II.1.4 A non-Hausdorff mini-twistor space

by

II.1.5 The 3-wave interaction from the self-dual II. 1.6 The

Bogomolny hierarchy

II.1.7

H-Space:

II.1.8

Integrable systems

II.1.9 Twistor

a

universal

higher

Yang

order

integrable system? by

integrability by

II.1.12 Harmonic II.1.13

II. 1.14

morphisms

More

Monopoles,

II.1.15 Twistor

theory

Chapter

2:

by

harmonic

and

correspondences

Applications

II.2.3 A theorem

split signature by

spinor

L.J.Mason 34

L.J.Mason 39

K.P.Tod 45

morphisms

by

K.P.

Tod 47

fields by P.Baird and J.C. Wood 49

by

62 M.G.Eastwood

between twistor spaces of Wolf spaces

to conformal

on

proof

II.2.7 A

simplified proof

of

description

conformally

a

by L.P.Hughston83

diagram’ by B.P.Jeffryes85

eight

dimensions

of Robinson’s theorem

II.2.6 A

75 L.J.Mason

by L.P.Hughston79

null fields in six dimensions

II.2.4 A six dimensional ‘Penrose

II.2.8 A twistor

geometry

by M.G.Eastwood, L.P.Hughston &

II.2.5 Null surfaces in six and

II.2.10 A

27 I.A.B.Strachan

66 P.Z.Kobak

II.2.1 Introduction

by

I.A.B. Strachan 20

Yang-Mills equations by

and mini-twistor space

II.2.2 Differential geometry in six dimensions

II.2.9 A

K.P.Tod 17

30 L.J.Mason

in

morphisms

equations by

and harmonic maps from Riemann surfaces

II.1.16 Contact birational

by

by

self-duality equations

on

harmonic

Mills

spectral problems by

II.1.10 On the symmetries of the reduced self-dual II.1.11 Global solutions of the

M.K.Murray13

23 L.J.Mason

and curved twistor spaces

and

theory

and

&

K.P.Tod 14

by L.P.Hughston87

by L.P.Hughston91

theorem of Sommers by L.P.Hughston93

of null self-dual Maxwell fields

by

96 M.G.Eastwood

invariant connection and the space of leaves of

T.N. Bailey 99

conformally

invariant connection

by T.N.Bailey106

a

shear free congruence

62

II.2.11 Relative

by

cohomology

series, Robinson’s Theorem and multipole expansions

T.N. Bailey 107

II.2.12 Preferred parameters

by

power

in conformal manifolds

on curves

T.N. Bailey & M.G.Eastwood 110

II.2.13 The Fefferman-Graham conformal invariant II.2.14 On the

weights

of

II.2.15 Tensor

products

conformally

invariant operators

of Verma modules and

II.2.16 Structure of the

112 M.G.Eastwood

by

by

M.G.Eastwood 114

conformally invariant

bundle for manifolds with conformal

jet

or

tensors

projective

by

R.J.Baston

structure

by A.R.Gover123 II.2.17

Exceptional

invariants

by

II.2.18 The conformal Einstein

127 A.R.Gover

equations by

II.2.19 Self-dual manifolds need not be

by T.N.Bailey &

3:

Chapter

II.3.3

cosmological

Cosmological

135 L.J.Mason

models

models in 𝕡5

by

by

on

conserved vectorial

by

R.Penrose138

142 T.R.Hurd

II.3.4 Curved space twistors and GHP II.3.5 A note

conformal to Einstein

general relativity

by L.P.Hughston &

II.3.1 Introduction II.3.2 Twistors for

locally

M.G.Eastwood 132

of

Aspects

131 L.J.Mason & R.J.Baston

by B.P.Jeffryes146

quantities

associated with the Kerr solution

L.P.Hughston148

II.3.6 Further remarks

on

conserved vectorial quantities associated with the Kerr solution

by L.P.Hughston152 II.3.7 Non-Hausdorff twistor spaces for Kerr and Schwarzschild II.3.8 More

on

the twistor

description

of the Kerr solution

II.3.9 An alternative form of the Ernst II.3.10

Light

II.3.11 Mass

by

rays

near

positivity

i0:

from

and the structure of

4:

problem

Quasi-local

in

general relativity by

by

R.Penrose163

space-like infinity

power series

quasi-local

momentum

K.P.Tod 177

theory

of 2-surface

by

V. Thomas174

mass

II.4.1 Introduction: two-surface twistors and

II.4.2 A

J.Fletcher157

potential by J.Fletcher160

mass-positivity theorem by

focussing

154 J.Fletcher

A.Ashtekar & R.Penrose169

II.3.12 The initial value

Chapter

a new

by

by

(‘superficial’)

twistors

by

R.Penrose180

& angular

momentum

120

II.4.3 The kinematic sequence II.4.4 Two-surface twistors

(revisited) by L.P.Hughston momentum flux and

angular

II.4.5 General-relativistic kinematics??

Spinors

II.4.9 The index of the 2-twistor of Pell’s

occurrence

Sparling 3-form,

equations by

equation

Symplectic geometry

by

II.4.15 ‘New

improved’ quasi-local

quasi-local

Quasi-local

mass

mass

by

W.T. Shaw 207

in twistor

theory by

K.P.Tod 212

general relativity

and

quasi-local

mass

II.4.18 Two-surface twistors for II.4.19 An

example

suggested

of

a

mass

Higher-dimensional Embedding

II.4.23

Asymptotically

224 by W.T.Shaw

and the Schwarzschild solution

Killing

by

R.Penrose232

by B.P.Jeffryes240

vectors

large spheres by

244 W.T.Shaw

two-surface twistor space with

II.4.22

220

229 K.P.Tod

further modification to the

II.4.21

II.4.24 Two-surface

by B.P.Jeffryes220

N.M.J.Woodhouse238

II.4.17 Two-surface twistors and

two-surface twistors

2-surfaces in 𝕔𝕄 anti-de Sitter

by

complex

quasi-local by

determinant

formula

by

by B.P.Jeffryes

R.Penrose251

R. Penrose252

R.Penrose255

space-times by R.Kelly257

pseudo-twistors by B.P.Jeffryes264

II.4.25 Two-surface twistors and

Index

204 M.G.Eastwood

of ℐ+ and 2-surface twistors

on

II.4.26 A

by

210 R.J.Baston

the Hamiltonian of

II.4.14 More

II.4.20 A

twistors

215 L.J.Mason

by

II.4.16

199 spacelike infinity by W.T.Shaw

quasi-local quantities by

II.4.12 Dual two-surface twistor space II.4.13

the Einstein-Maxwell

R.Penrose194

superficial

II.4.8 ‘Maximal’ twistors & local and

II.4.11 The

by

ZRM fields and twistors at

II.4.7 The ‘normal situation’ for

II.4.10 An

multipoles of

by W.T.Shaw 188 ℐ+

field at

II.4.6

& T.R.Hurd186

quasi-local

mass

hypersurface

construction with

twistors

by

R.Penrose266

positive energy by A.J.Dougan &

268 L.J.Mason

250

Editors and Contributors

Editors L.J.Mason , St. Peter’s

College

and the Mathematical Institute, Oxford.

e-mail: lmason @maths.ox.ac.uk

L.P.Hughston

,

.

Merrill Lynch International and

e-mail: lane @lonnds.ml.com

London.

Kings College,

.

P.Z.Kobak , Scuola Internazionale

Superiore

e-mail: [email protected]

di Studi

Avanzati,

Trieste.

.

Contributors A.Ashtekar , Center for Gravitational e-mail: [email protected]

.

Physics

and

Geometry, Pennsylvania State University,

M.F.Atiyah, Trinity College, Cambridge.

T.N.Bailey Department of Mathematics, University ,

P.Baird , School of

Mathematics, University

R.J.Baston , Paribas A.Dancer ,

A.J

Department

of

Capital

Applied

.Dougan ,

Edinburgh.

e-mail: [email protected]

.

of Leeds.

Markets, London.

Mathematics

The

of

and

Theoretical

Mathematical

Physics,

Cambridge.

Institute,

M.G.Eastwood , Department of Pure Mathematics, e-mail: [email protected] .

Oxford.

University

of

Adelaide.

J.Fletcher Fletcher & Partners, Salisbury, UK. ,

A.R.Gover ,

Department of Pure Mathematics, University

T.R.Hurd , Department of Mathematics e-mail: [email protected] .

B.P.Jeffryes ,

R.Kelly

Schlumberger

,

Cambridge

The Mathematical

M.K.Murray ,

R.Penrose ,

Department

Wadham

W.T.Shaw ,

and

Statistics,

Research

Ltd,

McMaster

Cambridge.

Institute, Oxford (but of

College

Pure

and

Oxford

see

e-mail:

note on

Mathematics,

the

of Adelaide.

University,

[email protected] .

p.257).

University

Mathematical

Systems

Ontario.

of

Institute,

Adelaide.

Oxford.

Solutions,

Oxford.

I.A.B.Strachan , Department of Mathematics and Statistics, University of Newcastle. e-mail: I. A. B. [email protected] . V.Thomas, The Mathematical Institute, Oxford. e-mail: [email protected] . K.P.Tod, St John's College and the Mathematical Institute,

Oxford.

J

.C.Wood ,

School

e-mail:

of

Mathematics,

[email protected]

University

of

N.M.J.Woodhouse , Wadham College e-mail: [email protected] .

Leeds.

and

e-mail:

the

.

[email protected] .

Mathematical

Institute,

Oxford.

Preface

It was in 1976 that a group of us at the Mathematical Institute, Oxford, began to circulate Twistor Newsletter, an informal publication consisting of short articles written mostly by members of Roger Penrose's research group, relating to active work going on in twistor theory. This was around the time of the publication of the original non-linear graviton construction. It has been said that the art of doing mathematics consists in finding that special case which contains all the germs of generality. The non-linear graviton was just such a 'special case', and as a consequence interest in the theory increased significantly, both among physicists ( especially relativists), but also among an increasing number of pure mathematicians. There was thus a small but steady demand from colleagues outside of Oxford and abroad for the informal communication of new ideas and the latest results, and Twistor Newsletter neatly fit the bill. It was no doubt an odd sort of journal, but it was successful in its own way, and after thirty-eight issues and eighteen years it continues to thrive.

By 1979 enough material had been stockpiled on this basis to warrant publication in a volume called Advances in Twistor Theory, edited by R.S.Ward and one of us (LPH). In it the collected articles were grouped into four broad (sometimes overlapping) categories: massless fields and sheaf cohomology, curved twistor spaces, twistors and elementary particles, and twistor diagrams. This proved to be a useful book, and so we were encouraged to gather together a further collection of subsequent Twistor Newletterarticles to edit for publication under the general title Further Advances in Twistor Theory. Volume I of this new series (which appeared in 1990) was called Applications of the Penrose Transform, and here we present Volume II, which is called Integrable Systems, Conformal Geometry, and Gravitation.

of

logically grouping

on

is

as

variety of related to basic

new

(let

follows. Volume I contains material of sheaf

higher

articles into volumes

improving

and

on

twistor contour

physics,

dimensional

directions;

cohomology as

well

analogues.

(whether

it be

alone

chapters!).

The scheme

primarily concerned,

in

one

we

way or

integral formulae.

as a

host of

loosely

are

properties

established

rigorously)

out and or

and

exploratory, launching

clarifying (and

conjectured.

in

some cases

a

applications to

out in

All of this material refers in

to the well known ‘Penrose

transform’, and thus

generic

a

characteristic feature of much of the analysis in Volume I is linearity. The contents include

or

applications

of

greatly

a

way

way

another, with

also various serious attempts at

speculative

straightening

already

are

simple

mathematically motivated interesting generalizations

Some of the articles

results

There

a

eventually settled

flat twistor space, and with the elucidation of the

whereas others set about

generalizing)

and it has not been easy to find

extensive,

The range of material in these volumes is

(1)

an

overview

(2)

(with

concrete

both mathematical

(i.e.

contour

approaches twistor

cohomology,

integral) approaches

to the Penrose

diagram theory

and

(Methods

For

transform, (4)

twistor

transform, (3)

theory

and

and

one

II,

contains

applications

studies non-linear structures

sometimes associate

space that

abstract

physics), (i.e. cohomological)

elementary particle physics, (5)

(6)

sources

and currents, relative

and non-Hausdorff twistor spaces.

when

example,

point.

to the Penrose

of flat twistor space to non-linear

that introduce ‘deformed’ twistor spaces will appear in volume

applied

one can

and the motivation for the ideas from basic

scattering amplitude evaluation,

The present work, Volume

be

background

local twistors

approximates

Another

example

a

flat twistor space in

are

most

lying a

either in

or on

natural way to

a

given

Flat twistor space

twistor space.

some

the flat space twistors associated to the

closely

III.)

problems. can

Alternatively,

otherwise non-linear

conformally flat

conformal manifold to second order at

object.

Minkowski some

given

arises with 2-surface twistors, which form the solution space of the linear

2-surface twistor equation. Chapter 1 contains articles on integrable or soluble non-linear equations. Here many of the non-linear constructions arise from the study of holomorphic vector bundles on twistor space, which give rise to solutions of the self-dual Yang-Mills equations by virtue of Ward's correspondence. This gives constructions for a wide variety of integrable equations. In fact it has emerged that most such equations arise as symmetry reductions of the self-dual Yang-Mills equations. Solutions of other non-linear equations can be shown to correspond to submanifolds of flat twistor space. Solutions of harmonic map equations and the equations for harmonic morphisms arise in this way.

Chapter 2 contains articles on conformal differential geometry. Here the linear twister spaces involved are the spinors in six dimensions. These are applied to the study of generalizations of the Kerr theorem and Robinson's theorem in six dimensions. Cartan's conformal connection in four dimensions has structure group SO( 4, 2). Its associated spin connection is the local twistor connection. This is used to study conformal invariants and invariant differential operators amongst other topics.

Chapter 3 contains articles on vanous aspects of general relativity. Flat space twisters are applied to cosmological models for which the underlying conformal structure is conformally flat. Space-times admitting solutions of the two-index twistor equation are also studied. In another range of applications, advantage is taken of the fact that the equations for space-times with two symmetries are equivalent to the self-dual Yang-Mills equations with two symmetries, so flat space twistor methods can be applied. We also include some 'exceptional' articles, viz., the Penrose-Ashtekar proof of the positive energy theorem and Thomas's study of the initial value problem for general relativity by exact sets and power series.

Finally, chapter This

uses

4 is concerned with the

2-surface twistors

as

essential

development

ingredients

in

of Penrose’s quasi-local a

mass

construction.

definition of the energy momentum and

angular momentum of the gravitational

and matter fields

threading through

a

space-like

two-surface

in space-time. The chapters start with introductions that give some of the background to the material in each chapter, and a summary of the contents of each chapter. We hope that this introductory material, together with that in volumes O and I, will help make these volumes relatively self contained, even for the non-expert.

Our warm thanks to Roger Penrose, Florence Tsou, their help in the production of this volume.

Debby

Morgan,

and

—L.J. Mason, L.P.

A note

on

global

article 1 in

chapter

preceeding

we mean

the

contributors

Hughston,

R.S.

Ward, editors, Pitman, 1979)

as

for

and P.Z. Kobak, November 1994.

structure and cross references. We refer to the

Theory (L.P. Hughston & volume

all

original

Volume

0,

Advances in Twistor

and

by ∮0.5.1

5 of that book. In the current Further Advances in Twistor the present book is Volume I:

article 3 of

chapter

the present work, Volume II:

2 of that book.

Applications of

By ∮∮II.4.5-8

the Penrose

we mean

theory series,

the

Transform. By ∮I.2.3

articles 5 to 8 of

Integrable Systems, Conformal Geometry,

we mean

chapter

and Gravitation.

4 of

Chapter

1

and soluble

Integrable

systems

§II.1.1 Introduction by L.J.Mason Nonlinear differential

analysis

of nonlinear

equations.

The most

differential

partial

correspondence

There

equations.

where the solutions of certain nonlinear one

impressive applications of twistor theory

partial

differential

with deformations of twistor spaces

constructions arises from the fact that the twistor For local solutions the twistor

original equations.

functions, and in general it

original equations

surfaces in R3 in terms of

free

a

with self-dual

Weyl

correspondence bundles

on

These

analogues

are

described in and

which

§§II.1.2

mostly based

-

11

on

also three articles

corresponds

In today’s

applications

§§II.1.12

-

spinor fields,

analyzes

description

to a solution of the

parlance we

would say that this free

holomorphic tangent

started with Penrose’s nonlinear

are

Yang-Mills equations

concerned with

integrable

Ward’s correspondence

on

§§II.1.7

-

them

.

using

given their

and

analogous

holomorphic

vector

in

systems and their twistor constructions.

(although

there

are some

8 ) and will be described in

harmonic morphisms

morphism

13 A detailed

is

graviton

is established between Ricci flat metrics

correspondence

to the intersection of a

harmonic

a

of these

be realized in terms of free

in minitwistor space, the

curve

construction noted in

graviton

ℙℕ. The basic idea of are

a

utility

tractable than the

is Weierstrass’ construction of minimal

( §§II.1.12

of the Kerr theorem. The Kerr theorem states that

Minkowski space

usually

can

more

to

(Ward 1977).

in twistor space

are

with the nonlinear

usually

are

one

equations directly.

between solutions of the self-dual

regions

There

in which

be shown to be in

curvature and deformations of twistor space. R.S. Ward then found an

In this chapter, articles

later.

Modern

can

structures thereon. The

description

general spirit

holomorphic

a

sphere.

construction, (Penrose 1976),

or

descriptions

holomorphic function.

function describes

bundle of the Riemann

number of twistor constructions

equations

be easier to pass from the twistor

can

than to solve these nonlinear

Probably the first construction in this

holomorphic

are now a

arise in the

a

-

geodesic

14 ),

which

connections more

are

detail

Euclidean

shear free congruence in

holomorphic hypersurface

in twistor space with

and its connections with the standard Kerr theorem

analysis of their properties, including connections

§II.1.14 Finally .

complexification

there is

and

an

an

article

article on

on

harmonic maps

minimal

spheres

in

with

monopoles

( §II.1.15 )

(symmetric)

Riemannian manifolds using generalizations of the Weierstrass construction ( §II.1.16 ).

DOI: 10.1201/9780429332548-1

1.

Integrable

and soluble systems

Twistor theory and with

a

circle of ideas

originating

Yang-Mills equations

in the

of nonlinear differential obtain

can

large

and leads to

(1987),

developed

concerning

equations that, despite

emerged

over a

their

nonlinearity,

theory

number of years

by

of

(1992).

There is

and soluble systems

are

still

book in

concerned

information about

general

Ward

following

an

outline of this

new

overview

Most of the technical details

self-duality.

also Ward 1977, Ward

&

Wells

Yang-Mills equations.

work with ‘low

technology’

algebra

of

classes of

(1985, 1986),

Hitchin

and in various lectures

preparation

on

this

topic by

by

Mason &

are

on

integrable systems arising from

omitted but there

are

ℝ4,

connection

some

It is convenient, when

sufficient references

DDa—=d∂

fixed group G

§1.4.0 (see

discussing integrable systems,

versions of the Ward transform. The self-dual

with coordinates xa,

on a

twistor

1989).

The self-dual

equations

one

(1995).

in the

on

so

This overview

to fill in the gaps. Further details of the basic Ward construction itself can be found in

defined

systems

paradigm for integrability,

a

including

are

relatively tractable,

Sparling (1989, 1992), a

are

completely integrable systems.

many workers

See also Witten

chapter

‘universal’ role for the self-dual

is that the twistor construction constitutes

Mason &

and

possible

precise analytic

(1988),

give

a

integrable systems. Integrable

Woodhouse & Mason

Woodhouse

theory

of

theory

substantial unification of the

a

M.F.Atiyah. I

with R.S. Ward

families of exact solutions and

solutions. What has

has been

The first few articles in this

integrable systems.

a —

=

0,1,2,3, and metric ds2

Aa where d∂

(usually

a

=

=

d∂d∂a and Aa

Mills

to

equations

are

dx0■·dxx + dx1 •·dx2. They

are

—

finite dimensional group of

Yang

Aa(xb)

£∈llwhere llis the Lie

matrices).

The connection Aa

is defined modulo the gauge freedom: a

The self-dual

Yang-Mills equations

[D0, D2] which

are

the

compatibility

are

=

[D1, D2]

conditions

[D0

[D0 D3]

=

+

[D1, D2]

+ 𻯑, D2 + λD3]

=

=

0

0 for the linear system of

equations

left-parenthesis upper D 0 plus lamda times upper D 1 right-parenthesis times psi equals left-parenthesis upper D 2 plus lamda times upper D 3 right-parenthesis times psi equals 0

(1) where

λ ∈ ℂ

and ψ

is

a

vector in some

The Ward construction of solutions of the self-dual In ‘low

provides

to one

Yang-Mills equations

technology’ language,

self-dual

representation

a one

Yang-Mills equations

one as

goes from

follows.

a

of l.

correspondence and

between gauge

holomorphic vector

bundle

on

equivalence

bundles

dual twistor space to

on a

regions

classes in ℙ𝕋*.

solution of the

§11.1.1 Choose local coordinates and ω2 for the

=

x3

—

region

λx2. One needs at least

near

λ

=

bundles have

a

Čech description

valued

‘patching’

or

! 9;𝕋

that the incidence relations

so

coordinate chart,

one more

large enough region

to cover a

corresponding

can

the bundle

‘clutching’

effectively

local solution

on

to the

point

at

variety of different representations; the

In the

function is

∞

of the line

complement

(λ,ω1,ω2)

can

be obtained from such

a

=

x1

—

case) by

covered is the vector

‘Čech’ and ‘Dolbeault’.

holomorphic SL(N, ℂ)

a

away from λ

=

0. The

patching

to gauge freedom. The

subject

λx0

(1/λ, ω1/λ, ω2/λ)

space-time). Holomorphic

this

(in

=

λ1

(the region

most basic are labelled

function but is

general

patching matrix. patching function P,

differentiation of the solution

by

in

P(λ, ω1, ω2) holomorphic

function

When the bundle is described in terms of a is reconstructed

in twistor space

infinity

be characterized

freely prescribable

a

(λ', ω'1, w'2)

are

Introduction

G±

of

a

the solution

Aa(xb)

on

space-time

parametrized Riemann-Hilbert problem:

up er G plus left-parenthesis x Superscript a Baseline comma lamda right-parenthesis equals up er G Subscript minus Baseline left-parenthesis x Superscript a Baseline comma lamda right-parenthesis up er P left-parenthesis lamda times comma x Superscript 1 Baseline minus lamda times x Superscript 0 Baseline comma x cubed minus lamda times x squared right-parenthesis period

(2) Here

G+

is

nonsingular

Note that the values in a

on

patching

SL(n, ℂ)

|λ|

> 0

including

function P is

at the

point λ

=

∞, and G_ is nonsingular

defined for ∞> |λ|

only

the Birkhoff factorization theorem tells

us

> 0.

Given

at least

that,

on

codimension-one subset of 𝕄, there exist solutions to the above factorization

unique

up to

premultiplication by

The self-dual that G± is

a

Yang-Mills

matrix function of xa alone

by

the

on

generic

|λ|

< ∞.

P with

complement

problem. They

of

are

virtue of Liouville’s theorem.

attempting

to find a connection

Aa such

solution of the linear system

(D0 We find that

a

connection is reconstructed by

a

we

+

λD1)G±

(D2

=

+ λD3)G±

=

0.

would have to have up er A 0 plus lamda times up er A 1 equals StartSet left-parenthesis partial-dif erential plus lamda partial-dif erential right-parenthesis times up er G Subscript plus-or-minus Baseline EndSet up er G Subscript plus-or-minus Superscript negative 1 Baseline comma

with

a

similar formula for A2 + λA3. It turns out that this

read off A0 and A1. The

consistency

equation

is consistent and allows

us

to

follows because

StartSet left-parenthesi partial-dif erential plus lamda partial-dif erential right-parenthesi times up er G Subscript minus Baseline EndSet up er G Subscript minus Superscript negative 1 Baseline equals StartSet left-parenthesi partial-dif erential plus lamda partial-dif erential right-parenthesi times up er G Subscript plus Baseline EndSet up er G Subscript plus Superscript negative 1

which is

a

consequence of

and A1 because the and the

expression

equation (2)

expression with

G+

with G_

has

a

and the fact that

implies

simple pole

extension of Liouville’s theorem, be linear in

satisfy

the self-dual

Yang-Mills equations

as

that the

at λ.

λ

=

right

∞, so

The

there

(∂0 + λ∂1)P

are

=

hand side is

the whole

corresponding

0. One

can

holomorphic

read off A0 for

expression must, by

connection must

solutions, G_ and

G+

to

|λ|


1 Phys. Lett. ,

,

,

-

.

Baxter , R. J. , Perk , J. H. H. and Au-Yang , H. ( 1988 ) New solutions of the the Chiral Potts model , Phys. Lett. A 128 (3,4 ), p. 138 - 142

star-triangle

relations for

.

§II.1.4 A Non-Hausdorff Mini-twistor Space by

K.P.Tod

(TN

This note is about another

complex

manifold

theory.

A mini-twistor space

Weyl satisfies the

example

space

of

funny a

Ж

non-Hausdorff

ways,

a

2-complex-dimensional

condition. Since it is defined a

mini-twistor space is

always

particularly simple Einstein-Weyl space

to be non-Hausdorff in a

fairly

tame way.

as a

space of

1990)

arising naturally

is the 4-real-dimensional space of directed

III, which becomes

Einstein-Weyl

wind around in an

example of a

30, June

geodesics

manifold if the

geodesics,

and

in twistor

of

a

Weyl

3-real-dimensional space

geodesics

can

liable to be non-Hausdorff. I will describe where the mini-twistor space

can

be

seen

Recall first that metric

[g]

a

Weyl

space III is

which is preserved

by

D.

a

manifold with

Given

a

between conformal metric and connection connection and

1-form ωa. Under

a

symmetric

a

choice gab of that

means

change-of-choice

connection D and

representative metric,

a

the

conformal

compatibility

define D in terms of the metric

we can

of representative metric

we

have

g Subscript a b Baseline right-ar ow normal upper Omega squared g Subscript a b Baseline times semicolon omega Subscript a Baseline right-ar ow omega Subscript a Baseline plus 2 nabla log normal upper Omega

(1) so

that

we can

e.g. Hitchin

think of

(1980),

Jones

The connection D has

symmetric. to the

The

Tod

& a

as

space

(1985)

the pair

metric.

condition

This

(gab,ωa) subject

and Pedersen

Riemann tensor and

Einstein-Weyl

(conformal)

representative

Weyl

a

a

Tod

&

For

.

more

details

see

(1993).

Ricci tensor, but the Ricci tensor is not

necessarily

III is that the symmetrised Ricci tensor be proportional

on

be written out

an

equation

metric and the 1-form ωa. In 3 dimensions the

equation

can

(1)

to

as

on

the Ricci tensor of the

is

up er R Subscript a b Baseline minus one-half nabla Subscript left-parenthesis a Baseline omega Subscript normal b right-parenthesis Baseline minus one-fourth omega Subscript a Baseline times omega Subscript b Baseline equals normal up er Lamda times g Subscript a b Baseline times comma some normal up er Lamda period

(2) This

equation is,

from its

generalisation satisfy (2)

since

of the Einstein

(1)

we can use

and

definition, conformally invariant

be

can

regarded

as a

conformally-invariant

Note that spaces conformal to Einstein spaces

equations.

examples

to eliminate ωa. These

can

be

recognised by

the fact that

ωa is exact.

The on

example I

want to consider comes about

flat space. Take the metric and 1-form

and

making identifications

S1 × S2

(this example is given

by conformally rescaling

as

a

and

conformally

rescale with

Ω

g =

Now

impose a periodicity

in Pedersen & Tod The

periodictity

in

(1993);

in X

exp(— ), defining

=

d

2

+

dθ2

to obtain an

+ sin2 θ

=

log r

:

dϕ2; ω=-2d

Einstein-Weyl structure

on

part of the interest of it is that this manifold has

corresponds

to

identifying

the radial coordinate

r

no

Einstein

with λr for

metric).

some

λ with

0 < λ < 1. As I said at the III is

a

beginning,

2-dimensional

the space of directed

complex

goedesics

space is the space of directed lines in ℝ3 which

real vectors

(a, b)

where

a

of

a

3-dimensional Einstein-Weyl

space

manifold Ж, the mini-twistor space of III. For flat space, the minitwistor

is unit and b is

orthogonal

can

to

a.

be

thought

of

Equivalently,

as

pairs

of 3-dimensional

this is T
;1, the tangent

bundle of the complex this

For the example to be considered here

geodesic

in the S1 × S2

Einstein-Weyl

line which, when it hits the outer

making the same angle which is

limiting one

radially

inwards and closed. In

a

point

points

through p (I example).

am

r

grateful

through

this

means

that in the

that there

future,

modify

are

at

r

=

λ

tends to

a

which

‘shadows’ in the space:

given

a

might

by geodesics

be shadows in this

We shall return to these shadows below.

two closed radial

geodesics

Next the non-radial

bringing a

geodesic

straight

limiting one

that there

suggestion

to the zero-section of T
;1, ie. to lines in ℝ3 defined

leaving

the

a

sphere

the other side cannot be reached

on

to Paul Gauduchon for the

basically

to the inner

To construct the mini-twistor space Ж, consider first the closed radial

of

shall need to

1. It is

while in the past is tends to

means

p but

diag.

1, is brought back

=

closed,

particular,

the diameter

on

at

outwards and

radially p,

sphere

structure is as shown in

with the radius vector. This

is

are

we

little.

a

A

projective line.

for each radial

geodesics:

this back from the outer

alone but

rescaling b,

ie. b

geodesic

think of

sphere λb,

a

in

line in

with λ

as

flat-space sphere

so we

as a

figure

at the

doubled-up-zero-section,

since any

geodesic

‘near to’ the continuation of it to the other side A point p in the

corresponds

Einstein-Weyl

in the mini-twistor space. The

space is

specification

to

before.

which is ‘near to’

as a

zerosection.

Then the process

above

identify

b with λb in the

of the zero-section back. It is non-Hausdorff at the radial

then put two

copies

need to double the

pair (a, b).

in the

This is then the mini-twistor space: delete the zero-section from T&#x;1;

fibres;

These correspond

by pairs of the form (α,0), but there

flat-space

to the inner

geodesics.

a

radially outgoing

represented by

a

radially ingoing one

ie.

is also

one.

holomorphic

of this twistor line

geodesics,

includes,

curve

at some

(a

‘twistor

stage,

a

line’)

choice of

which of

pair

of

pair of doubled-up points

a

to take. Then any twistor line

through the

in the mini-twistor space will correspond to

doubled-up points

a

other of the relevant

point of the Einstein-Weyl

space in the ‘shadow’ of p. A

complicated example

more

‘Berger sphere’ Einstein-Weyl to ones

in the

example

with non-Hausdorff-ness of it

as a

non-Haudsorff mini-twistor

on

the

3-sphere.

(1985)

along

generators of the

‘weighted projective space’

a

special

was

done

a

set of

goedesic

seems

provided by

geodesics

tends to

one

the

This corresponds

(1993).

like the radial of them in the

to be a sort of deformed

quadric

Henrik Pedersen and I have

family.

same

but it is

space should be

and Pedersen & Tod

There is

The mini-twistor space

two

Like my article II.1.13 , the work for this in Odense, and I

Tod

above with the property that any other

future and another in the past.

description

a

space, Jones &

left-invariant metric

a

of

a

little obscure. visit to Henrik Pedersen

during a most pleasant received.

gratefully acknowledge hospitality

References

( 1980 ) in proceedings of Twistor geometry and non-linear systems, Primorsko Bulgaria, 1980 , eds. Doebner and Palev (Springer Lecture Notes in Mathematics 970).

Hitchin , N.

Jones , P. E. & Tod , K. P.

( 1985 )

Pedersen , H. & Tod , K. P. pp. 74 109

Class.

( 1993 )

Quant. Grav.

2 , pp. 565 577 -

Three-dimensional

.

Einstein-Weyl geometry

,

Adv. in Math. 97 ,

-

§II.1.5

(TN

.

The 3-Wave Interaction from the Self-dual

a

& Segur

1981). According

self-dual

Yang-Mills equations.

a

Mills

by

Equations

K.P.Tod

33, November 1991)

There is

them

Yang

of

completely integrable systems to current twistor

this route. I also found,

by

slightly The

family

While

though

different route with similar

starting point

equivalent

trying

called ‘the

dogma, to do

these

n-wave

equations

interaction’

(see

eg Ablowitz

should be reductions of the

something different,

I found

somewhat later, that Chakravarty

&

a

way of

Ablowitz

getting

(1990)

had

end-points.

is the self-dual

Yang-Mills equations

with 2 null

symmetries. aThres

to the commutation relation left-bracket upper D 1 comma upper D 2 right-bracket equals 0

(1)

where up er D 1 equals partial-dif erential minus up er A 1 plus zeta up er B 1 up er D 2 equals partial-dif erential minus up er A 2 plus zeta up er B 2 times comma

(2) the Ai and Bi

Substituting (2) into (1) term is

functions of x1 and x2 only, and ζ is

complex matrices,

are n × n

and

equating separate

powers of

ζ

to zero

gives

a

complex

constant.

3 equations. The O(ζ2)

just left-bracket upper B 1 comma upper B 2 right-bracket equals 0

(3) Mason & n-th

Singer (1991) (see

generalised

K dV

diagonalisable by

are

also II.1.7 ) solve this by

equation. the

The

Yang-Mills

taking

opposite

freedom,

gauge

the Bi to be nilpotent and arrive at the

extreme, which I shall take, is to suppose that the Bi which is

up erBSubscriptiBaselineright-arowup erGSuperscriptnegative1Baselineup erBSubscriptiBaselinetimesup erGsemicol nup erASubscriptiBaselineright-arowup erGSuperscriptnegative1Baselinetimesleft-parenthesi up erASubscriptiBaselinetimesup erGhyphenpartial-difer ntialup erGright-parenthesi period

(4) where G is

an n × n

complex

matrix

depending

on

x1 and x2. Now the O(ζ) term in

(1)

is

partial-dif erential Subscript 1 Baseline upper B 2 minus partial-dif erential upper B 1 plus upper A 2 times upper B 1 plus upper B 2 times upper A 1 minus upper A 1 times upper B 2 minus upper B 1 times upper A 2 equals 0

(5) The

diagonal

entries in

(5) imply

that there is

a

‘potential’ for the

Bi:

up er B Subscript i Baseline equals partial-dif erential Subscript i Baseline up er C

(6) while the

off-diagonal

that I shall write out

entries

imply

explicitly

that the

entries of the

off-diagonal

below. Before that,

we

consider the

Ai

O(1)

are

proportional

term in

(1)

in

a

way

which is

partial-dif erential Subscript 2 Baseline upper A 1 minus partial-dif erential upper A 2 plus upper A 1 times upper A 2 minus upper A 2 times upper A 1 equals 0

(7) The

diagonal

analogous and

can

to

entries in

(6)

(7) imply

that the

A gauge transformation

.

be chosen to

remove

the

diagonal

To summarise the situation at this

point

the Bi

are

diagonal

(ii)

the Ai

are

purely off-diagonal

(iii) finally (7) imposes At what is

essentially

this

and arrive at the

n-wave

made constant

a

shall

see.

by

some

and

with

(4)

entries of the

diagonal

Ai

have

G preserves the

potentials diagonality

in

a

way

of the

Bi

entries of the Ai.

in the argument:

and derived from

(i)

diagonal

can

differential

potential

a

be

expressed

as

in

(6) ;

in terms of C and each other

using (5) ;

equations.

point, Chakravarty & interaction. This is

gauge transformation

C

(4)

,

Ablowitz a

(1990)

specialisation

take the matrices Bi to be constant

in that the B’s can’t in

but it leads to the

same

general

equations eventually

be

as we

Now it is necessary to resort to

taking components so

for

simplicity I

will restrict to 3×3 matrices.

Set upper B 1 equals diag left-parenthesis alpha comma beta comma gamma right-parenthesis equals partial-dif erential upper C semicolon times upper B 2 equals diag left-parenthesis lamda comma mu comma nu right-parenthesis equals partial-dif erential Subscript 2 Baseline upper C

(8) and

alpha minus beta equals partial-dif erential up er P times beta minus gamma equals partial-dif erential Subscript 1 Baseline up er Q gamma minus alpha equals partial-dif erential Subscript 1 Baseline up er R lamda minus mu equals partial-dif erential up er P times mu minus nu equals partial-dif erential Subscript 2 Baseline up er Q times nu minus lamda equals partial-dif erential Subscript 2 Baseline up er R

(9) so

that upper P plus upper Q plus upper R equals 0 period

(10) We will

eventually

switch to

With the choices

using

for the

(8)

two of

Bi,

P,Q,

we can

R

as

solve

independent

(5)

for the

variables.

Ai

in terms of another

off-diagonal

matrix E. Set A1 = then

(aij), A2=(Ebeij),

(5) implies a 12 equals left-parenthesis alpha minus beta right-parenthesis times e 12 times semicolon b 12 equals left-parenthesis lamda minus mu right-parenthesis times e 12

(11) and the 5

equations

Finally, equations

can

obtained from this

substitute

we

(11)

into

by

(7)

the obvious

permutations.

to obtain differential

equations

all be written with the aid of the Poisson bracket in

which the other 5 follow

by permutations,

Thesif erentieal dE.

on

(x1 x2).fotTyrpnhicema,l ,

is

StartSet e 12 comma upper R EndSet equals e 13 times e 32 times StartSet upper P comma upper Q EndSet

(12) We

can

break the symmetry between P,

Write ‘dot’ and

‘prime’

Q, R by adopting P

and

Q

for differentiation with respect to P and

as new

independent

Q respectively

coordinates.

and set

up erEequalsStart3By3Matrix1stRow1stColumn02ndColumnup erH3rdColumnup erV2ndRow1stColumnup erW2ndColumn03rdColumnup erF3rdRow1stColumnup erG2ndColumnup erU3rdColumn0EndMatrix

then

(12)

becomes the system StartLayout1stRow1stColumnup erFprime qualsminusup erVup erWtimes2ndColumnup erUprime qualsup erGup erHtimes2ndRow1stColumnModifyngAboveup erGWithdot imesequalsup erWup erUtimes2ndColumnModifyngAboveup erVWithdot imesequalsminusup erFup erHtimes3rdRow1stColumnModifyngAboveup erHWithdotminusup erHprime qualsminusup erUup erVtimes2ndColumnModifyngAboveup erWWithdotminusup erWprime qualsup erFup erGtimesEndLayout

( 13) which is

equivalent

to the 3-wave interaction.

The further reduction ‘dot

integrable

=

minus

prime’ leads,

Hamiltonian h

=

p1p2q3

+ q1q2p3

after

some

manipulating

of constants, to the

References Ablowitz , M. J. &

Segur

,

H.

Chakravarty S. & Ablowitz preprint PAM ,

of Colorado

Mason , L. J. &

Singer

,

M. A.

Math.

Phys.

§II.1.6

The Bogomolny

,

( 1981 ) M. J.

Solitons and the Inverse

( 1990 ) On

Scattering Transform

reductions of self-dual

,

SIAM Philadelphia.

Yang-Mills equations University ,

62.

( 1994 ) The

twistor

theory of equations

Hierarchy and Higher

Order

of KdV type, to appear in Comm.

Spectral

Problems

by

I.A.B. Strachan

(TN 34, May 1992) The

starting point

for the construction and solution of

the

equation as the integrability condition ;;1
λ &#x is the spectral parameter):

a

wide range of

integrable

models is to write

for the otherwise overdetermined linear system

(where

partial-difer ntialSubscriptxBaselinesequalsminus p erUleft-parenthesi lamdaright-parenthesi periodtimes com apartial-difer ntialSubscript Baselinestimes qualsminus p erVleft-parenthesi lamdaright-parenthesi periodtimes period

(1) The

integrability

condition for

(1)

is partial-dif erential Subscript x Baseline upper V hyphen partial-dif erential upper U plus left-bracket upper U comma upper V right-bracket equals 0 comma

(2) and

equating

powers of λ

of those systems which

(if

are

U and V

are

known to have

SG and N-wave equations) arise from

a

polynomial a

in

twistorial

λ) yields

=

λA

+

equation

description (such

so-called first order U

the

as

the

spectral problem,

in

question. Many

KdV, mKdV, NLS,

with

Q(x,t),

up er V equals normal up er Sigma Underscript i Endscripts times lamda Superscript i Baseline up er A Subscript i Baseline times left-parenthesi x comma t right-parenthesi period

In this article the matrices will be taken to be

where h is the Cartan

i.e. A ∈ h and

Q ∈ k,

of Mason &

Sparling,

terminology

the fields

are

a , with a and

sl(2, ℂ)-valued, subalgebra

of type β; type

a

and k is the

complement.

In the

fields will not be considered here.

A

higher

order

spectral problem

is

one

for which U and fpV gener are uonlycntoi mnaisl,

namely:

The

simplest example (p

nothing and

secondly

The

to

n

λp A + λp-1 Q1

V

=

λnV0+ λn-1.V1+Vn.

4 and

=

reduction of the

a

generalise

Q2

is due to

(2

this

.

(often

called

λ yields,

on

projecting

onto

U

=

that

w

k

,

U

u

=

V

=

=

(λpωAω-1)+

λp

Sparling (1992)

their

integrability.

Qp, V0 ...,Vn for these higher order

.

A

,

λn.A.

so

is gauge

invariant,

so

if w(x,

defined

that U and V involve

the λ0

term)

and

V

,

ωtω-1

,

by decomposing ω as ω

by

t)

is

a

λ-dependent

w

h.k

=

,

only non-negative

negative

powers of

powers of

λ the equations ,

=(λnωAω-1)+, =

(λpωAω-1)-

.

where h

,

hi(x,t)

h and

k. One then has

ki(x,t)

V

,

Let An-i denote the coefficient of λ-i in the

later),

are

ωvω-1 -ωtω-1,

positive (including

=(λpkAk-1)+

will become apparent

retaining

systems

by

is chosen

ωxω-1 =(λpωAω-1)further

Q1,...,

Schrödinger

=ωuω--1ωxω-1,

V

.

while

‘dressing transformation’),

a

U

satisfy (2) Assuming

to show how such

introduced in Mason &

l)-dimensions

+

equation

with wi ∈ sl(2, ℂ), then U and V, defined

will also

firstly

Let

trivially satisfy (2) However,

simplify

Qp,

+

Bogomolny hierarchy

these systems to

Crumey (1992).

gauge transformation

These

...

results in the derivative non-linear

0)

=

+

method to generate the matrices

following

problems

These

than

,

=

The purpose of this article is two-fold:

(or DNLS) equation. more

2

=

U

=(λnkAk-1)+

expansion

of k Ak-1

.

(the

reason

for this skew choice

i.e.

up erASubscriptnminusiBaseline qualsnormalup erSigmaUnderscriptrequals1OverscriptiEndscriptsStartFraction1Over factorialEndFractiontimesnormalup erSigmaUnderscriptleft-parenthesi StartSetsSubscriptjBaselineEndSetcol n ormalup erSigmatimes SubscriptjBaseline qualsirght-parenthesi Endscripts imesleft-bracketkSubscripts1Baselinetimescom aleft-bracketkSubscripts2Baselinecom aperiodperiodperiodtimescom aleft-bracketkSubscriptsSubSubscriptrSubscriptBaselinetimescom aup erAright-bracketperiodperiodperiodright-bracketright-bracketperiod

From this are

procedure

one

obtains the

matrix valued fields. The

fields with their above

equations,

general

form of the functions U and V

integrable equation

spacial derivatives), together or

equivalently, equation (2)

itself

with the .

(which

.

The matrices k1,...,

kp

connects the time evolution of these

remaining matrices,

may be found

using

the

found the

Having the

general

form of U and V it remains to show how these

Bogonolny hierarchy. Assuming

m

≡n

—

p

≥ 0, the matrix V

are

contained within

may be written in the form

up er V equals lamda Superscript m Baseline period up er U plus normal up er Sigma Underscript i equals 0 Overscript m minus 1 Endscripts times lamda Superscript i Baseline times up er A Subscript i Baseline times comma

and hence the

original system (1)

may be rewritten as

partial-difer ntialSubscriptxBaselinesequalsminusStarSetnormalup erSigmaUnderscriptiequals0Overscript Endscripts imeslamdaSuperscriptiBaselinetimesup erASubscriptmplusiBaselineEndSetscom apartial-difer ntialsminuslamdaSuperscriptmBaselinepartial-difer ntialSubscriptxBaselinesequalsminusStarSetnormalup erSigmaUnderscriptiequals0Overscriptm inus1Endscripts imeslamdaSuperscriptiBaselinetimesup erASubscriptiBaselineEndSetsperiod

(3)

Recall from Mason & Sparling (1992) that given the minitwistor the Riemann

sphere

of Chern class

{[∂zi where

Ai,

together

and

Bi+1

with Bi

eliminating

+

≥ 1, the Ward construction

λ[∂zi+1+ Bi+1]}s

-

sl(2,
)-valued

are

0,i

=

Ai]

n

=

1,...,, n

—

gauge

=

(3)

over

rise to the linear system

0, -1=0,. n i

potentials.

1 , Bn ≡ An+1 ,A0

the other variables results in

gives

the line bundle

O(n),

space

With the symmetry A

=

,

relabelling

z0

generated by ∂Zn, =

t,

zm

=

x

,

and

:

StartLayout 1st Row left-bracket partial-dif erential negative lamda partial-dif erential right-bracket s times equals negative up er A period times 2nd Row elipsi elipsi 3rd Row left-bracket partial-dif erential negative lamda partial-dif erential right-bracket s times equals minus up er A Subscript m minus 1 Baseline times period times EndLayout right-brace left-bracket partial-dif erential minus lamda Superscript m Baseline partial-dif erential right-bracket s right double ar ow minus StartSet normal up er Sigma Underscript i equals 0 Overscript m minus 1 Endscripts times lamda Superscript i Baseline times up er A Subscript i Baseline EndSet period times com a

Thus these

higher

Solutions of the

O(4)

order

an

𝕋m,P and

~

elegant generalisation

(3) by λm∂y

systems. Thus the DNLS

=

given

a

one

equation

twistorial

{(Z0, Z1, Z2, Z3)}/

is the

equation, correspond

Bogomolny hierarchy.

to bundles over the space

symmetries.

These systems have the term λm∂x in

These may be

that of the DNLS

simplest example,

with certain

may all be embedded within the

spectral problems

equivalence

~

,

naturally

has the

to

(2+1)-dimensions, Strachan (1993). By replacing

obtains

examples

of

(2

+

1)-dimensional integrable

following generalisation:

i∂tψ

=

∂xyψ + 2i∂x[V.ψ],

∂xV

=

∂y|ψ|2.

description by introducing

where Z0 , Z1

are

coordinates

a on

weighted

twistor space defined

the Riemann

relation

(Z0,Z1,Z2,Z3)

~

(μZ0,μZ1,μmZ2,μpZ3),

∀μ

𝕄;𝕡1.

sphere, Z2 Z3 ,

by 𝕄,

Reimposing

the symmetry ∂x

vector field on space

O(m

to recover

𝕋m,P

 02y corresponds

=

to

factoring

p), exactly analogous

+

out

by

a

non-vanishing holomorphic

to the construction of the minitwistor

from standard twistor space.

O(2)

References Mason , L. J. & of

Geometry

Crumey preprint. ,

Sparling G. A. J. ( 1992 ) Physics 8 243 271 ,

and

A.

-

,

Phys.

,

( 1993 ) Some 34 , 243 259 -

§II.1.7 H-Space: The

a

aspects of the ideas

Integrable

universal

have not been fulfilled yet

There is

a

large

forest of

as a

Riemann surfaces

for

play

an

Two gaps in the story the KP and some

of the solution of

equations gap is

a

these equations,

example

however, be somewhat nontrivial,

is

Kac-Moody Algebras and their twistor

,

Leeds

description

,

may

(TN 30,

never)

June

1990)

but I feel that the concrete

are

intriguing.

integrable systems. R.S.Ward, amongst others, of

question

as

are

reductions of the self-dual

bookkeeping,

the inverse

scattering

it

gives

a

substantial

has

pointed

Yang-Mills into

insight

transform for these systems

can

be

symmetry reduction of the Ward construction for solutions of the self-dual Yang-Mills

equations. (See

systems into

L.J.Mason

(and

integrable systems

equations. This observation isn’t just

understood

(2 + 1)-dimensions

of interest and the various relations involved

are

theory underlying

models in

integrable system? by

out that many, if not indeed most

the

and

.

following speculations

Motivation.

for the Soliton Hierarchies , Journal

Correspondences

Integrable Hierarchies, Homogeneous Spaces

( 1992 )

Strachan , I. A. B. J. Math.

Twistor

.

more

see

essential are as

Mason & in

Sparling

particular

1989 and 1992; the symmetry reduction can,

Woodhouse & Mason 1988 in which non-Hausdorff

role).

follows.

Firstly

Davey-Stewartson equations.

there appear to be There is little

kind of twistor framework if the inverse a

Riemann-Hilbert

subtle and

particularly irritating

requires

problem.

difficulty

scattering

However the inverse

the solution of

a

genuine in

difficulties in

transform is realised

scattering problem

‘non-local Riemann-Hilbert

in view of the theoretical importance that the KP

with their relations to the

theory

incorporating

incorporating integrable by

means

for the KP

problem’.

equations

of Riemann surfaces and infinite dimensional

This

have

acquired

grassmanians

and

so on.

The second gap is that there appears to be little role for the self-dual

and its twistor construction, Penrose’s nonlinear

out, is

graviton

not based on the solution of a Riemann-Hilbert

state the

vacuum

equations

construction—this, it should be either.

problem

pointed

However I should like to

following conjecture:

CONJECTURE. The KP and

Davey-Stewartson equations

are

reductions of the self-dual Einstein

equations. The circumstantial evidence is with

with metric of

space-times

LEMMA 1. KP reduced

by

two

orthogonal null

∑2, S Diff(∑2)

can

be

translations.

The Lie

algebra

by

of Mason & Newman

results would self-dual

If it

imply

a

∞ of the

SL(n)

Yang-Mills equations

extends the results of Mason

(This of the

equations

self-dual

preserving diffeomorphism

area

that of SL(n)

as n

to the self-dual

equivalent

are

Sparling 1989).

&

group of

—>

a

surface

∞

Yang-Mills equations

Diff(∑2). (This

extends the results □

the

were

that

case

SL(n)

that all 2-dimensional least by

Yang-Mills equations (at

yields

taken to be concerned

1989).

mark. However, my current still

—>

orthogonal null translations with gauge group S

two

REMARK.

as n

approximated arbitrarily closely by

LEMMA 3. The self-dual Einstein reduced

are

signature (2,2 ).)

be obtained in the limit

can

(Hoppe, J.)

LEMMA 2.

(The self-duality equations

follows.

as

opinion

reasonable class of

subgroup

were a

models

integrable is

SL(n)

only

and

integrable systems

are

SL(∞)

=

obtainable

Diff(∑2)

S

then these

reductions from the

as

Hence the title of this note and the

translations).

is that

of

a

subgroup

certainly

the

of S

more

Diff(∑2)

for

famous

ones

n

question

=

2. This

such

as

the

KdV, nonlinear Schrödinger and sine-Gordon equations. Proof of lemma 1. See for instance

approach.

The

I shall

Segal &;

equations

solution Vd to the

(∂t2,

use

the

(1985)

Wilson

of the KP

following system -

(Q2)+)ψ

=

0,

of the KP

presentation in the

hierarchy

of linear

(∂t3

-

proceedings are

the

partial

hierarchy

of the I.H.E.S for

consistency

differential

(Q3)+)ψ

=

differential operators

u

by

(Qr)+

the are

equations.

description

of this a

equations

0,..., (∂tr

-

=

determined in terms of

a

conditions for the existence of

(Qr)+)ψ

(∂x)r (Qr)+ is an rth order O.D.E. in the x variable, Qr u(x,t2,t3,...) is the subject of the KP hierarchy equation and wr is

where

due to Gelfand and Dickii.

+

=

0,

ru(∂x)r-2

some

...

+

...

+ wr and

function which will be

The notation is intended to indicate that the

the differential operator part of the

ordinary

pseudo-differential operator Q

raised to the rt h power where

pseudo-differential operator

Q

∂x +

=

defined

by

u(∂x)-1+ (lower order)

and where

(∂x)-1

is

a

formal

the relation

left-parenthesi partial-difer ntialSubscriptxBaselineright-parenthesi Superscriptnegative1Baselineftimesequalsftimesleft-parenthesi partial-difer ntialSubscriptxBaselineright-parenthesi Superscriptnegative1BaselineplusModifyngAbovenormalup erSigmaWithinf ityUnderscriptiequals1Endscripts imesleft-parenthesi minuspartial-difer ntialSubscriptxBaselineright-parenthesi SuperscriptiBaselinetimesfleft-parenthesi partial-difer ntialSubscriptxBaselineright-parenthesi Superscriptnegativeiminus1Baselineperiod

The

original for

variables

(∂t2

KP equation is the —

(Q2)+)ψ

symmetries

are

=

u(x,t1,t2) that (Q3)+)ψ 0 alone.

equation

0 and

(∂t3

of the basic

—

on

=

equations (and

each

nth time variable tn, then the reduced system is referred

(n

=

2

gives

the standard KdV

hierarchy

The idea is that the operators on

L2(∝)

where

is

x

a

coordinate

(Qr)+ on

(n

—

(∂t2

can

be

=

(Q2)+)ψ

=

with respect to

x.

The evolution in the

other).

If

to as the

one

∂tn ψ

imposes

higher

conditions time

invariance in the

nth generalized KdV hierarchy

of

as

infinite dimensional matrices

approximate this by =

consistency

Boussinesq).

thought

can

(setting

3 the

in the n-dimensional solution space of this

1)-derivatives —

n

∝. One

symmetry in the nth time variable since

only ψ lying

and

follows from the

λψ)

we

have

n× n

matrices

(Qn)+ψ

equation, represented,

With this reduction

we

=

λψ and

say,

by

acting

by imposing we

a

consider

ψ and its first

have:

0 reduces to

StarSetparti l-difer ntialminusStar 6By6Matrix1stRow1stColumn2u2ndColumn03rdColumn14thColumn05thColumn elips 6thColumn02ndRow1stColumndot2ndColumn2u3rdColumn04thColumn15thColumn elips 6thColumn elips 3rdRow1stColumndot2ndColumndot3rdColumn2u4thColumn05thColumn elips 6thColumn04thRow1stColumndot2ndColumndot3rdColumndot4hColumn elips 5thColumn elips 6thColumn15thRow1stColumndot2ndColumndot3rdColumndot4hColumn elips 5thColumn elips 6thColumn06thRow1stColumndot2ndColumndot3rdColumndot4hColumndot5hColumndot6hColumnleft-parenthesi 2minusnright-parenthesi uEndMatrixpluslamdatimesStar 5By5Matrix1stRow1stColumn02ndColumn03rdColumn04thColumn elips 5thColumn02ndRow1stColumn elips 2ndColumn elips 3rdColumn04thColumn elips 5thColumn03rdRow1stColumn02ndColumn elips 3rdColumn elips 4thColumn elips 5thColumn elips 4thRow1stColumn12ndColumn03rdColumn elips 4thColumn elips 5thColumn05thRow1stColumn02ndColumn13rdColumn04thColumn elips 5thColumn0EndMatrixEndSetModifyngBelowpsiWith̲equals0

and

(∂t3

—

(Q3)+)ψ

where 0r is the and

can

be

seen

r

=

0 reduces to

× r zero matrix. This matrix linear

to be the linear

system of

a

system is linear in the spectral parameter λ

reduction of SDYM with 2 null

symmetries. NOTE. A

orthogonal

translation □

large

gap in the above discussion is that the linear system is shown to be contained

within the SDYM linear systems, but I have not characterised those SDYM solutions with the 2

orthogonal null symmetries

that

give

rise to the nth KdV system.

Proof of Lemma 2. of

diffeomorphisms fields

corresponding

coordinates

HA

=

on

These ideas

a

by using

torus

the torus such that the

{HA, HB} For

SL(N)

where A

plane

=

A basis for the Lie

the

=

form is dθ1 Λ dd2, then

diagonal Uij

algebra

of

constructed

ξVU where

SL(N)

=

(A Λ B)

where

algebra

ξiδij

ξN

area

preserving

representing

their Hamiltonians. Let θ1 and θ2 be

(A1, A2) ℤ× ℤ The

=

algebra of the form and

symplectic

as a

Diff(∑2) by

(A Λ B)HA+B

relations: UV

ξ down

form

area

area

basis for the Lie

we use a

with powers of

the

to elements of Lie S

exp{2πi(A1θ1 + A2θ2)}

the quantum

standard. One presents the Lie

are

a

and V

a

Lie bracket is the Poisson bracket:

A1B2

=

using

a

pair

shift matrix

is then furnished

angular

basis for the Hamiltonians is

-

A2B1.

of matrices U, V

satisfying

1. An explicit representation has U

=

vector

Vij

=

δi(j+

diagonal

1mod N).

by

up er T Subscript up er A Baseline quals up er N xi Superscript StartFraction up er A 1 times up er A 2 Over 2 EndFraction Baseline up er U Superscript up er A 1 Baseline times up er V Superscript up er A 2 Baseline period

The commutators

are

then

given by:

left-bracket up er T Subscript normal up er A Baseline times comma up er T Subscript normal up er B Baseline right-bracket equals up er N sine StartFraction 2 times pi normal up er A normal up er Lamda times normal up er B Over normal up er N EndFraction up er T Subscript normal up er A plus normal up er B Baseline times left-parenthesi normal up er A times normal up er Lamda times normal up er B right-parenthesi times normal up er T Subscript normal up er A plus normal up er B Baseline

which gives the same commutation relations as above for HA in the limit as N ∞. → □

Proof of lemma 3. paper it

symmetries volume

This is

a

corollorary

shown that if you take the

was

on

the self-dual

equations.

Yang-Mills equations conditions

on

Lemma 3 4

on

algebraic

relations obtained

of

can

some

4-manifold then,

be reformulated

with metric ds2

connection components

=

by imposing

(1989).

In that

four translational

and take the gauge group to be the group of

Yang-Mills equations

preserving diffeomorphisms

vacuum

of the results in Mason & Newman

du

so as

dy

roughly speaking,

to be a

+ dv dx

one

obtains the self-dual

special

case

of this. The self-dual

(signature

2,2 )

are

the

integrability

(Au, Av, Ax, Ay) in the Lie algebra of the gauge group for the

the linear system

{∂u + Au {∂v When G is S on

Diff(∑2)

the coordinates

on

the system

{Vu + λVX}ψ

on

=

+ Av +

0

the coordinates =

{Vv + λVy }ψ on

preserving diffeomorphism

Yang-Mills equations diffeomorphisms

+

on

the

are

symmetries

quotient, ℝ2. are

ℝ2 × ∑2 and

so

group.

0.

all vector fields

where the V’s

with 4 translational

of ℝ2 × ∑2.

= 0

Ax)}ψ

λ(∂y + Ay)}ψ =

two translational

fields preserve the natural volume form of the volume

λ(∂x

the connection components

ℝ4). Impose

components depend only

+

on

the ℝ4

∑2 (depending also

so

that the connection

The linear system then reduces to

vector fields on ℝ2 × ∑2. These vector

determine elements of the Lie

The linear system is

symmetries

on

precisely

algebra

that for the self-dual

and gauge group the volume

preserving

Concretely

introduce coordinates

and suppose the the coordinates

hx

symmetries (u,v)

etc.. The field

on

ℝ2. Represent

equations

hu

=

 02;q

and

∂qg

for

Ay some

=

∂p.

that the

so

area

the vector fields Ax

λ2[AX, Ay]

The term

0

=

proportional

form is the

symplectic form dpΛdq,

on

∑2 by

depend only

on

their Hamiltonians denoted

so

to

that

we

can

choose coordinates

λ implies ∂qhv

=

∂phu,

on

that hv

so

∑2 =

so

that

∂pg

and

The final equation yields in terms of g

≡ g(u, v, q,p).

g

∑2

Au + λ Ax ,∂v + Av + λAy] = 0

+

The first implication of this is that

=

on

are

[∂u

Ax

(p, q)

to be in the x and y directions so that the variables

g which

is

Plebanski’s

Thanks to G.A.J.

second

Sparling and

heavenly

equation.

□

E.T. Newman for discussions.

References Mason , L. J. & Newman , E. T.

( 1989 )

Comm. Math. Phys. 121 659 668

A connection between the Einstein and

Yang-Mills equations

,

-

,

,

Mason L. J. h

.

Sparling G.A.J. ( 1989 ) Non-linear Schrodinger Yang-Mills equations Phys. Lett. A. 137 29 33 ,

and KdV

are

reductions of the self-dual

-

,

,

,

.

Mason L. J. & Sparling , G. A. J. Phys. 8 , 243 271

( 1992 ) Twistor correspondences

Ward , R. S. ( 1985 ) Soc. A 315 , p. 451

and solvable systems and relations among them , Phil.

for the soliton hierarchies , J. Geom.

-

.

Integrable

Woodhouse , N. M. J. & Mason , L. J.

Nonlinearity

Trans. R.

.

1 , 73 114

( 1988 ) The

Geroch group and non-Hausdorff Riemann surfaces ,

-

,

.

§II.1.8 Integrable Systems

and Curved Twistor Spaces

by

I.A.B.Strachan

(TN

35, December

1992) One of the ways in which the self-dual Einstein equations may be understood is chiral model with the gauge fields

taking

values in the Lie

algebra

as a

two dimensional

sdiff(∑2)of volume preserving

sdiff(∑2),

∑2 (Q.

of the 2-surface

diffeomorphisms

solutions of certain

Han Park

integrable

1990). Moreover,

systems associated with

the geometry of the nonlinear

graviton (Mason §II.1.7).

rank

algebras,

subalgebras

such

a

which

description

not

are

may be achieved.

of

Another

dimensional gauge groups is that the

This

sdiff(∑2). reason

equations

for

often

since

sl(2, ᵔ ; )

sl(2,𝕄)

description

is

a

subalgebra

of

may be encoded within

breaks down for

higher

However, by generalising the algebras

studying integrable systems

simplify

,

and in

with infinite

some cases even

linearise,

(Ward 1992). Let

{

}

,

be

Poisson bracket

generalised

a

some

manifold N

,

satisfying the

conditions:

(antisymmetry)

{f,g} = -{g,f}

{f, gh}

acting on

{f, g}h

=

{f, {g, h}} With respect to

+

+

cyclic

{f, h}g

basis xi, i

a

(derivation) (Jacobi identity)

0

=

=

1,... dimN, ,

one

may take

StartSetfcom agEndSet qualsnormalup erSigmaUnderscripticom ajEndscriptsup erGSuperscriptijBaselinel ft-parenthesi xright-parenthesi timesStartFractionpartial-difer ntialfOverpartial-difer ntialxSuperscriptiBaselineEndFractionStartFractionpartial-difer ntialgOverpartial-difer ntialxSuperscriptjBaselineEndFractioncom a

(1) where Gij

(x)

is constrained

by

the

equations Gij + Gji

=

0

normalup erSigmaUnderscriptlequals1Overscriptdimensiontimes Endscriptsup erGSuperscriptliBaselineStarFractionpartial-difer ntialup erGSuperscriptkjBaselineOverpartial-difer ntialxSuperscriptlBaselineEndFractionplusup erGSuperscriptljBaselineStarFractionpartial-difer ntialup erGSuperscriptikBaselineOverpartial-difer ntialxSuperscriptlBaselineEndFractionplusup erGSuperscriptlkBaselineStarFractionpartial-difer ntialup erGSuperscriptjiBaselineOverpartial-difer ntialxSuperscriptlBaselineEndFractionequals0period

(2) Such

Poisson structures

generalised

Given such fields. Let

a

bracket

(2)

.

LfLg

algebra

H am of Hamiltonian vector

and -

algebra

Lg

as

may be defined in two

different, but equivalent,

ways:

differential operators, and define the Lie bracket for the

algebra by

LgLf,

Lg as vector fields on N and define the Lie bracket for the algebra be the Lie of vector fields [Lf, Lg]Lie. cases L{f,g} The fact that this forms a Lie algebra follows trivially from (1) and [Lf, Lg] and

=

The idea

now

is to

study

the self-dual

in this infinite dimensional Lie Let

Lie.

up erLSubscriptfBaseline qualsnormalup erSigmaUnderscripticom ajEndscripts imesup erGSuperscriptijBaselinetimesleft-parenthesi xright-parenthesi tmesStarFactionpartial-difer ntialfOverpartial-difer ntialxSuperscriptiBaselineEndFractionStarFactionpartial-difer ntialOverpartial-difer ntialxSuperscriptjBaselineEndFractionperiod

Regard Lf In both

by Sophus

H am, where

Lf

Regard Lf

[Lf ,Lg]

first studied

structure one may define an associated Lie

The Lie bracket for the

=

were

yAA' be spinor

condition).

with gauge

potentials taking

values

algebra.

coordinates for

The self-dual

Yang-Mills equations

ᵔ ; 4 (or perhaps ℝ2+2

Yang-Mills equations

are

the

etc.

depending

compatibility

on a

choice of reality

condition for the otherwise

overdetermined linear system: script up er L Subscript up er A Baseline normal up er Psi equals pi Superscript up er A prime Baseline StartSet StartFraction partial-dif erential Over partial-dif erential y Superscript up er A times up er A prime Baseline EndFraction plus up er A Subscript up er A up er A prime Baseline EndSet normal up er Psi comma up er A comma up er A prime equals 0 comma 1 times comma pi Superscript up er A prime Baseline times element-of double-struck up er C double-struck up er P Superscript 1 Baseline period

(3)

The AAA'(y) are Lie algebra valued functions known as gauge potentials. In what follows it will be assumed that these take values in the Lie algebra H am constructed above. Thus the AAA'’s are represented by vector fields AAA' ↔ LfAA', where the functions fAA' depend on both the coordinates on 𝕄4 and on N

With this, the linear operators

LAare

Ǖ 4;4 × N,

vector fields on

now

scriptup erLSubscriptup erABaseline qualspiSuperscriptup erAprimeBaselinel ft-braceStarFractionpartial-difer ntialOverpartial-difer ntialySuperscriptup erAtimesup erAprimeBaselineEndFractionplusnormalup erSigmaUnderscripticom ajEndscripts imesup erGSuperscriptijBaselinel ft-parenthesi xright-parenthesi StarFractionpartial-difer ntialfSubscriptup erAtimesup erAprimeBaselineOverpartial-difer ntialxSuperscriptiBaselineEndFractionStarFractionpartial-difer ntialOverpartial-difer ntialxSuperscriptjBaselineEndFractionright-braceperiod

(4) to the

Owing the

equivalent

definition of the Lie bracket, the

conditions for the distribution

(Frobenius) integrability

surfaces of this distribution may be surfaces The

regarded

curved twistor space, fibred

as a

converse

construction involves

the Riemann

over

studying

(1991)

which also

developes

(4)

,

i.e.

are a

[L0, L1]Lie

=

special

0. The

of

case

integral

curved twistor surfaces, and the space of such

as

sphere.

appropriate Riemann-Hilbert problem

an

dimensional group. Similiar ideas have been

& Takebe

self-duality equations

to the

applied

the notion of

a

SU(∞)-Toda equations

for the infinite

in Takasaki

τ-function for this system and its associated

hierarchy. As mentioned at the

of this

beginning

curved twistor space construction to certain it in the

algebra sdiff(∑2).

The

structure constants for the Lie

.

e

From this

one

Mason in

integrable systems

§II.1.7

shows that

associated with

,

with respect to

may define

some

generalised

a

basis ei ,i

=

one

could

give

algebra

g. Let the

1,..., dim g be Cijk

Poisson bracket

a

sl(2, ℂ) by embedding

is true for any finite dimensional Lie

same

algebra g

article,

,

so

by setting

up er G Superscript i j Baseline left-parenthesi x right-parenthesi equals normal up er Sigma Underscript k Endscripts times c Superscript i j Baseline Subscript k Baseline times x Superscript k

(the

conditions

(2)

are

automatically

satisfied due to the

let the associated infinite dimensional Lie

original

Lie

algebra

is

now a

subalgebra

algebra

properties

of the structure

functions),

of Hamiltonian vector field be denoted

by g̃.

and The

of g̃, since

left-bracketup erLSubscriptxSubSuperscriptiSubscriptBaselinetimescom aup erLSubscriptxSubSuperscriptjSubscriptBaselineright-bracket qualsnormalup erSigmaUnderscriptkEndscripts imescSuperscriptijBaselineSubscriptkBaselinetimesup erLSubscriptxSubSuperscriptkSubscriptBaselinetimesperiod

Thus any solution to the self-dual encoded within the structure of Another

(1992), direct

approach

in which

is to

higher

a

use a

Yang-Mills equations

curved twistor space deformation of

order derivatives

geometrical interpretation

are

Han Park ,

( 1990 ) Phys.

by

sdiff(∑2)

first

a

of the results is absent.

Lett. B238 , 287 290 -

.

finite dimensional algebra may be

embedding

known

present. This leads

References

Q.

with

as

the

to some

g in g̃.

Moyal algebra,

Strachan

interesting results,

but

a

Mason , L. J.

( 1990 )

Twistor Newsletter 30 , 14 17 and §11.1.7. -

Ward , R. S. ( 1992 ) J. Geom.

Phys.

Takasaki , K. & Takebe , T. Lett. Math.

Strachan , I. A. B.

Twistor

§II.1.9

( 1991 23 , 205 214

Phys.

) SDifF(2)

Toda

( 1992 ) Phys.

Lett. B282 , 63 66

theory

integrability by

give

Tau Function and

equations Hierarchy,

Symmetries

,

.

and

mostly

-

of certain

.

L.J.Mason

and

speculative

in my 5 minute contribution to the

In this note I wish to make the

point

integrable systems

should lead to

and

results in the

hopefully

new

In various articles it has

being

.

(TN 33,

conjectural

special

November

1991)

comments that I gave or would

twistor

workshop

to celebrate the 60th

of the founder of many of these ideas.

birthday

systems

-

-

This note consists have liked to

8 , 317 326

are

new

that the

recently

techniques of

theory

established links between twistor

and results in twistor

theory

emerged that,

hierarchy. Furthermore,

the KP

reasonably

a

correspondence

direct way

with

a

for the self-dual

much of the as

well

as

unification

most

integrable

integrable systems. small number of

exceptions,

symmetry reductions of the self-dual Yang-Mills equations, the

understood in

as

theory and

theory

most notable

exception

and structure of these equations

can

be

various features of the symmetry reductions of the Ward

Yang-Mills equations.

See Ward

(1986),

Mason &

Sparling (1992)

and references therein. As far as

as

the

equations

reductions from self-dual

are

concerned,

Yang-Mills

it appears that

in 4-dimensions

by

we can

a

gauge group,

b)

a

symmetry group

c)

a

normal form for the gauge and the various constants of

a

possible

discrete

most

integrable systems

choice of:

a)

(with

classify

component), integration

that arise in the reduced

equation. For

large

example,

classes of

the Drinfeld Sokolov systems

can

all be understood in this way

as can

various other

integrable systems.

The standard

theory

of the

transform and realizations of the

equations partial

consists of such constructions

differential

equations

as

flows

on

as

the inverse

grassmanians.

scattering These

can

be understood bundles

on

as

various ansatze and normal forms for the

symmetry reductions of the Ward transform for the self-dual However, these ideas from the theory of twistor ideas, and in others

completely

are

of methods from the theory of The

following conjectures

in twistor

corresponding

in many

are

refinements of

cases

There is therefore the possibility

theory.

used to solve problems in twistor theory.

integrable systems being

and connections include

vector

Yang-Mills equations.

integrable systems

new

holomorphic

that arise from the

appropriate symmetry properties

twistor space with the

data of the

patching

examples

where twistor

theory

may benefit from

this interaction. Inverse

1)

space of

integrable partial

for the Ward bundle

equation

we

twistor space

on

and Sk is the

the

linearizing

cartesian

equations (for

ℂ*

One would expect this pattern

signature.

Minkowski space with

equations with

a

perhaps

is

So

are

which

parametrization of the solution

a

can

be used to build

patching

the attractive nonlinear

linear

a

to be

complex

The first factor are

Schrodinger

the Fourier

are

numbers, II is the

disjoint

solutions that

would expect

transform)

one

the Radon transform. A similar

but the second factor

for solutions of the self-dual on

the

that the solution space of the

picture

union

are

analogue.

generic

be the

presumably

that the first factor

data

where D is

non-zero

Yang-Mills equations

compactified

can

(2,2 ) analogues

be understood

as a

4-dimensional

self-dual

SU(n)

of maps from ℝ𝕡3 to unit determinant Hermitean

product

mentioning

the

they

soliton type sector, which would worth

example for

would expect for example that

one

signature (2,2 )

Cartesian

a

For

product.

the soliton solutions which do not have

in indefinite

The parameters

equations. directly.

provides

space identified with: M

complex plane ℂ,

symmetrized

transform

scattering

differential

get the solution

the unit disc in the

from

The inverse

scattering.

Yang-Mills

n× n

matrices

of instantons.

nonlinear

It is

generalization

should hold for the symmetry reductions to

equations

of in

2+1 dimensions and other 1 + 1 dimensional systems.

2) that of

The inverse

scattering

distinguish it clearly

incorporating

transform in 2+1 dimensions such

from

existing twistor correspondences

it into the above framework.

framework for the KdV

equations

It is

so

perhaps

role in the KP

that it leads to worth

some new

remarking that

equations

also

problem—the patching operation by integration against Another

point

a

kernel

was

just

the

a

coherent inverse

hope, then,

that there a

natural

scattering

seems

has features

little real

generalization can

hope

of the

transform based

that the transform

pseudo-differential operators

on

a

be articulated

in

one

given by pseudo-differential in the KP inverse

scattering

that

play such

a

prominent

of Penrose’s earlier discussions of the

a

as

is that the inverse

so

hierarchy

category of twistor constructions.

naturally

arose

for the KP

Nevertheless, it is

and leads to

non-local Riemann-Hilbert transform. One may

geometrically

as

operator that

can

be

googly

represented

scattering problem.

transform does work for many other field equations

in

dimensions but is

higher

solution

practical

no

generation

longer implementable by

methods. It may nevertheless lead to

general relativity using spin 3/2 twistors

3)

(see

It is

KdV type

possible

equations

to

use

the Ward

certain

workable framework for understanding

asymptotic

work).

theory

of free

Fermions, developed by

holomorphic sections

of the KdV

equations

of the Ward bundle

and the quantum field theoretic Greens function for the

are

given by amplitudes

associated to flows

twistor space restricted to

on

in

amplitude

∂̄-operator. Finding

the line which is the

on

the

Segal & Wilson (1985) and Witten (1988). Solutions (at least those acting on

vectors in the free Fermion Fock space. The link is that the free Fermions

special

vector bundle

on

to understand the connections between the

correspondence

and the 2-dimensional quantum field

reflectionless)

are

a

fields and Penrose’s elemental states based

Penrose’s article in 𝕋ℕ 10 to appear also in volume III of this

Japanese school and described in that

linear procedures and hence does not lead to

question

is the

2-point

in

obtaining

equivalent

the self-dual

the

complex projective line,

function that

the Greens function is

key step

a

are

to

gives

rise to the

trivializing

Yang-Mills

the

field in terms

of the bundle. One may ask the constructions such

as

question

2-dimensional quantum field

by Ooguri & 4)

graviton In

theory.

Vafa between N

There is much scope for how to

then of whether its

the nonlinear

=

2

possible

construction this

particular

string theory

might explain

ideas from the quantum inverse

using

complicated

twistor

complicated, perhaps interacting the remarkable link discovered

and the self-dual Einstein

twistor methods in the context of

use

to realize other more

as a more

equations. transform to understand

scattering

integrable quantum

field

theory.

In

particular

the Russian school’s introduction of the R-matrix to describe the Poisson bracket structure should

directly

pass over

have

managed

hope

to

on

5)

quantize

for

on

The

using

(1989)

scattering

patching

transform survives

existing theory

is still in need of further

data. Other workers

quantization

twistor space and then transform the results to obtain

on

has

attempted

the

knot

studying

a

unification of the

theory.

polynomials. Unfortunately

spectral parameter

satisfactory understanding

as a

a

Chern-Simons quantum field

understanding

this

the Poisson bracket relations for the twistor

insights

a

so

that

one can

quantum field theory

that twistor

theory

may be

provide.

Witten

models

give

to show that the inverse

space-time.

able to

to

quantum field

theory

that is of

so

This

of

integrable

produces

it does not

crucial to

integrable

of self-dual

theory

/i-matrices that

provide

integrability.

statistical mechanical

the

dependence

So it is not

statistical mechanics. One may

Yang-Mills

are

possible

sufficient of the R-matrices to

conjecture

regard

that

by

reduced to 3-dimensions this gap would be

remedied. It is

(see

perhaps

their article

also worth

§ 11.1.3 ).

drawing attention

to the

Atiyah-Murray conjecture

also in this context

Author’s comment

This will appear in

projects.

and

theory

integrability to

be

forthcoming book by the

significant

progress

of the above

on some

author and N.M.J.Woodhouse called Twisior

published by Oxford University

The first has been realized by

1)

Minkowski space with small open

neighbourhood

Ward bundle

operator, leads to

on

Press. More immediate

developments

on

! D;𝕡3.

generalized

the

direct

a

copies

of ℂ𝕡3

analogue

compactified

glued together

of the Inverse

over

scattering

to be

as

equations

(see Ablowitz,

replaced by

on

general

a more

linear differential

parameters, with non-trivial index. This

based

on

the ‘d-bar’

approach

M. & Clarkson, P.A.

1991).

due to Fokas &

For further details

(1994).

quantum field still in

so

sphere depending

There has also been progress in

3)

two

twistor space for the

type of twistor construction in which the the ∂̄-operator of the

Ablowitz and Zakharov & Manakov Mason

appropriate

These leads to

construction for the KP

a new

that the

in the form alluded to above. See §II.1.11 for further details.

a new

twistor space is

Dirac operator

a

of

Yang-Mills

The second has led to

2)

observing

signature ( 2, 2 ) is non-Hausdorff being

transform for self-dual

see

been

follows.

are as

some

a

already

There has

(1994):

and twistor

theory

clarifying

theory.

the connections between Grassmanians and 2-d

This is to appear in Mason &

Singer (1994).

Part II is

preparation.

References Ablowitz , M. & Clarkson , P. A.

( 1991 ) Solitons,

LMS lecture note series 149 , CUP Mason , L. J.

( 1994 )

proceedings

of the Seale Hayne conference

Mason , L. J. & Math.

Generalised twistor

Singer

,

M. A.

( 1994 )

Nonlinear Evolution Equations and Inverse Scattering,

.

d-bar

correspondences, on

and the KP

problems

equations

Twistor Theory , ed. S. Huggett , Marcel Dekker

The twistor

theory

of

of KdV type part I

equations

,

,

in

.

Comm.

Phys.

Mason , L. J. &k

Sparling G.A.J. ( 1989 ) Korteweg de Yang-Mills equations Phys. Lett. A ,

Mason , L. J. &k

Physics

Sparling ( 1992 )

8 , 243 271

137 ,

#1,2,

29 33

Schrodinger

are

reductions

-

.

for the soliton hierarchies , J.Geom.&k

correspondences

.

Wilson , G.

Ward , R. S.

( 1986 )

,

,

Twistor

,

-

,

Segal G. k Strings

Vries and nonlinear

,

of the self-dual

( 1985 ) Loop

groups and

Multi-dimensional

eds. H.J. de

Vega

equations

integrable systems

,

of KdV type , Publ. I.H.E.S. 65 , 5 65 -

in Field

and N. Sanchez , Lecture Notes in

Witten , E.

( 1988 )

Quantum field theory

Witten , E.

( 1989 )

Nucl. Phys. B.

and

grassmanians

,

Physics

and

Theory, Quantum Gravity 246 Springer Berlin ,

Comm. Math.

,

Phys.

.

113 , 529 600 -

,

.

.

§II.1.10 by

On the

L.J.Mason

One of the remarkable features of reductions of the self-dual

to

systems in

of

space-time symmetries)

projection one

is

two dimensions is that the

priori,

one

is much

on

&, Sparling (1992) it

(1987)

+

2-plane

on

was

as

opposed

of the gauge group.

to

(or

2-dimensions An a

a

rotation and

equations

totally

a

than

also has

complex

an

just

the

Yang-Mills invariant

actually conformally (a priori

one

would

only expect In Mason

plus scalings).

Yang-Mills by

2 translations

spanning

infinite dimensional symmetry group of the Galilean group in

underlying

examples

null ASD

just GL(2)),

2-plane

so

this result and state it

of this

phenomena,

called

one

independently

being

the

where the symmetry group is the

and the other

being

the reduction

in which the symmetry group is the

of Hitchin’s result is that it makes it

give

they considerably

alternate ways of

Yang-Mills Higgs equation on

self-dual

one

enrich the

hyperbolic

transferring

We

use

with

(z, w, z̃, w̃)

z̃

=

z̃ etc.

as

a

for Euclidean

theory

on

ℝ4

that

signature.

of

geometric

Riemann surface.

coordinates

possible

to transfer the

by

two

group in

holomorphic

equations

are

I first

Yang-Mills

structures.

give

independent

a

brief review of Hitchin’s

and real for

We start with the Lax

pair

signature (2, 2)

formulation of the

Yang-Mills equations.

The self-dual

Yang-Mills equations Lo

=

are

Dz

—

the

compatibility

λDw̃, L1

=

conditions for the

Dw + λDz̃.

to

vector bundles.

different reductions of the self-dual

to 2-dimensional surfaces endowed with different

equations. or

two translations are

the geometry

clarify

translation)

Riemann surface where

The above results

The

a

equations

infinite dimensional.

to be

the linear Galilean group in 2-dimensions..

just

(rather

being

context

(SL(2, ℝ) .

important corollary

general

by

I also discuss two other

group

often

observed that the reductions of the self-dual

SL(2)—nonlinear analogues

by symmetries spanning

diffeomorphism

rotations

was

observed that reductions of self-dual

The purpose of this note is to

whole

it

which the metric has rank

0)-dimensions

reduction

expected,

in the residual 2-dimensional space

sense

at least when the gauge group is

(1

have

to be invariant under the 2-dimensional Euclidean group

equations

a

might

one

in 4-dimensions that normalize the invariance group that

Euclidean 4-dimensional space

in the infinite dimensional the

than

larger

symmetries

In Hitchin

Yang-Mills equations

symmetry group of the reduced equations (in the

would expect the symmetry group of the reduced

of those conformal

reducing by.

equations

Yang-Mills equations

(TN 34, May 1992)

Introduction.

A

of the reduced self-dual

symmetries

pair

of operators:

where λ

ℂ is

an

auxiliary complex parameter

connection in the direction

directions. In

(w, w̄), Dw leave the

=

pair

∂/∂w

we

start in Euclidean

invariant gauge

an

φ̄and '

+

of operators

Yang-Mills

∂/∂z.

For Hitchin’s equations

∂/∂w

and Dz is the covariant derivative of some

cc.

(with

and

and impose symmetries in the

in which the gauge

one

potentials

are

∂/∂w

and

independent

throw away the derivatives with respect to

we can

little

a

(i.e.

signature

(w, w̄)

of to

rearrangement):

up er L 0 equals up er D Subscript z Baseline minus lamda normal up er Phi prime comma up er L 1 equals up er D Subscript z overbar Baseline plus StartFraction 1 Over lamda EndFraction normal up er Phi overbar prime period

We

can

make this

geometric by multiplying L0 by

more

dz and L1

by

dz and

defining

φ = φ'dz.

We then obtain the one-form valued operator: upper L equals d z circled-times upper L 0 plus d z overbar circled-times upper L 1 equals upper D minus lamda normal upper Phi plus StartFraction 1 Over lamda EndFraction times normal upper Phi overbar period

The

Yang-Mills Higgs equation

Riemann surface

on a

the

are

consistency

conditions for these

operators: D2

=

φ Λφ 4;,

= Dφ 0,

Where D is the covariant exterior derivative. These group in two dimensions

they only require

Alternatively,

these

symmetries.

equations depend only

The data consists of

a

equations

are

invariant under the conformal

bundle with connection and

a

Φand Φ̄. One solution will be transformed

define

of the

as

= 0. Dφ̄

on

complex

a

to another if z

↦ z'(z) and Φ

the *-operator

on

connection D

on a

1-forms

bundle, E and

and D

on

a

the

structure to

pull

back.

quotient

space

section Γ = φ + φ̄ of

Ω ⊗ End(E). The operator L is up er L equals up er D plus left-parenthesis minus lamda StartFraction 1 minus i asterisk Over 2 EndFraction plus StartFraction 1 plus i asterisk Over 2 lamda EndFraction right-parenthesis normal up er Gamma period

Thus the field the

equations arising

diffeomorphisms preserving

The Galilean one

analogue.

non-null symmetry

from the consistency conditions of this operator

x

=

(z

+

z̃)

and

If, in (2,2 ) signature,

along ∂/∂z

we

=

have

∂/∂z̃ we

Dx

—

to the

trick, multiplying L0 by L

—

=

we

impose

obtain the Lax

λφ, L1

reorganized

part of the Higgs field associated the above

invariant under

*, i.e. the conformal transformations in 2-dimensions.

L0 where

are

=

Dw

+

null symmetry

+ ψ)

the covariant derivative in the

by

dxLo + dwL1

dw to and

=

D+

along  02;/ 02;w̃ and

pair:

λ(Dx

symmetry in the ∂z

dx and L1

one

—

x

direction to include

 02z̃ direction. We

adding together

λ(Γ + dwDx)

can

again perform

to obtain

where Γ = φdx + ψdw. To write this that

can

be

of

thought

geometrically,

more

we

introduce

a

degenerate *-operator

map from 1-forms to 1-forms:

as a

asterisk equals d w StartFraction partial-dif erential Over partial-dif erential x EndFraction comma alpha right-ar ow from bar alpha times left-parenthesi StartFraction partial-dif erential Over partial-dif erential x EndFraction right-parenthesi times d w period

The operator L then becomes: L The field equations

arising from

the

D2 where D above is

acting

as

equations. Geometrically with

a

D+

+

Γ).

consistency equations

-

0, DΓ

=

0, D

*

for this system

these

equations

determine

are:

Γ + Γ Λ Γ = 0

a

so

that the

equations

flat connection D

on a

are

all 2-form

bundle E,

together

section Γ of ω1 ⊗ End(E).

will be invariant under

equations arising

diffeomorphisms

of ℝ2

are

h(w)

These

from the

preserving

consistency the

the nonlinear Galilean transformations referred to

(w, x) ↦ (w', x') where

λ(*D

the covariant exterior derivative

It is clear, now, that the field

These

=

and

g(w)

equations

to

higher

are

=

rank gauge groups

system. At least in the SL(2)

degenerate *-operator,

dw ⊗ ∂/∂x.

previously:

(h(w), (∂wh(w))x + g(w))

free functions except that ∂wh

embed the nonlinear

conditions for this operator

Schrodinger

(the Drinfeld

≠ 0.

and KdV

equations

etc.)

into

completely

fixed

Sokolov hierarchies

case, this coordinate freedom is

and most of their a

generalizations

Galilean invariant

by

the reduction to

KdV and NLS. The

totally

null

case.

In the

case

where the

symmetries

span

an

again

a

anti self-dual null

2-plane

we

obtain the linear system L where field

again

D is

equations

a

flat connection

on a

equations

the ‘zero’ These

D + λΓ

bundle E and Γ is

section of Ω1 ⊗ End(E). The

are now:

D2 These

=

are now

=

0, DΓ

=

0, Γ Λ Γ

=

0.

invariant under the full 2-dimensional

diffeomorphism

group

another way of

writing

(preserving

*-operator). equations

are

therefore

‘topological’

equations (Strachan 1992).

and indeed

are

Their reductions include the

n-wave

the Wess-Zumion-Wit en

equations

and

those parts of the Drinfeld-Sokolov hierarchies not obtainable from the Galilean reductions. These

further reductions constant. In the

that their exists coordinates and

require

SL(3)

case one can

and

by

2 rotations gave the

SL(2,ℝ);

same

field equations

difficult to To

z

see

(1990)

as

it

was

one

preserving

hyperbolic

a

translation

unexpected symmetry;

These equations a

observed that the

the reduction by

are

invariant under

metric. While this

was

it

was

plane

and

(1987),

space-time.

on

impose

we

z̄and

+

these additional conditions.

clear from the reduced twistor correspondence in Woodhouse & Mason

this

see

=

However,

is true.

more

the group of motions of the residual space

some sense

set x

by using

rotation. This fact alone endows the 2 rotation reduction with

a

the residual translation symmetry.

in

gauge in which the components of Γ are

In Fletcher & Woodhouse

Stationary axisymmetric systems. reduction of SDYM

a

fix the coordinate freedom

impose

a a

rotational invariance with respect to θ in the symmetry in the ∂z

—

w

=

y exp(iθ)

 02;z̄ direction. We obtain the linear system:

up er D Subscript x Baseline minus i up er A plus lamda e Superscript i theta Baseline left-parenthesis up er D Subscript y Baseline plus StartFraction i Over y EndFraction left-parenthesis partial-dif erential negative up er B right-parenthesis right-parenthesis comma e Superscript minus i theta Baseline left-parenthesis up er D Subscript y Baseline minus StartFraction i Over y EndFraction left-parenthesis partial-dif erential negative up er B right-parenthesis right-parenthesis minus lamda left-parenthesis up er D Subscript x Baseline plus i up er A right-parenthesis period

We cannot

just

throw away the ∂θ as there is

explicit dependence

connected with the fact that the Lie derivative of work

independently

this is the

θ in the operators. This is

spinor and hence λ along ∂θ is

not zero.

of θ and to avoid derivatives with respect to the ‘spectral parameter’

use, instead of λ the parameter

as

a

on

simplest

function

we

To

must

gamma equals StartFraction y e Superscript i theta Baseline lamda Over 2 EndFraction plus x minus StartFraction y Over 2 e Superscript i theta Baseline lamda EndFraction

on

the

spin

bundle that is both invariant and constant

along

the

twistor distribution. nf

we

introduce the

complex

coordinate ξ =

+ iy,

x

a

bit of massage

yields

the

following form

for

the linear system: 2 up er D Subscript xi Baseline plus i StartRo t StartFraction gamma minus xi overbar Over gamma minus xi EndFraction EndRo t left-parenthesi up er A plus StartFraction i Over y EndFraction up er B right-parenthesi comma 2 up er D Subscript xi overbar Baseline plus i StartRo t StartFraction gamma minus xi Over gamma minus xi overbar EndFraction EndRo t imes left-parenthesi up er A minus StartFraction i Over y EndFraction up er B right-parenthesi period

In order to

out the invariance

bring

first operator

by dξ

coordinates γA

=

(γ0,γ1)

complex conjugate reduces,

after

and the second

some

with γ =

properties

by dξ̄ and

γ1/γ0

and

of ξA and denote the skew

of this system,

add them

similarly

we can

together.

first of all

Then introduce

for ξA. Define ξ̄A to be the

product γ1ξ0 ξ0 —

=

multiply

the

homogeneous

componentwise

γ ξ. The linear system then •

further massage, to: 2 up er D plus i StartRo t StartFraction gamma dot xi overbar Over i xi dot xi overbar gamma dot xi EndFraction EndRo t normal up er Phi xi dot d xi plus i StartRo t StartFraction gamma dot xi Over i xi dot xi overbar gamma dot xi overbar EndFraction EndRo t imes normal up er Phi overbar xi overbar dot d xi overbar

a put have we where

It on

can now

be

ξA preserving

seen

the

that the linear system is invariant under

reality

structure ξ&#pxer03br0ol4ic;A ↦ hyξA the hence and met . m

SL(2, ℝ);

the Mobius transformations

The

conditions

integrability

Γ(O(—1) ⊗ E)

that is

a

The field equations

are

dual

equations for

spinor valued

connection D

a

section of

bundle E and

on a

a

section φ

End(E).

are

up er D squared equals left-bracket normal up er Phi comma normal up er Phi overbar ight-bracket StartFraction xi dot d xi tmes normal up er Lamda times xi overbar times dot d xi overbar Over xi dot xi overbar EndFraction comma partial-dif erential normal up er Phi equals StartFraction normal up er Phi overbar Over 2 xi dot xi overbar EndFraction comma partial-dif erential normal up er Phi overbar equals minus StartFraction normal up er Phi Over 2 xi dot xi overbar EndFraction

where ∂ and  02;̄ here denote the ‘eth’ operator and its of the covariant derivative

Remarks. above to

in Hitchin’s case,

as

and

(1,0) parts

Dirac field and satisfy the

a

provides

might hope

one

background coupled

the curvature of the connection. to be able to transfer the other

equations

case, the

hyperbolic

equations

clearly

can

be transferred for g > 2

using

the

unique

metric

the Riemann surface with curvature —1. For the Galilean

instead of

analogue,

endow it with

might

one

the Riemann surface with

endowing

measured foliation which

a

boundary

*-operator introduced above. *-operator has

an

They

both determine

affine structure

the

on

whereas the measured foliation has

a measure

leaves. Nevertheless,

that

diffeomorphisms is still

the linearized a

Further

one

in the

might hope

global

context

of these

analogues

condition that will have

analysis

give

rise to any

case

for which

Thanks to

is

required

difficulty

more

the

as

analysis

Jorgen

is

a

leaves,

but

no

structure;

relations is the Thurston

concept

as

the

degenerate

structure transverse to the

transverse to the an

leaves, but

equivalence

good

existence

no

leaves,

structure on the

between the two, modulo

are

for solutions of this

theory

have *Γ covariant constant

solutions when the leaves

for the other

equations

same

complex

kind of uniformization result. Even if this is feasible, it

equations no

complex structure,

foliation of the Riemann surface, but the degenerate

could prove

one

as a

not clear that one can obtain a

perhaps

foliation,

equivalence

for Teichmuller space. It turns out that this is not the

a

to a limit of a

corresponds

indeed the space of measured foliations modulo certain

as

(0,1)

Riemann surface also.

a

In the on

Just

constitute

Their commutator

equation.

the

respectively.

‘Higgs fields’ φ and φ̄ together

So the

massive Dirac

complex conjugate,

cases.

The

totally

are

along

equation

the leaves of the

dense.

null reduction will

presumably

underdetermined anyway. This leaves the

not

Hyperbolic

required.

Andersen for conversations.

References Fletcher , J. &. Woodhouse , N.M.J. of Einsteins

equations

,

( 1990 )

L. M. S. Lecture Notes Series 156 , CUP Hitchin , N. J.

( 1987 )

The

Twistor characterization of

in Twistors in Mathematics and

stationary axisymmetric solutions Physics eds. Bailey T. N. & Baston ,

,

.

self-duality equations

on a

Riemann surface Proc. L.M.S. ,

,

Mason , L. J. & 243 271

Sparling

,

G. A. J.

( 1992 )

Twistor theory of the soliton hierarchies, J.Geom

.

Phys.

,

8,

-

.

Mason , L. J. &

Singer

,

M. A.

theory

The twistor

( 1994 )

I , to appear in Comm. Math.

Phys.

Strachan , I. A. B.

Lett. B282 , 63 66

§II.1.11 by

(TN

35, December

Introduction.

globalization the

global

self-duality equations

of twistor

in

a

in

particularly simple In

particular

Yang-Mills equations

description

examples

of

in

split signature

in

conformally

a

This is obtained

and

of the space of

parametrization

signature ( 2,2 ). Despite

by examining some

unusual features of

appealing parametrization of the

split signature on

leads to the non-linear

S2 × S2 which specializes

anti-self-dual metrics described

desire to understand the

and see

thereby give also

§11.1.9

When

one

to

twistor

considers

α-planes

give

by

of the Penrose transform in

of the Radon transform and the inverse

one

might try

compactification of ℝ4, S2 × S2/ℤ2

in S2 × S2

S2 × S2 (where

boundary

are

not

one can

simply

ask the

would be to as

the

require

arose

from

signature (2, 2 )

scattering transform,

equations

are

conformally invariant.

question

as

are

rectified if

we

so

Affine (2,2 ) Minkowski space 𝕄0 is ℝ4 with metric of

compactification

by adjoining

a

‘light

cone

at

that

one

cannot

go to the double cover,

to which fields are invariant under the

𝕄, obtained

We will

theory. Furthermore,

connected and have fundamental group ℤ2

problems

equations on ℝ4 in

that solutions should extend to

conditions eliminate all solutions in linear

Penrose transform. These

global geometry.

The conformal

the twistor

K.P.Tod in his article

conditions for solutions of conformally invariant

the first condition

immediately apply the

2. The

description

boundary

see, however, that these

the

globalization of other aspects

global

for further discussion.

signature (2, 2 ), the conformal

a

solution

graviton construction

which will appear in volume III of this work. Part of the motivation for this construction a

global

the appropriate

the treatment of the Ward construction for

for anti-self-dual metrics of signature ( 2,2 ) of the

give

signature ( 2,2 ).

correspondences

self-duality equations.

solutions of the self-dual

de Vries type part

Korteweg

1992)

duality equations

geometry, it leads to

space of the

of

.

The purpose of this note is to

solutions of the self

equations

-

Global solutions of the

L.J. Mason

1.

( 1992 ) Phys.

of

ℤ2 action).

signature ( 2, 2 ).

infinity’

to

𝕄0, is the

projective quadric in ! D;𝕡5 given by conformal structure is determined with the tangent To w

and y

planes

that the

see

as

that y · y

=

topology

of

Q

by asserting

=

w ·w

Q 0; this yields S2 × S2 ! D;𝕡5 is S2 × S2/ℤ2.

on

of the round

simply

realized

metric dΩ2

sphere

quadratic

that the

light

in ℝ6 of

form

Q

cones

of 𝕄 are the intersections of 𝕄

S2/ℤ2 diagonalize Q using

ℝ6 such that Q

on

set of a

The

signature (3,3).

of 𝕄.

of 𝕄 is S2 ×

The conformal structure is

pullback

zero

on

y

—

on

pair

a

y. Set the scale

•

the double

each factor and

of Euclidean 3-vectors

by requiring

! D;𝕡5, (w, y)

in ℝ6. However, in

=

0 in

=

points

topology

coordinates 1 also

of

the

cover

w ·w

𝕄̃ = S2 × S2

1

=

(—w, —y)

~

so

so

the

by taking

the

the difference

taking

d

where p1,P2 and

so

the

are

projections

global correspondence.

which the fields for these

are

(or

cases

defined

analytic

components of

This is ℤ2 invariant

U

correspond

to

Points of Lx

by

null self-dual U

𝕡𝕋(U),

the

space-time U,

complexification

we

shall

on

correspondence

assume

of U which will also,

𝕡𝕋(U)

by

an

all fields abuse of

to be the space of connected

in U. For Z

2-planes (α-planes)

𝕡𝕋(U)

we

will denote the

by Ẑ.

can

be

equivalently defined

the twistor distribution

α-planes

are

small

original region

consider the

now

metrics).

the U. We then define its twistor space

corresponding α-plane in

over

on a

where the

cases

ᵔ ;̃. We will

cover

to define the twistor space of a

by

totally

Twistor space,

We will be interested in the

𝕄 and its double

and consider fields

be denoted

notation,

are

deformations thereof for ASD

As, usual, in order

𝕡𝕋(U)

respectively.

descends to 𝕄.

3. The

are

onto the first and second factors

in U.

the

quotient

spanned by πA'▿AA'.

Conversely, points

α-planes through

as

U

x

of the

projective spin bundle

By definition, points

correspond

to

𝕔𝕡1's

of

denoted Lx in

𝕡𝕋(U) 𝕡𝕋(U).

x.

correspondence for 𝕄. Just as compactified complexified Minkowski space 𝕔𝕄is the complex lines in Ǖ 4;𝕡3 via the complex Klein correspondence, � is the space of real lines

3.1. The space of

in ℝ𝕡3 via the real Klein 𝕄 are

correspondence.

In the context of the

complex lines in Ǖ 4;𝕡3 that intersect ℝ𝕡3 in

in 𝕔𝕡3 that

are

mapped

into themselves

by

the

a

real line.

complex correspondence, point

Alternatively, they

complex conjugation

Zα

—>

are

complex

Z̄β given

by

of

lines

standard

complex conjugation, component by component. According Z

to the definition above we have

𝕡𝕋(ᵔ ;)

=

ℝ𝕡3 and any real line in ℝ𝕡3 through Z corresponds

intersects 𝕄 in

an

𝕔𝕡3. Given to a

point

Z

𝕋𝕡3,

then if Z

=

Z̄,

in 𝕄 on Ẑ. In this case, Ẑ

ℝ𝕡2. If Z ≠ Z̄ then the complex line through Z and Z̄ is real and corresponds

point of 𝕄. In fact the complex α-plane Ẑ intersects 𝕄 in the unique point corresponding

to a

to

this line. 3.2. Linear theory.

The linear

by

X-ray

Fritz-John

wave

using

equation

on

the

problem

completely

was

solved in the

transform. In twistor notation, the

ℝ4 satisfying appropriate boundary

case

general

conditions

can

of the

equation

wave

solution of the hyperbolic

be obtained from the

integral

formula phi tmes left-parenthesi x Superscript up er A times up er A prime Baseline right-parenthesi equals contour-integral f times left-parenthesi x Superscript up er A up er A prime Baseline pi Subscript up er A prime Baseline times com a pi Subscript up er A prime Baseline right-parenthesi times pi Superscript up er A prime Baseline d pi Subscript up er A prime Baseline period

Here f is

a

freely specifiable equation

wave

One

might naively

by

This

oriented lines in ℝ𝕡3. This is ᵔ ;̃ the double orientation of the line and of the

point

wave

there is

Actually,

is that these solutions

weight

there

choices for

as

far

of the

as

two

possible

the twistor

tautological

which is wrong down

as even

as

equation possible

a

are

the inverse conformal are

bundle from as

functions

O[— 1],

ℝ𝕡5.

is the

𝕡𝕋(ᵔ ;̃)

a

,

point

in this

boundary

the closure of the any

Proof.

a

has

a

by

just

topologically

on

of ℝ𝕡3

ᵔ ;̃ a small

are

the

! D;𝕡3,

integration,

defined

one

the space of

on

under reversal of

that there

no

are

solutions

of the

wave

or

Grgin phenomena—the

equation

are

sections of

O[— 1],

S2 × S2/ℤ2

on

p

real

the Mobius bundle. The correct choice

are

actually

as

this is

just

the restriction

sections of the trivial bundle

So

one can

simply

write them

O[— 1].

complex thickening

correspondences

for ᵔ ;̃. We

of ᵔ ;̃. We have:

by gluing together two copies of 𝕔𝕡3, thickening of ! D;𝕡3 using the identity map.

space obtained

small those

on

the

boundary

of the

glued

down

region—each

the other copy of 𝕔𝕡3 with which it would be identified if were

glued

down.

Any

open set of one such

point

intersects

the other sheet.

on

complement

see

to the

the metric

odd sections of

some

points

partner

of its partner

Points in the

intersect 𝕄 in

will

We must therefore study twistor

together along

neighbourhood

neighbourhood

we

owing

The solutions above

as

ultrahyperbolic

𝕄.

confusion here

(non-Hausdorff)

Remark. The non-Hausdorff

perform

Clearly ϕ changes sign

the trivial bundle

S2 × S2 but

solution of the

the space of lines in

actually

the twistor correspondence is concerned.

on

on

is concerned is the Mobius bundle

shall abuse notation and denote also Lemma 3.1.

on

function

that ϕ is

of 𝕄.

anti-Grgin. Solutions

3.3. The correspondence for ᵔ ;̃.

denoted

a

lines and in order to means

a

integral sign.

naturally

cover

bundle. Given

correspondence

far

ℝ𝕡3. That ϕ is

on

does not descend to 𝕄. Indeed,

so

conformally invariant

Remark.

ϕ is

by integrating f along

orientation of the line.

an

O(—2)

differentiation under the

think that the function

𝕄. However, ϕ is defined needs to have

smooth section of

follows

of

trivial

(the thickening of) ℝ
D561; in Ǖ 4;𝕡3 correspond

region

and these

are

necessarily

to α-planes that

covered by two components

in the double small

points

ᵔ ;̃. Whereas,

cover

topology ! D;𝕡2

𝕄 with

thickening of)

(thickening of) ! D;𝕡3 correspond

in the so

that when

takes the double

one

to

a-planes

in

(the

cover

the α-plane has

! D;𝕡3

and the double

topology S2.

𝕡𝕋(ᵔ ;̃)

Thus

covering

is

double

covers

glued together

over

𝕔𝕡3 the

with

one

piece lying given by the

multiplying 90

in f

𝕡𝕋(ᵔ ;̃)

lines in

complex

and the other in

the intersection of

P

complex

.

This

thickening

that

yields

! D;𝕡3

line with

from the intersection with C

arrow

of the

complement thickening of ℝ𝕡3.

We reconstruct ᵔ ;̃ as the space of

the line is

the

over

are

of

cut into two

pieces by ᴡ D;𝕡3

the space of oriented lines in

ℝ𝕡3;

and the orientation is determined

to that with C

by

by i, thereby rotating it by

degrees.

The non-Hausdorffness arises ᵔ ;̃ and 3.4.

twistor space is

as

deforms

as one

Complex conjugation.

a

a

quotient

leaf of the foliation, it

Complex conjugation on

can

[Z]

goes to

correspond

to

[Z̄]

points

P

and the real lines of the

equation

wave

means

of the

fact that

complexified)

(the slightly

break into two disconnected leaves.

thickening

of ᵔ ;̃ sends α-planes to

covers

the standard

complex conjugation

that it

interchange

k

conjugation

are

l

and

.

Thus

those described above that

of ᵔ ;̃.

4. The X-ray transform. of the

bundle of

spin

the small

α-planes and hence leads to a conjugation on 𝕡𝕋(ᵔ ;̃). This on Ǖ 4;𝕡3 that fixes ! D;𝕡3 and is lifted to 𝕡𝕋(ᵔ ;̃) by requiring P

of the

on

We

can now

ᵔ ;̃ correspond

Meyer-Vietoris

H1(𝕔𝕡3,O(-2))

=

=

transform in this context. Solutions

X-ray

H1(𝕡𝕋(𝕄̃), O(—2)). These can be studied by covering of 𝕡𝕋(ᵔ ;̃) by P and P Using the

to elements of

sequence 0

understand the

using

the

.

H0(𝕔𝕡3, O(-2))

we

find

up erHSuperscript1Baselineleft-parenthesi double-struckup erPtimes left-parenthesi ModifyingAbove Withtilderight-parenthesi right-parenthesi equalsup erHSuperscript0Baselineleft-parenthesi double-struckup erCtimesdouble-struckup erPSubscriptplusSuperscript3Baselineintersectiondouble-struckup erCtimesdouble-struckup erPSubscriptminusSuperscript3Baselinetimescom atimes left-parenthesi negative2right-parenthesi right-parenthesi equalsup erHSuperscript0Baselineleft-parenthesi double-struckup erRtimesdouble-struckup erPcubedcom a left-parenthesi negative2right-parenthesi right-parenthesi

and the formula for the Penrose transform

5. ASD

fields and the inverse

Yang-Mills

fields

on

ᵔ ;̃,

we can

Theorem 5.1 There is group G

(a)

a

=

SL(n,ℝ)

c2(E) (b)

for G P

or

holomorphic =

=

k/2,

using

obtain a

an

representatives

scattering

Yang-Mills

and second Chern class

vector bundle E

on

𝕋𝕡3

with

of the

connections

C2(F)

is

precisely

=

k

vanishing

on

general on

the X-ray transform.

For anti-self-dual

transform.

analogous parametrization

1-1 map from ASD

SU(n)

these

a

Yang-Mills

solution.

bundle F with structure

ᵔ ;̃, and pairs consisting of

first and third Chern

classes,

and

and

SL(n,ℝ),

a

nondegenerate with P

=

P̄t

map P

with P̄ = P-1 or, for G

=

SU(n),

Remark. Here Ē denotes the the bundle F

Given

or

a

space-time, F̄

bundle

will be Ē* for

F when G

=

The Ward transform

Proof. C

on

on

𝕡𝕋(ᵔ ;̃),

gives

restrict it to C

we can

F̄ = F* when G

or

1-1 map from ASDYM fields

a

by

For

SU(n).

=

ᵔ ;̃ to bundles

on

Ǖ 4;𝕡3.

to obtain the bundle E on

shows that it must be

conjugation

E.

𝕡𝕋(ᵔ ;̃).

on

The restriction to

Ē for

G

=

SL(n, ℝ)

SU(n) by complex conjugation.

The rest of the data of the bundle

given

SL(n, ℝ)

=

different bundle, but the complex

a

vector bundle defined

conjugate holomorphic

(b).

in part

For

compatibility

𝕡𝕋(ᵔ ;̃)

on

with the

is encoded in the

patching

complex conjugation,

P

P must

over

satisfy

𝕔𝕡_ ∩ Ǖ 4;𝕡+

as

the conditions

stated. Note that this illustrate the

1)

implies that

general

When k

Ē Remark. Mason & a

E

or

are

two extreme

examples

of the above that

Ē

=

P

satisfying all

over

E*

=

Ǖ 4;𝕡3,

depending

P̄-1

or

1992 it

was

=

patching function

on

came

bundle E

and the

transform is

scattering one

effectively by paradigm.

could realize the inverse

patching function

‘scattering

data’

consisting

parametrizing the radiative/dispersive modes in linear

theory.

In this four dimensional

is

played by

the bundle E

is

played by

the map P that

Ǖ 4;𝕡3

on

scattering

generalizes

Deformations of

could be factorized into

an

‘algebreo geometric’ analogue

in

of C∞ functions of the real twistor coordinates

we see

linearized

that the role of the

analogue

algebro-geometric part

and the role of the

scattering

transform

straightforward

as

requires

scattering

data

it will

the examination of symmetry

change

the

𝕡𝕋(𝕄).

The nonlinear on

graviton

construction

boundary

S2 × S2 correspond

to

implies

(small)

conditions

preserved,

the

gluing map

P from

some

open set in g

that

(small)

deformations of

Since Ǖ 4;𝕡3 is rigid, the only deformable part is the gluing along ℝ𝕡3. In order structure is

no

global structure.

deformations of the conformal structure

reality

as

not clear how the normal form for the

was

the Radon transform from linearized theory.

reductions of this framework. This is not

6.

no

transform

In

of the solution—these reduce to the Fourier transform

analogue

that has

A direct connection with the inverse

and hence the details of the

Ǖ 4;𝕡3 satisfying

on

G.

shown that

about. The

a

part which parametrized the ‘solitonic’ degrees of freedom of the solutions and have

theory

to

P̄t on ℝ𝕡3.

all the information is contained in

The connection with the inverse

Sparling

P

coordinate realization of the Ward transform, but it

linearized

serve

0, the bundle E is necessarily trivial, and all the information is contained in the

=

When P extends =

There

even.

case.

matrix function

2)

k is

to

ASD

𝕡𝕋(ᵔ ;̃).

guarantee that the

to one in C

must be

with the

compatible P-1

Take a

small

a

small

analytic

embedding also

sending and is

the

ρ̄ of U into

are

the

point

and acts

we

such real

gluing

in Ǖ 4;𝕡3

or

the condition that

follows:

P

P

point

glued

down twistor space

can

This

conjugation clearly

fixes the

.

1

—

maps

map{ℝ𝕡3

ρ&#o x0304;

=

is

ρ-1.

by image of ! D;𝕡3

are

divided into two parts

by

the

by constructing down

glued

region

.

above. These —>

conjugate

then be defined

1 map from ASD deformations of the conformal structure

as

the

ρ has

globally.

and half in C

a

that

is then done with the map P

to C

of the deformed

so

complex conjugate embedding (it

with deformed ASD conformal structure is then reconstructed

P

have

of the

complexification

lines in the deformed space that

half in

arranged

yields

embedding of ! D𝕡3 into Ǖ 4;𝕡3 ! D;𝕡3 in 𝕔𝕡3. Then we also have

U of

gluing from

to the

C

space-time

Thus

ℝ𝕡's

which is the

The deformed

be

can

as

ρ of the standard

neighbourhood

complex conjugation map

complex

and

Ǖ 4;𝕡3

a

This

.

map. This

deformation

extension to

antiholomorphic

The

conjugate

analytic

holomorphic).

The

that sends g

conjugation

P̄ where P̄ is the

=

be

can

Ǖ 4;𝕡3} / Dif {ℝ𝕡3}

thought as a

of

as

the space of

diffeomorphism

of

S2 × S2 and

on

(analytically)

! D;𝕡3

embedded

does not affect the final

P. The

Examples. a

basic idea is to take

gluing

a

some

use

real slice V

=

fixed amount

2 × 2 matrices

! D;𝕡3

sits inside

iλA∂/∂λA

quotient by

the

—

obtained from

of LeBrun’s

the

along

version of the construction of Jones & Tod Use

are

hyperbolic Gibbons-Hawking ansatz, LeBrun (1991). The global holomorphic vector field on Ǖ 4;𝕡3 that is real on ! D;𝕡3 and to drag the

split signature analogue

standard

that K.P.Tod writes down in his article in volume III

examples

PSU(2)

as

with

iμa∂/∂μA corresponds

S2. On ᵔ ;̃ the symmetry

the

λA

can

be

on

twistor space with columns of μA.

SU(2) complex conjugate

a

global

represented

quadric Q

with coordinates

that it rotates

so

just

one

(λA,μA).

The

The vector field

right multiplication by diagonal SU(2)

to

vector field is the

complexified

of the vector field. This is

(1985).

coordinates

homogeneous

as

imaginary part

matrices. The

([λA], [μa„])

and real slice

of the S2 factors and leaves

the other invariant. Choose some

small

a

real

analytic

thickening

(exp(f)λA, exp(—f)μA)

function

The

by gluing

space-time

one

S2, f([λA],[μA] defined

of the real slice. in

P

out the lines λA = 0 and μA obtained

on

where =

we

and

[μA]

given

quadric Q+

are close to

μ̂A and continue it

to another

a

in

being conjugate. Thus,

complex Qover

line bundle

some

in standard form in Jones & Tod d s squared equals d times normal upper Sigma 3 squared plus left-parenthesis d phi plus omega right-parenthesis squared slash upper V squared

=

identify (exp(—f)λa,exp(f)μA)

0, the twistor space 𝕡T is

copy of the

metric is

[λA]

Then

for λA

over

as

we

with take

the space

thickening of the

(1985)

P if

to

Q̃

real slice.

where here d

,

is the

Einstein-Weyl

and

and V

hyperbolic space

u

corresponding

space

to d

It is

conformal

a

4dζdζ̄/sin2 θ(1

+

Discussion.

It is

rescaling (the

|ζ|2)2

possible

respectively orthogonal

*3 is the

hodge

dual with

for the components of

f(wAπA, πA)

be put into this —

cuts the space in half. This uses the same construction but

generated by πA∂/∂wA

(wA, πA). has

can

trivial part is to show that the 3-metric d&#xθ2 /sin2 θ

non

hyperbolic metric).

S2 ×S2 but whose null infinity

translation symmetry

f

*3dV where

Lorentzian

to write down metrics that are Ricci flat with conformal structures that

with

where

main

the Lorentzian

is

over

:=

=

exercise to show that the metrics in Tod’s article

extend a

dw

are

just

.

straightforward

a

form after

satisfy

which is

quadric

the parts of an ASD Maxwell field that

are

and tangent to the symmetry direction and thus

respect

to the

The

gluing

homogeneity

where the real slice is

identifies wA

zero

and is

ifπA

—

rapidly decreasing

given by

with wA +

g

on

now

as

wAπA/(

real values

if

ℝ is harmonic iff φ o

f

by

K.P.Tod

(TN 29,

November

N of Riemannian manifolds M, N with the :

M

→

ℝ is.

As

a

concrete

example,

1989)

following

take M to be

ℝ3

with coordinates x,y,z and N to be

ℝ2

with coordinates u,v. The map

φ is

defined

by giving

u(x,y,z), v(x,y,x) satisfying nabla u equals nabla upsilon equals 0 equals nabla u dot nabla upsilon times semicolon StartAbsoluteValue nabla u EndAbsoluteValue squared equals StartAbsoluteValue nabla upsilon EndAbsoluteValue squared

(1) In this

Baird & Wood

case

the tangent bundle of the defined harmonic

φ is locally

find that

(1988)

line.

complex projective

They

go

in this case, and also in the

morphisms

defined

by

a

holomorphic

to use this fact to

on

of

theory of this property

of N

points

give

curves

harmonic

morphisms

shear-free

congruence.

Now T𝕡1

corresponds there is In

a

is

3, dim N

=

equivalent

to the condition that this congruence

of the flat metric

geodesics

parameter surface in T𝕡1. As

mini-Kerr theorem that this surface is

particular,

when dim M

case

this leads to

an

on

ℝ3 and

holomorphic

a

of

so a

might anticipate

one

T𝕡1,

classify globally

anything,

curve

is the relation

2, the inverse images

=

in M. One purpose of this note is to observe that the

is ths space of to a 2-real

of φ. In the

in

S3 → surface and H3 → surface.

cases

Since T𝕡1 is the mini-twistor space of ℝ3 it is natural to wonder what, if to twistor

curve

defining property of

curves

be

a

geodesic

congruence of

and

geodesics

from the Kerr theorem,

iff the congruence is shear-free.

formula for such congruences: if the generator is

explicit

up er L equals StartFraction 1 minus alpha alpha overbar Over 1 plus alpha alpha overbar EndFraction StartFraction partial-dif erential Over partial-dif erential z EndFraction plus StartFraction alpha plus alpha overbar Over 1 plus alpha alpha overbar EndFraction StartFraction partial-dif erential Over partial-dif erential x EndFraction minus StartFraction i left-parenthesi alpha minus alpha overbar ight-parenthesi Over 1 plus alpha alpha overbar EndFraction StartFraction partial-dif erential Over partial-dif erential y EndFraction

then

α(x, y, z)

is

f(x( 1 for

given implicitly by —

α2) + iy( 1 + α2)

arbitrary holomorphic f

Baird & Wood

or

—

2αz, α)

holomorphic

=

and

As Baird & Wood remark, to find solutions of

I

am

this

or

in

spinors

homogeneous

F(xABαAαB, αc) F

(a

=

0

formula similar to this is in

(1988)).

into the class of non-linear more on

0

(1)

was

set as a

problem by Jacobi.

solvable

differential-geometric problems

by

twistor

This

theory (see

now

falls

II.1.13 for

subject).

grateful

to John Wood and Paul Baird for

telling

me

about harmonic

morphisms.

References Baird , P. & Wood , J.

( 1988 )

Math. Ann. 280 , p. 579 603

Baird , P. & Wood , J.

( 1987 )

Ann. Inst. Fourier, Grenoble 37 , p. 135 173

Baird , P. k, Wood , J.

( 1989 )

-

-

Harmonic

space-forms University ,

2.

.

morphisms

.

and conformal foliations

by geodesics

of threedimensional

of Melbourne Department of Mathematics Research Report

§II.1.13 More

on

I wish to make

harmonic

morphisms by

observations which

some

and to describe what I think is twistor theory

(though

might

way of

a new

geodesic

a

(TN 30,

June

1990)

make my Twistor Newsletter article II.1.12 , clearer,

looking at

in this last connection

Recall that the generator of

K.P.Tod

see

the Kerr theorem

appropriate

to Riemannian

Hughston & Mason 1988).

shear-free congruence is

up er L equals StartFraction 1 minus alpha lpha overbar Over 1 plus alpha lpha overbar EndFraction StartFraction partial-dif erential Over partial-dif erential z EndFraction plus StartFraction alpha plus alpha overbar Over 1 plus alpha lpha overbar EndFraction StartFraction partial-dif erential Over partial-dif erential x EndFraction minus i StartFraction left-parenthesi alpha minus alpha overbar ight-parenthesi Over 1 plus alpha lpha overbar EndFraction StartFraction partial-dif erential Over partial-dif erential y EndFraction

(1) where

α(x,

y,

is

z)

given implicitly by f left-parenthesis x left-parenthesis 1 minus alpha squared right-parenthesis plus i y left-parenthesis 1 plus alpha squared right-parenthesis minus 2 alpha z comma alpha right-parenthesis equals 0 period

(2) In the interest of

(a)

If α =

u

clarity,

I should have added in II.1.12 that:

+ iv then

▯2u ie. the

(b) α(x,

(c)

For

complex function α(x, z)

y,

is constant

given

a

along

▿2v

=

y,

L

constant value of

=

0

=

▿u·

|▿u|2

▿u;

z) obtained from (2)

as

given by (1)

α,

the real and

an

‘elementary’ interpretation

of

(2)

as

|▿u|2

defines the harmonic

morphism of II. 1.12

.

.

imaginary parts

L is then tangent to the line of intersection of the two There is

=

follows:

a

of

(2)

each define

a

plane

and

planes.

line in 𡇓 is

given by

an

equation

of the

form r times normal upper Lamda times a equals b

(3) where

a

is

number

α

a

unit vector and b is

according

to

(1)

,

then

orthogonal

(3)

can

Parametrise the unit vectors with the

to a.

complex

be written

x left-parenthesis 1 minus alpha squared right-parenthesis plus normal i y left-parenthesis 1 plus alpha squared right-parenthesis minus 2 alpha z equals beta

(4) where

β parametrises

the

arbitrary holomorphic ‘direction’,

Argand plane orthogonal to

function of α, ie. the

both understood

as

complex

a.

‘intercept’

is

Now an

(2)

is

equivalent

to

arbitrary holomorphic

morphisms’

in Riemannian twistor

is also the

theory (when

be

an

function of the

variables. When this function is linear, the congruence is

3-dimensional picture of the Kerr congruence and is sketched in Baird & Wood ‘Harmonic

taking β to

answer

to the

there

are no

a

(1988).

question ‘what do you get from the Kerr theorem shear-free

geodesic congruences)?’

To

see

this,

take

a

twistor function

homogeneous holomorphic

in twistor space which is ‘real’ in the

F(Za)

appropriate

sense

and intersect the zero-set of F with

to Riemannian twistor

This

theory.

a

line

gives

upper F left-parenthesis a plus b zeta comma minus b overbar plus a overbar zeta comma 1 times comma zeta right-parenthesis equals 0

(5) writing (1,ζ)

for the

π-spinor (rather

coordinates

complex

than

(1,α)

which I used at the

beginning).

Here

a

and b

are

ℝ4 and the metric is

on

d s squared equals d a times d a overbar plus d b times d b overbar period

(6) Solving (5) gives

function

a

ζ(a, b, ā, b̄)

with

StartFraction partial-dif erential zeta Over partial-dif erential a overbar EndFraction plus zeta StartFraction partial-dif erential zeta Over partial-dif erential b overbar EndFraction equals 0 semicol n StartFraction partial-dif erential zeta Over partial-dif erential b EndFraction minus zeta StartFraction partial-dif erential zeta Over partial-dif erential a EndFraction equals 0

(7) from which it follows that ζ has

vanishing Laplacian

and null

gradient

in the metric

(6) :

StartFractionpartial-difer ntialzetaOverpartial-difer ntialapartial-difer ntialaoverbarEndFractionplusStartFractionpartial-difer ntialzetaOverpartial-difer ntialbtimespartial-difer ntialboverbarEndFractionequals0semicol nStartFractionpartial-difer ntialzetaOverpartial-difer ntialaEndFractiondotStartFractionpartial-difer ntialzetaOverpartial-difer ntialaoverbarEndFractionplusStartFractionpartial-difer ntialzetaOverpartial-difer ntialbEndFractiondotStartFractionpartial-difer ntialzetaOverpartial-difer ntialboverbarEndFractionequals0

(8) If

think of

we

conditions for

(5)

it

implies that,

if

ζ satisfies (8)

as

the

ζ as

to define a harmonic

well

then

as

so

holomorphically-related For the

stereographic defining

does any harmonic

suppose that

converse

a

coordinate

on

the

morphism from ℝ4

harmonic

morphisms η satisfies

to

(8)

is

𝕊2. However, on

2-planes,

and is constant

on

flat

easily

(7)

flat

function ofζ In this sence,

constant on flat

(8)

then

ζ is constant

morphism,

holomorphic

sphere,

one

seen

to be the

is stronger in that Note that

2-planes.

ζ defines

a

of which satisfies

2-planes.

of

family (7)

.

Define ζ by

StartFraction partial-dif erential eta Over partial-dif erential a overbar EndFraction plus zeta StartFraction partial-dif erential eta Over partial-dif erential b overbar EndFraction equals 0 times s o times t h a t imes a l s o times StartFraction partial-dif erential eta Over partial-dif erential b EndFraction minus zeta StartFraction partial-dif erential eta Over partial-dif erential a EndFraction equals 0

then it follows from the conditions

(7)

and

(8)

on

η that ζ is

a

holomorphic function

of η and

so

in turn satisfies

.

To summarise, constant on

a

family

2-planes

of

holomorphically-related harmonic morphisms from ℝ4

defines and is defined

by

a

holmorphic function

to

𝕊2 which

in twistor space. This

can

are

be

called ‘the Riemannian Kerr theorem’. This view of the Kerr theorem

arose

in discussions with Henrik Pedersen.

References Baird , P. & Wood , J.

Hughston 5 , p. 275

.

,

( 1988 )

Math. Ann. 280 , p. 579 603

L. P. and Mason , L. J.

-

( 1988 )

A

generalized

.

Kerr-Robinson theorem , Class.

Quant.

Grav.

harmonic

§II.1.14 Monopoles, October

(TN 31,

In this paper and

morphisms

is

a

real

satisfying harmonic

again

curve

spinor

fields defined

means

of the

morphisms

real structure

a

a

in TS2

curves

equations,

conformality condition,

one

second order

equations. Furthermore,

therefore,

description the

express

may

give

viewed

to

Now let

φ be

φ we are able to

γ1,γ2

a

harmonic

interpret

of twistor bundles.

harmonic

morphisms

(Baird &

Wood

φ. However, the that

they

from

1988),

morphism

the

of the

Higgs

we are

continuously

equivalent

to an

γ1 ,γ2

A direct

objects.

of

a

from

a

a

(m 2)-dimensional

satisfying

(see

section

1)

domain in ℝ4 to in terms of

description a

between

on

ℝ3 and

simple first

a

the

region

morphisms some

examples

of

physical

on

ℝ3

particular

surface.

ℝ3. This

on

At

such

as

interest

may also be

direction.

regular points

holomorphicity properties

of

of sections

of the second author for submersive

surface

advantage

some

field

For instance, in

Unlike the ℝ3

(Wood 1986).

may have ‘unremovable’

critical

morphisms

an

first order

relationship

(multiple-valued) spinor

monopoles.

a

equations explicitly. Indirectly,

field. Note that harmonic

has the great

over

morphism from —

complex

field to the curvature of

valued function with

able to describe harmonic

in terms of

Another connection between harmonic harmonic

a

same

Riemannian 4-manifold to

spinor formulation

extend

M4)

in

in terms of spinor fields

spinor equations

the sections

ideas,

TS2,

studied

we

gradient

This ties in with the a

surface

their classification,

(see below), obtaining

ℝ4 invariant under translation in

on

complex

line of

curve

the

elliptic equation

spinor

algebraic

Thus, remarkably, these solutions

illuminating picture

of the

This

1988).

relating

monopole

k in the

degree

write down all solutions to these

we

morphisms

of

τ)

apparently independent

solutions of Prasad & Rossi,

singularities

harmonic

as

static

a more

axially symmetric

corresponds

so

a

(Hitchin 1982).

from

canonically

&. Wood

4-dimensional Minkowski space

we

an

classified in terms of the

are

k may be constructed

(Baird

solutions remains obscure. However, here

order

charge

Ansatz Ak

Atiyah-Ward

invariant under

algebraic

SU2-bundle, the other

on

of

monopole

from domains in Euclidean 3-space

different

quite

ℝ4 (also

3- and 4-spaces.

on

certain other technical conditions. In

in terms of

to two

by

(that is,

curve

by P.Baird and J.C. Wood

draw attention to the connection between static monopoles, harmonic

we

Hitchin has shown how every static

algebraic

fields

1990)

Introduction.

an

spinor

and

morphisms

singularities

at critical

solutions

that the

spinor

physical

fields is the

can

case

points

of

be chosen

points.

morphisms

and

Riemannian m-dimensional manifold M to conformal foliation of M

by

following: a

surface is

A submersive

locally

minimal submanifolds. In the

case

m

3,

=

we can remove

foliations

are

the Riemannian

rest mass fields

zero

Such conformal

by geodesics.

of the well-known shear-free, null

analogue

relativists in connection with

by

much studied

the restriction ‘submersive’ and the foliation is

geodesic

congruences,

Baird & Wood 1991, 1992

(see

and II.1.12 , II.1.13 ).

Throughout, enables

us

to consider

1. Harmonic φ : M

we use

spinors

spinors

morphisms

described in the

as

defined

vector space with metric of

on a

from ℝ3 in terms of

This

arbitrary signature.

Let M ⊂ ℝm be

spinors.

(1986).

open subset and

an

horizontally conformal if

𝕔 a smooth map. Then φ is said to be

—>

of Penrose & Rindler

Appendix

normalup erSigmaUnderscriptaEndscriptstimesleft-parenthesi StartFractionpartial-difer ntialphiOverpartial-difer ntialxSuperscriptaBaselineEndFractionright-parenthesi squaredequals0com a

(1.1) and harmonic if

normalup erSigmaUnderscriptaEndscripts imesStartFractionpartial-difer ntialphiOverleft-parenthesi partial-difer ntialxSuperscriptaBaselineright-parenthesi squaredEndFractionequals0period

(1.2) Here

coordinates

(x1,... xm) are standard ,

is called

a

harmonic

if both

morphism

on

(1.1)

ℝm and summation is

and

hold. There

(1.2)

mappings

over a

which

pull

characterized

as

functions to local harmonic functions

(see Fuglede

1978, Ishihara 1979).

harmonic

example,

mappings

morphisms

which send Brownian

The two latter definitions then

paths

to Brownian

generalize

manifolds, the equations (1.1) (1.2) becoming ,

Ishihara

1979).

Note that in the

case m

which

are

many more, see below. For all m the

holomorphic

in the range, that is, if φ : M where ρ : V

→

(1.1)

(1.2)

—>

only

equations

V ⊂ 𝕔 is

a

mappings

complicated solutions to

functions

are

φ(x1

in this

morphisms

with respect to

local

solution, then

complex

so

is the

coordinate

a

on

the

spinor

covariant derivatives DAB D n

i

i

±-holomorphic

case m

≥ 3 there

map. In

are

particular

M → Ǖ 4; we

may

N; these will satisfy

N.

x Superscript Baseline Superscript a Baseline equals left-parenthesi x Superscript 1 Baseline comma x squared comma x cubed right-parenthesi left-right-ar ow x Superscript up er A up er B Baseline equals StartFraction 1 Over StartRo t 2 EndRo t EndFraction Start 2 By 2 Matrix 1st Row 1st Column x squared plus i x cubed 2nd Column minus x Superscript 1 Baseline 2nd Row 1st Column minus x Superscript 1 Baseline 2nd Column minus x squared plus i x cubed EndMatrix period

∂/∂xa,

are

composition ρ o φ :

Riemann surface

spinors by the correspondence

=

1979).

Riemannian

(see Fuglede 1978,

in the

We consider ℝ3 with its standard Euclidean metric. Vectors xa may be

Writing  02;a

are

invariant under conformal transformations

φ : M → N with values in a

case

,

ix2);

& Davie

arbitrary

(1.1) (1.2) ±

The map φ

Equivalently they

between

𝕔 is any weakly conformal (equivalently ±-holomorphic)

consider harmonic and

or

2 the

antiholomorphic)

(by

we mean

=

more

to

m.

back local harmonic

paths (see Bernard, Campbell

the notion

,

equivalent definitions,

other

are

are

for

1,...

=

given by

expressed

in terms of

(see

also Sommers 1980

⊂ ℝ3

Let M

conformal if and

be

equation (14) ). open subset and

an

only

M

φ :

is

▿φ is

a

spatial

null vector

field),

But the symmetry of DAB φ then

implies

conformal if and

horizontally

φ

Then

φ is horizontally

0

=

and this holds if and

DAB φ

that φ is

mapping.

if det DAB

(that

smooth

→ 𝕔 a

=

that ηB

if

only

ξAηB.

—

λξB for

scalar function λ. So

some

we

deduce

if

only

up er D Subscript up er A up er B Baseline times phi equals xi Subscript up er A Baseline times xi Subscript up er B Baseline

(1.3) for

some

If

spinor field ξA

defined

φ is horizontally conformal,

morphism, if and

only

M.

over

so

that

holds, then φ is harmonic, and

(1.3)

so

is

a

harmonic

if up er D Subscript up er A up er B Baseline times xi Superscript up er A Baseline xi Superscript up er B Baseline equals 0 period

(1.4) Conversely,

as

in Sommers

i

given by

.

THEOREM 1.1. between harmonic

(1980), given

Combining this

Let M ⊂ ℝ3 be

morphisms φ :

a

a

spacial

null vector field va ↔ μAμB, then curl va is

with equation

simply

M → 𝕔 and

(1.4)

obtain

we

,

There is

connected open subset.

spinor fieldsξA on

M

satisfying

a

correspondence

the

spinor equation

up er D Subscript up er A up er B Baseline times xi Superscript up er A Baseline xi Superscript up er C Baseline equals 0 period

(1.5) REMARKS 1.2.

(i)

This

correspondence

φ̃ = φ + c, for

(ii)

The

spinor

morphism,

2)

some

field

ξA

then at

the derivative

a

is one-to-one if

identify

we

complex

number c,and define the

extends

continuously

critical

point (i.e.

a

over

point

components),

then ǁDAB

φǁ2

=

|ξA|4,

where

A

spinor field ψAB

time

independent

a

field up to

points

of

sign only.

φ. For

if

φ is

a

harmonic

where the differential of φ has rank less than

collapses completely (Fuglede 1978),

follows that ξA extends with the value 0 at

(iii)

spinor

critical

to be the Hilbert-Schmidt norm of the derivative

its

morphisms φ, φ̃ whenever

two harmonic

(the ǀξAǀ2

that is DAB φ ≡ 0. sum =

Setting ǁDAB φǁ2

of the squares of the moduli of

|ξ0|2

+

|ξ1|2.

Since φ is smooth, it

critical point.

= ξAξBsatisfying equation (1.5) may be interpreted solution to Maxwell’s equations. This is clear

as a

null,

source

by expressing ▿φ

=

free,

E + iB

in real and

(nullity), (iv)

Then horizontal

imaginary parts.

curl E

curl B

=

0 is automatic and

=

In Baird & Wood

(1988)

corresponds

holomorphic

lines in

ℝ3,

to

a

it should be

on

ℝ3.

thought

of

corresponding

as

where

of the harmonic

line field

a

morphism.

ℝm, is determined by

an

Such

div E

=

In fact such

a

div B was

=

=

defined. This

an

oriented line

morphism. Globally

harmonic

morphism,

infinitesimally close) correspond multiple-valued

0.

the space of oriented

to a harmonic

multiple-valued

to a

equation

TS2,

ǀEǁ ǁBǁ

0 and

=

determines

a curve

corresponds

lines become

nearby

·B

E

generalized harmonic morphism

a

in the mini-twistor space

curve

Locally such

envelope points ( points

z(x),x

the notion of

harmonicity gives

with its natural complex structure.

field defined

points

conformality implies

harmonic

where

to branch

morphism

z

=

of the form

upper P left-parenthesis x comma z right-parenthesis equals 0 comma

(1.6) where P is

holomorphic in

and

z

harmonic

a

morphism

in x, that is

normalup erSigmaUnderscriptaUnderUnderscriptEndscripts imesleft-parenthesi StartFractionpartial-difer ntialup erPOverpartial-difer ntialxSuperscriptBaselineSuperscriptaBaselineEndFractionright-parenthesi squaredequals0timesandnormalup erSigmaUnderscriptaUnderUnderscriptEndscripts imesStartFractionpartial-difer ntialup erPOverleft-parenthesi partial-difer ntialxSuperscriptBaselineSuperscriptaBaselineright-parenthesi squaredEndFractionequals0period

In

general

for each

P is

polynomial

(1.6)

coincide.

generalization surface for Given

a

a

theory,

in

z

=

z(x) correspond

in which

z

higher

for

(1987, 1990)

harmonic =

morphism φ : unit

Branch where

points

∂P/∂z

of the

=

0.

resulting

Frequently

to values x for which two roots z of

harmonic

morphisms

examples

as

a

natural

functions of Riemann

(multiple-valued) analytic

some

1988).

easily

checked that in the chart

In fact

The

M → 𝕡, M open in

positive tangent

Wood

γ extends smoothly

equation (1.5)

ℝ3,

to the fibre of

across

critical

and Gudmundsson & Wood

Harmonic

morphisms φ :

classified in Baird h Wood smooth foliation F of M

M

we can

associate

φ through

has the

points (Baird &

(1993)

—>

𝕔 from

(1988).

and, locally, up

Ǖ 4;, φ is given implicitly by

an

to

are

composition

Baird 1987, Baird &

Wood

γ is

represented

of the fibres

ℝ3

have been

(parts of) straight

with

Then it is

1992).

→ 𝕔 ∪ ∞,

of

open subsets of Euclidean space

Indeed the fibres

Gauss map γ : M →

(i) minimality (Baird 1987, Wood 1986).

simple interpretation map

a

(see

x

given by stereographic projection S2

now

const., and (ii) horizontal ±-holomorphicity of the Gauss

subset of

(x,z)

multiple-valued

dimensions of the

Baird

see

points

they correspond

case

We may think of such to

to

general theory.

S2, given by γ(x)

by ξ0/ξ1.

this may have several roots.

ℝm

x

multivalued function

a

weakly

φ

completely

lines which form

conformal map

=

on an

a

open

equation alpha Subscript a Baseline left-parenthesis phi left-parenthesis x right-parenthesis right-parenthesis x Superscript a Baseline equals 1

(1.7)

where

alpha equals StartFraction 1 Over 2 h EndFraction times left-parenthesis 1 minus g squared comma i left-parenthesis 1 plus g squared right-parenthesis comma minus 2 g right-parenthesis

and g,h

horizontal is

conformality

local leaf

space.)

Note that

writing

a

φ is

to

functions

meromorphic

are

given

z

condition

(1.2)

certain Riemann surface N.

a

the foliation F has

,

φ(x), equation (1.7)

=

M

at x

on

a

is of the form

(Indeed,

because of the

transverse conformal structure and N

(1.6)

The solution to

.

(1.5) corresponding

by xi Subscript up er A Baseline quals StartFraction 1 Over StartRo t 2 Superscript 1 slash 2 Baseline alpha prime period times x EndRo t EndFraction left-parenthesi StartFraction 1 Over StartRo t h EndRo t EndFraction com a StartFraction g Over StartRo t h EndRo t EndFraction right-parenthesi com a

(1.8) where g, h

evaluated at

are

in Baird & Wood

generally

more

(1988),

in

π(x),

Riemann

a

conformal map of 𝕔. In this

h(z)

z, and

=

[ξA],

Note that, where

by

a

infinitesimally

close

field ξA has

The

these

of

morphism

to

a

case

as

k

given by

=

1). g(z)

radial

(1991).

envelope

=

see

=

z.

C,

see

=

1

(z

N)

of this

by

weakly

͌ 𝕔, g is constant and

φ away

defining

sis

where

family

a

from

family

nearby

At such

1993).

Baird

This corresponds to

(1987),

of the fibre is the

points

of lines

lines get

points

the

a

two-valued harmonic

Baird k Wood

through

the point

(1988),

Gudmundsson

(0,0,—1),

with both

single point (0,0, —1).

two-valued harmonic

morphism.

also

corresponds

One branch of this is called the

Davie 1979, Baird 1987, Baird k Wood

1988).

In this

is the circle

through

:

x3

=

0, (x1)2 +

(x2)2

the interior of this circle and the

as we cross

the

(x1, x2)-plane

Gudmundsson k Wood

=

1.

corresponding tangent

outside the circle C.

this example where the line field is discontinuous circle

Ǖ 4;(or,

a

(1982, 1983):

z,h(z)

projection,

C

discontinuous

with values in followed

has smooth solutions

envelope points

Hitchin

example (Bernard, Campbell k

Lines twist

result

monopole of charge k=2). g(za)=z, This

rotationally symmetric the

a

By

is constant.

The fibres consist of lines

envelope

Prasad-Rossi

disk

orthogonal projection

equation αa(z)xa the

globally on ℝ3

choice of coordinates, N

Theorem, (1.7)

points are

are

charge

orientations. The

(ii) (The

defined

the leaf space N.

singularity.

known

& Wood

given by

an

onto

Baird & Wood 1988, Gudmundsson & Wood

simplest monopoles

(i) (Monopole

of the

projection

morphisms

appropriate

Function

Thinking

(see

a

after

are

projectivized spinor field,

N)

z

surface)

case

Implicit .

(parametrized by spinor

the

the

π being the natural

the only harmonic

(1993).)

as we cross

the

(There

is also

(x1, x2)-plane

line field is a

version of

inside

the

2.

Harmonic

from

morphisms

Euclidean metric. Vectors xa may

ℝ4

in terms of be

now

expressed

We consider ℝ4 with its standard

spinors. in terms of

spinors by

the

correspondence

x Superscript a Baseline equals left-parenthesi x Superscript 0 Baseline comma x Superscript 1 Baseline comma x squared comma x cubed times right-parenthesi times left-right-ar ow x Superscript up er A up er A prime Baseline equals StartFraction 1 Over StartRo t 2 EndRo t EndFraction Start 2 By 2 Matrix 1st Row 1st Column i x Superscript 0 Baseline plus x Superscript 1 Baseline 2nd Column x squared plus i x cubed 2nd Row 1st Column x squared minus i x cubed 2nd Column i x Superscript 0 Baseline minus x Superscript 1 EndMatrix

Let M ⊂ ℝ4 be

open subset.

an

only if the gradient ▿φ is

a

Then,

complex

as

ℝ3,

for

a

map

φ :

M → 𝕔 is

null field and this holds if and

horizontally

only

conformal if and

if

nabla phi equals xi Subscript upper A Baseline times eta Subscript upper A prime Baseline

(2.1) for

some

▿AA'

=

fields

spinor

ξA,χA'

defined

on

M, where the spinor covariant derivatives

are

given by

∂/∂xAA'.

REMARK. We

always

have the freedom left-parenthesi xiSubscriptup erABaselinetimescom aetaSubscriptup erAprimeBaselineright-parenthesi right-ar owfrombarleft-parenthesi lamdaxiSubscriptup erABaselinetimescom alamdaSuperscriptnegative1Baseline taSubscriptup erAprimeBaselineright-parenthesi com a

(2.2) for any

non-zero

Suppose a

now

scalar function λ. that

φ is horizontally conformal,

so

that

holds. Then

(2.1)

φ is

harmonic and

so

is

harmonic morphism, if and only if nabla Superscript up er A up er A prime Baseline xi Subscript up er A Baseline times eta Subscript up er A prime Baseline equals 0 period

(2.3) Conversely, given

they We

determine

require

a

a

of spinor fields

pair

harmonic

morphism.

to be zero. This is

t

ξA,ηA'

Now the

would like conditions which

we

product ξAη)A'

to the

equivalent

M,

on

pair

of

determines

ensure

null vector field va.

a

spinor equations:

StartLayout Enlarged left-brace 1st Row nabla times eta Subscript up er B prime Baseline equals 0 2nd Row nabla xi Superscript up er A Baseline equals 0 period EndLayout

(2.4)

Combining (2.3)

and

THEOREM 2.1.

Let M ⊂ ℝ4 be

between harmonic

(2.4)

we

obtain:

morphisms φ:

a

simply

M → 𝕔 and

spinor equations

connected open subset.

pairs

of

spinor

fields

There isa corespondence

(ξA,ηA')

satisfying M the

on

StarLayoutEnlargedleft-brace1stRow nabl Subscriptup erAup erAprimeBaselinexiSupersciptup erABaseline taSupersciptup erBprimeBaseline quals02ndRow nabl xiSupersciptup erB aseline taSupersciptup erAprimeBaseline quals0EndLayout

(2.5) Proof.

It is clear that

equations (2.5)

with A'

=

(2.5) implies (2.3) B'

=

nabla xi Superscript 0 Baseline eta Superscript 0 prime Baseline plus nabla xi Superscript 1 Baseline eta Superscript 0 prime Baseline equals one-half left-parenthesis nabla xi Superscript 0 Baseline eta Superscript 0 prime Baseline plus nabla xi Superscript 1 Baseline eta Superscript 1 prime Baseline plus nabla Subscript 10 prime Baseline xi Superscript 1 Baseline eta Superscript 0 prime Baseline plus nabla xi Superscript 0 Baseline eta Superscript 1 prime Baseline right-parenthesis comma

0. Then

and

(2.4) Conversely, .

suppose

we

consider the first of

by (2.4)

.

But this

equals

by (2.3)

zero

The other equations

.

obtained

are

similarly.

REMARKS 2.2. The

(i)

correspondence

constant and

to critical are now

(iii)

points. For choose

=

α(x),β(x)

case)

on

extend

to 0 at a critical

quadric Q2 ⊂ 𝕔P3, This

Q2.

and

(1984),

determines

[ηA'(x)]

an

x

to ℝ4)

a

(they =

point. point

M

x

a

β-plane β(x).

Then

the identification of Q2

origin in 𝕔4.

the

2-plane through

to the fibre of φ through

(the tangent

ηA'

(ii), ǁ▿AA',φǁ2

at each

determines

real

a

a-plane a(x) (a complex

point corresponds (under

This

the vertical space at

thatǀξAǀ =

continuously

the

point

λ such

of Penrose & Rindler

by

extend continuously

in Remarks 1.2

projectivized spinor [ξA(x)] of

ξA,ηA'

as

the

a

(2.2)

which differ

morphisms .

spinor Helds

Then

1).

=

2-planes in

x),

translated to

origin.

A harmonic

morphism on ȑ3 may be

under

translation. Thus

some

Examples.

Particular

examples

with respect to

±-holomorphic

are

ξA,ηA’

geometric description

intersect in

plane is

two harmonic

any continuous scalar function

with the Grassmannian of oriented

the

(iv)

identify

way that the

a

equivalence |λ|

and both

▿aa'φ i1 0,

line in this

3.

|ξA|4

In terms of the where

in such

λ in (2.2)

defined up to

ǀξA|2|ηA'|2

we

subject the spinor fields to the equivalence

We may choose

(ii)

is one-to-one if

from the standard

one

viewed

as

equation (1.5) of harmonic

one

is

harmonic a

special

morphism on ℝ4 of equations

case

morphisms φ : ℝ4

of the Kähler structures

by identifying ℝ4

obtained

a

with 𝕔×

on

→ 𝕔 are

which is invariant

(2.5)

.

given by

ℝ4. Each Kähler

Ǖ 4;, by composing

maps which

structure arises

with

an

of

isometry

ℝ4. Use coordinates in

ℝ4,

is

(z, w)

holomorphic

which holds if and

for 𝕔× 𝕔,

if and

only

only

so

that

z

=

x0

+

ix1,

clearly

det

=

x2

+

ix3. Then φ : M → Ǖ 4;, M

open

StartFractionpartial-difer ntialphiOverpartial-difer ntialzoverbarEndFractionequalsStartFractionpartial-difer ntialphiOverpartial-difer ntialwoverbarEndFractionequals0com a

if

▿00'φ=▿10oφ? Then

w

if

▿AA,φ

=

=

0.

0 and nabla Subscript up er A up er A prime Baseline phi equals Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column asterisk 2nd Row 1st Column 0 2nd Column asterisk EndMatrix equals StartBinomialOrMatrix asterisk Cho se asterisk EndBinomialOrMatrix left-parenthesi 0 times asterisk right-parenthesi

so

that ηB'

some

=

(0, λ),

scalar μ. We

for

now

some

scalar λ.

Similarly

consider the effect of

an

φ is

anti-holomorphic

isometry

on

the

if and

only

if

ξA

spinor decomposition

=

of

(μ, 0)

for

▿AA'φ.

There is define

φ̃

action

on

=

a

well-known double

φo θ.

Then

SU(2) × SU(2)

cover

If

▿φ(θ(x))oθ.

SU(2) × SU(2)

(A, B)

that θ

Suppose

→ SO(4).

and

then the induced

θ,

covers

SO(4)

given by

is

spinors

φ̃(x)

=

ξAηA'↦ AξAηA'B* , where

B

ηA'(x)B*. θ. In

(θA,ηA')

that

so

,

Note that under the

particular

we see

from the standard

one

that

by

φ :

an

↦

equivalence (2.2) M → 𝕔 is

only

if and

if

𝕔P1 is

[ξA]

one

EXAMPLE 3.2. Let

to a

may compute the

—>

ℝ4) on

is

a

spinor fields ηA',ξA

harmonic

(A, B) covering

𝕔P1 is

[ηA']

by an

morphism,

constant.

orientation

then Æ is

[ηA']

(after identifying

zw

=

reversing

have

ℝ4 if and only if either =

if

only

obtained

we

η̃A'(θ(x))

to a Kähler structure obtained

with respect if and

and

A(ξA(x))

=

of the choice of

independent

Summarizing

given by ℝ(z,w)

𝕔 be

ξ̃A'(θ(x))

Kähler structure

constant.

of the Kähler structures

テ : ℝ4

this is

preserving isometry

THEOREM 3.1. If φ : N → 𝕔 (M open in with respect to

,

±-holomorphic

orientation

Similarly φ is ±-holomorphic with respect isometry

that is

(AξA,ηA'B*),

[ξA]

or

00B1;-hoiomorphic is constant.

ℝ4 with

𝕔2).

Then

we

to be

eta Subscript up er A prime Baseline equals left-parenthesis 0 comma 1 right-parenthesis comma xi Subscript up er A Baseline equals StartRo t 2 EndRo t imes StartBinomialOrMatrix z Cho se minus i w EndBinomialOrMatrix

In this

case

φ is holomorphic

by [ηA'1] (see Section 4). smoothly

over

this

with

examples

implicitly by

globally,

an

single

critical

point

N

are

those which have

a

origin

and both

on

ℝ4 represented

spinor

fibres. These

totally geodesic

ℝ4,

let N denote the leaf space of the fibres.

can

be

given

the structure of

fields extend

a

is

a

are

classified in

harmonic

Locally,

morphism

and in favourable

smooth Riemann surface and φ is

given

equation

αa(φ(x))xa where

structure

complex

at the

also II.1.13 ). If 㳆 : M → 𝕔, M open in

(1988) (see

totally geodesic fibres,

circumstances

a

point.

Another class of Baird & Wood

with respect to the standard

There is

=

1,

and f, g, h

functions. In this

case

it is

easily

checked that

,

ξA'ηA'

are

:

N

—>

𝕔 ∪ ∞; are meromorphic

given by

xi Subscript up er A Baseline equals StartFraction 1 Over StartRo t 2 Superscript 1 slash 2 Baseline left-parenthesi alpha prime dot x right-parenthesi h EndRo t EndFraction times StartBinomialOrMatrix 1 Cho se normal i g EndBinomialOrMatrix times eta Subscript up er A prime Baseline equals StartFraction 1 Over StartRo t 2 Superscript 1 slash 2 Baseline left-parenthesi alpha prime dot x right-parenthesi times h EndRo t EndFraction left-parenthesi times i comma f right-parenthesi comma

(here, f,g

and h

Note that, in are

not

are

evaluated at

general,

±-holomorphic.

π(x),

where π is the natural

neither of these is

More

generally,

projectively

projection

onto

N).

constant and so the harmonic

morphisms

the second author described all harmonic morphisms from

open sets of

ℝ4 locally

classification into

Locally

a

in terms of

spinor fields,

holomorphic

we can

harmonic morphism

μ

:

functions

(Wood (1992)).

express all solutions to

→

M

(M 5 4;

open in

ℝ4)

When

equations (2.5) is

as

we

translate that

follows:

given implicitly by

equation

an

psi times left-parenthesis z minus mu times w overbar comma w plus mu z overbar comma mu right-parenthesis equals 0 comma

(3.1) or an

equation psi times left-parenthesis z minus mu times w comma w plus mu z comma mu right-parenthesis equals 0 comma

(3.1') where

ψ

Function

ψ(u1,u2, μ)

=

is

holomorphic

a

Theorem, equation (3.1)

function of three

complex

variables.

By the Implicit

μ if

has smooth solutions

up er P identical-to minus w overbar times StartFraction partial-dif erential psi Over partial-dif erential u 1 EndFraction plus z overbar times StartFraction partial-dif erential psi Over partial-dif erential u 2 EndFraction plus StartFraction partial-dif erential psi Over partial-dif erential mu EndFraction ot-equals 0 period

(3.2) Corresponding spinor fields

are

given by

xi Subscript up er A Baseline equals StartFraction 1 Over StartRo t up er P EndRo t EndFraction times StartBinomialOrMatrix negative i partial-dif erential psi slash partial-dif erential u 2 Cho se minus partial-dif erential psi slash partial-dif erential u 1 EndBinomialOrMatrix comma eta Subscript up er A prime Baseline equals StartFraction 1 Over StartRo t up er P EndRo t EndFraction left-parenthesi negative mu comma negative i rght-parenthesi

(or

any

pair equivalent

the rôles of

in the

sense

of

(2.2) ).

We have

a

similar

description

in the

(3.1')

case

with

ξ and η reversed.

REMARKS 3.3

(i)

The

setting

(ii)

with

case

μ

=

totally geodesic

-f

and

ψ(u1, u2, μ)

It follows from Wood

apart from those with 4.

equations (2.5) Let V be

a

the second author Wood

corresponding orthogonal

orientations for each

Vx,Hx,x

changing

M,

so

in

assuming

complex

V,

2-planes

an

2h. not have any solutions

Here

relate

we

our

globally

defined

spinor description

in terms of twistor bundles, thus of Gauss sections. We

on

ℝ4

to the

interpreting

briefly

summarize

oriented 4-dimensional Riemannian manifold M,

2-dimensional distribution.

structures

the orientation of Vx

that Jv and JH

The Gauss section of bundle of oriented

general description by

γ : M →Gti=Vx,mabyγ(d&rn#ahexs03fmtxipan;e2o(Tsd)Min,

in TM.gtcJasTorlumionhHvVbpdesxt

We may

that the combined orientation of Vx ⊗ Hx

All -JH). , (-JV the results below will be

generality

(1986)

more

(1986).

of M. We then define almost Note that

gu2

—

holomorphicity properties

2-dimensional distribution in

and let H be the

+π/2.

u1

-

(1992) that (2.5) does [ξA] or [ηA'] constant.

in terms of

the results of Wood

=

in terms of twistor bundles.

Interpretation

description given by the

fibres may be obtained from this

are

Jv,JH

on

changes

that of Hx, and

independent globally

=

change,

so

choose

TxM is that

each Vx,Hx to be rotation

of this

chosen.

locally

through

replaces (Jv, JH) by

that there is

no

loss of

almost-complex

and J

structures J

Tx M. We have thus defined

compatible

complex

with the orientation; these distribution V defines

that, if M is and there is

Let Z+

incompatible.

is all metric almost

surface,

to a Riemann

(see

1986). Conversely,

(Wood 1986) Let

THEOREM 4.1.

M

→

:

M

→ Z-

is

Now let

φ :

(1986),

a

orthogonal complements

distribution H

on

the

a

M. At each

G̃2(ℝ4)

Q2

Then,

if ▿AA'ϕ =

charts

on

͌

Z±, γ1

to the almost

opposite complex

x,

Write W

=

▷φ,

is the

point

x

The

=

M× S2,

minimal and conformal

integrable,

locally

in this way. We recall

on an

oriented 4-dimensional

structure

J2

only

on

complex structure J1

[ηA'), γ2

=

[ξA].

then

M,

left-bracketup erWright-bracketequals eft-bracketminusileft-parenthesi StartLayout1stRow xiSuperscript0BaselineEndLayoutetaSuperscript0primeBaselineplusStartLayout1stRow xiSuperscript1BaselineEndLayoutetaSuperscript1primeBaselineright-parenthesi com aStartLayout1stRow xiSuperscript0BaselineEndLayoutetaSuperscript0primeBaselineminusStartLayout1stRow xiSuperscript1BaselineEndLayoutetaSuperscript1primeBaselinecom aStartLayout1stRow xiSuperscript0BaselineEndLayoutetaSuperscript1primeBaselineplusStartLayout1stRow xiSuperscript1BaselineEndLayoutetaSuperscript0primeBaselinecom aileft-parenthesi StartLayout1stRow xiSuperscript1BaselineEndLayoutetaSuperscript0primeBaselineminusStartLayout1stRow xiSuperscript0BaselineEndLayoutetaSuperscript1primeBaselineright-parenthesi right-bracketel ment-ofup erQ2period

if the section

M and the section

on

M.

structure was used on the twistor space Z-

with respect to J1.

morphism

from

an

open subset M of 𝕔4

to the fibres determine an oriented

.

Then

2-dimensional

Hx is given by

spinor decomposition,

up erWSuperscriptaBaselineleft-right-ar owStartFraction1OverStartRo t2EndRo tEndFractiontimesStart2By2Matrix1stRow1stColumniup erWSuperscript0Baselineplusup erWSuperscript1Baseline2ndColumnup erWsquaredplusiup erWcubed2ndRow1stColumnup erWsquaredminusiup erWcubed2ndColumniup erWSuperscript0Baselineminusup erWSuperscript1BaselineEndMatrixequalsStartBinomialOrMatrix iSuperscript0BaselineCho seStartLayout1stRow xiEndLayoutSuperscript1BaselineEndBinomialOrMatrixtimesleft-parenthesi etaSuperscript0primeBaselinecom aetaSuperscript1primeBaselineright-parenthesi com a

and at each

an

complex

to the almost

tangent planes

point

gives

Q2 ⊂ 𝕔P3,

is the standard identification of the Grassmannian with the

ξAηA' =

trivial: Z±

are

minimal and conformal if and

[ 1ϕ, ϕ, 3Õ, 4ϕ] where

1985).

oriented Riemannian four-manifold

2-dimensional distribution

submersive harmonic to the

an

any such distribution arises

being antiholomorphic

M → 𝕔 be

Salamon

(Eells&.

the twistor bundles

M → N from

integrable,

holomorphic with respect

which resulted in γ2

compatible (respectively incompatible)

distribution:

Z+ is holomorphic with respect

:

REMARK. In Wood

the

a

V be

Riemannian manifold M. Then V is

γ1 γ2

x

2. Note

space𝕔®4,

morphism

characterization of such

following

J1e2,

M whose fibre at

M→► Z+ and

:

the tangent spaces to its fibres

Wood

over

holomorphic bijectionG̃j2ℝR4)͌« S2× S2.

submersive harmonic

a

be the fibre bundle

the well-known twistor bundles of M

are

Note that J1 is

oriented basis of the form e1, J1 e1, e2,

an

structures on xXM which are

sectionsγ71

well-known

structures on the manifold M.

(respectively -~)

open subset of Euclidean

an a

distribution the

complex

with the orientation, i.e., there exists

whereas J2 is

Given

two almost

each tangent space

on

a

direct

computation

complex quadric.

verifies that in suitable

But the

hand side is the

right

𝕔P1 × 𝕔P1

with

Thus

Q2.

𝕔P1

each

identifying

↦ξ0/ξ1,[η0'η , 1'] ↦η0'/η1',

[ξ0,ξ1]

([ξ0,ξ1] ,[η'η1'])

of

image

under the standard identification of

with 𝕔∪ ∞by the

stereographic projections

find that

we

gam aSuperscript1Baseline qualsStartFractionup erWsquaredminusnormaliup erWcubedOveriup erWSuperscript0Baselineminusup erWSuperscript1BaselineEndFractionequalsStartFraction ormaliup erWSuperscript0Baselineplusup erWSuperscript1BaselineOverup erWsquaredtimesplusnormaliup erWcubedEndFractioncom agam asquaredequalsStartFractionup erWsquaredplusnormaliup erWcubedOveriup erWSuperscript0Baselineminusup erWSuperscript1BaselineEndFractionequalsStartFraction ormaliup erWSuperscript0Baselineplusup erWSuperscript1BaselineOverup erWsquaredtimesminusnormaliup erWcubedEndFractionperiod

In order to show that consider

W2

—

a

iW3

point =

x

0 at

equations (2.5) imply

and suppose, without loss of x.

By

horizontal

the

holomorphicity that

generality, in

conformality

∂2,∂span 3 Vx.

neighbourhood

a

results of Theorem 4.1,

of

x we

Then W2

we

+iW3

have∑(Wa)2

=

0

=

so

that left-parenthesis up er W Superscript 0 Baseline plus normal i up er W Superscript 1 Baseline right-parenthesis times left-parenthesis up er W Superscript 0 Baseline minus normal i up er W Superscript 1 Baseline right-parenthesis equals minus left-parenthesis up er W squared plus normal i up er W cubed right-parenthesis times left-parenthesis up er W squared minus normal i up er W cubed right-parenthesis comma

(4.1) so, at x, either

W0 + iW1

W0 +iW1

so

=

0,

that W0

(W0 for any

a

=

0,1, 2,3,

-

so

0

=

or

W0

—

iW1

iW1)∂a (w0

equations (2.5) (with A'

computation (with

A'

—

+

1, B'

vertically antiholomorphic. Similarly,

holomorphic

=

0, B'

have

=

0,

0

=

=

1) implies

∂1)(W2 + iW3)

=

0)

=

Conversely, fields

0;

—

i∂3)(W2

equations (2.5)

shows

These conditions combine to

directly

conformal distribution determines

a

ξA,ηA'. by

If the

Theorem 4.1

gives

over

corresponding Gauss

a

that the

an

null vector

maps

that theorem the distribution is

ξA,ηA' satisfy equations (2.5)

points).

(∂2

give

iW3) that γ2

+

the

=

is

0

so

γ1

that

horizontally

holomorphicity assertions

spinor equations (2.5) imply

the

be described

by

holomorphicity conditions

of

assertions of that theorem.

Theorem 4.1, then

Theorem 2.1

shows that

the second of

vertically holomorphic.

and

holomorphicity

critical

W1)

-

of Theorem 4.1. We have therefore shown

spinor

have that

with the formula thimolrizmplonriptesahliscy. ig is γ1 hthat

A similar is

we

we

that

(—i∂0 together

(4.1)

choice of orientation

by

iW1) + (W0 + iW1)∂a (W0- iW1)

+

 02;a (iW0 at x. Now the first of

0. In fact,

=

iW1 ≠ 0. Differentiating

—

satisfy

integrable

field, the

which

can

and minimal, and the

spinor

fields

.

interpretation

of the

spinor equations

Theorem 4.1 is that it is valid for

arbitrary

in Theorem 2.1. The

harmonic

advantage

morphisms (i.e.,

Indeed, for the harmonic morphism φ of Example 3.2, which has

an

of

those with

isolated critical

point (▿φ the as

0)

=

origin,

projectivized spinors,

in Remark 2.2

extend at least 5.

at the

(iv),

γ1

the distributions V, H do not extend =

[ηA']

extends

with suitable

continuously

Minkowski space.

across

across

the

normalization,

critical

We consider

a

origin,

the

over

whereas

this critical

γ2

[ξA]

=

unprojectivized spinor

point. Regarding

does not.

However,

fields ξA,ηA'

always

points. U → Ǖ 4;, where U is

φ :

map

an

open subset in Minkowski

M4, satisfying the equations

space

left-parenthesi partial-dif erential Subscript 0 Baseline phi right-parenthesi squared minus left-parenthesi partial-dif erential Subscript 1 Baseline phi right-parenthesi squared minus left-parenthesi partial-dif erential Subscript 2 Baseline phi right-parenthesi squared minus left-parenthesi partial-dif erential Subscript 3 Baseline phi right-parenthesi squared equals 0

(5.1) partial-dif erential Subscript 0 Superscript 2 Baseline phi hyphen partial-dif erential phi hyphen partial-dif erential phi hyphen partial-dif erential phi equals 0 period

(5.2) Such

map is

a

functions

a

harmonic

morphism in

Parmar 1991, Lemma

(see

φ is conformal

on

standard Lorentz

the

that it

sense

pulls back

2.1.7). Equation (5.1)

the horizontal space

harmonic functions to harmonic

may still be

(the orthogonal space

to the

interpreted

fibre,

now

as

that

saying

with respect to the

metric), while (5.2) is simply the wave equation—the harmonic equation in

Lorentz

metric. The

spinor correspondence

is

given by

x Superscript a Baseline left-right-ar ow x Superscript up er A up er A Super Superscript Superscript prime Baseline quals StartFraction 1 Over StartRo t 2 EndRo t EndFraction Start 2 By 2 Matrix 1st Row 1st Column x Superscript 0 Baseline plus x Superscript 1 Baseline 2nd Column x squared plus i x cubed 2nd Row 1st Column x squared minus i x cubed 2nd Column x Superscript 0 Baseline minus x Superscript 1 Baseline EndMatrix period

Proceeding exactly

as we

did for the ℝ4

THEOREM 5.1. There is

(i) mappings φ : (5.2)

U

—>

a

case we

correspondence

Ǖ 4;, U ⊂ M4

obtain

between

open and

simply connected, satisfying equations (5.1)

and

and

,

(ii) pairs of spinor Helds (ξA ,ξA' )

on

U

satisfying

the

spinor equations:

StarLayoutEnlargedleft-brace1stRow nabl Subscriptup erAup erAprimeBaselinexiSupersciptup erABaseline taSupersciptup erBprimeBaseline quals02ndRow nabl Subscriptup erAup erAprimeBaselinexiSupersciptup erB aseline taSupersciptup erAprimeBaseline quals0EndLayout

(5.3)

REMARK. Given

thought either a

of as

as

a

time

a

static

monopole

(x0)-independent

solutions to

there

(5.3) independent

of x0,

direction. This contrasts with

particular

corresponds

solution of (5.3) or as

.

a

harmonic

Thus

we

solutions to

describing

them

as

have

morphism on ℝ3. a

This

can

be

different view of monopoles,

(2.5) independent

of translation in

time-independent Yang-Mills fields.

(1994). Harmonic morphisms were studied in the late (1979), as mappings between Riemannian manifolds which

by Fuglede (1978)

Comments

1970’s

Ishihara

preserve harmonic functions.

Their investigation

was

properties

taken up almost ten years later

associated to

a

harmonic

by

Baird and Wood,

morphism

with values in

a

who, noting surface

and

the holomorphicity

(Wood

1986,

Baird

1987),

1991,

1992).

Wood

classified the harmonic morphisms from 3-dimensional manifolds (Baird & Wood 1988,

(1992)

extended this

study

4-manifolds, classifying

to

Einstein 4-manifolds in terms of

holomorphic

the harmonic

Baird

by

anti-self-dual

relativity theory

was

(1992).

With the natural introduction of between static

on

sections of twistor bundles. The direct correspondence

between harmonic morphisms and shear free ray congruences of observed

morphisms

monopoles and

geometrical elegance

spinor fields, morphisms

and harmonic

simplicity

of harmonic

the article in this volume defined

morphisms

the links

explores

3-dimensional Euclidean space. The

on

may lead to

a

greater understanding of

monopoles.

References Baird , P. ( 1987 ) Harmonic morphisms onto Riemann surfaces and Ann. Inst. Fourier, Grenoble 37 : 1 , p. 135 173

generalized analytic

functions ,

-

.

Baird , P.

Harmonic

( 1990 )

Grenoble 40 : 1 p. 177 212

morphisms

and circle actions

on

3- and 4-manifolds , Ann. Inst. Fourier,

-

,

Baird , P. J. Math.

.

( 1992 ) Phys.

Riemannian twistors and Hermitian structures 33 ( 10 ), p. 3340 3355

on

low-dimensional space forms ,

-

.

Baird , P. & Eells , J.

( 1981 ) A conservation law for harmonic maps , Lect. Notes in Math. 894 , (Springer-Verlag ), p. 1 25

Geometry Symp. Utrecht,

Proceedings,

-

.

Baird , P. & Wood , J. C. Ann 280 , p. 579 603

( 1988 ) Bernstein

theorems for harmonic

morphisms

from ℝ3 and S3 , Math.

-

.

Baird , P. & Wood , J. C.

( 1991 ) Harmonic morphisms and

space forms , J. Australian Math. Soc.

( 1992 )

Baird , P. & Wood , J. C. Proc. London Math. Soc.

(3 )

Harmonic

(A )

morphisms,

64 , p. 170 196

conformal foliations

51 , p. 118 153

by geodesics

of threedimensional

-

.

Seifert fibre spaces and conformal foliations ,

-

.

Bernard , P. , Campbell , E. A. k Davie , A.M. ( 1979 ) Brownian motion and inner functions , Ann. Inst. Fourier, Grenoble. 29 : 1 , p. 207 228

generalized analytic

and

-

.

Eells , J. & Salamon , S. Ann. Scuola

Fuglede

,

B.

( 1985 ) Twistorial constructions of Norm. Sup. Pisa ( 4 ) 12 p. 589 640 ,

( 1978 ) Harmonic morphisms between

28 : 2 , p. 107 144

harmonic maps of surfaces into fourmanifold,s

-

.

Riemannian manifolds , Ann. Inst. Fourier, Grenoble

-

.

Gudmundsson , S. & Wood , J. C.

(1993 )

Multivalued harmonic

morphisms

,

Math. Scand. 72

(to

appear). Hitchin N. J. ( 1982 ) Monopoles and geodesics Comm. Math. Phys. 83 ,

,

Hitchin , N. J. Ishihara , T. Math.

Kyoto

( 1983 ) ( 1979 )

p. 579 602 -

,

.

On the construction of monopoles Comm. Math. Phys. 89 p. 145 190 -

,

A

mapping

,

.

of Riemannian manifolds which preserves harmonic functions , J.

Univ. 19 , p. 215 229 -

.

( 1991 ) Harmonic morphisms between

Parmar , V. Leeds

semi-Riemannian manifolds ,

Thesis, University of

.

Penrose , R. &. Rindler , W.

( 1986 ) Spinors and Space-Time, vol. II: Spinor Space-Time Geometry (Cambridge University Press Cambridge ).

and Twisior Methods in

,

Sommers , P.

( 1980 ) Space spinors

(1986 )

Wood , J. C.

Harmonic

,

J. Maths.

p. 2567 2571 -

foliations and Gauss maps ,

morphisms,

American Math. Soc. 49 , p. 145 183

Phys. ( 21 ) ( 10 ),

.

Contemporary Mathematics,

-

Wood , J. C.

( 1992 )

Harmonic

.

and Hermitian structures

morphisms

J. Maih. 3 , p. 415 439

Einstein 4-manifolds , International

on

-

§II.1.15 by

Twistor

M.G.Eastwood

Suppose

Theory

(TN

M and N

are

M is constrained to lie

attempt cases

and

to attain an

equilibrium

parametrized

is

.

and Harmonic Maps from Riemann Surfaces

15, January 1983) oriented Riemannian manifolds with M made of rubber and N of stone. If on

N

by

means

a

smooth

equilibrium configuration.

always possible.

minimal surfaces

review article. To be

of

more

precise,

If ϕ is in

are

mapping ϕ :

This may be

See Eells

define the energy

E(ϕ)

impossible,

it is called

equilibrium

examples.

M → N and then released it will

of

harmonic map. Geodesics

a

& Lemaire

i.e. M snaps. In many

(1978)

for

comprehensive

a

ϕ by

up er E left-parenthesis phi right-parenthesis equals integral Underscript up er M Endscripts one-half StartAbsoluteValue d phi EndAbsoluteValue squared d times vol

where on

N,

|dϕ|

is the Hilbert-Schmidt

ǀdϕV0;2

=

a

critical

compact subdomains of M with

ϕ is

of dϕ : TxM

harmonic iff it satisfies the

point

where ▾ is the connection

on

of E

(if M

on

i.e. in local coordinates xi M and h is the metric

on

is not compact then E must be variations

allowed).

on

M, yα

N. Then

computed

In other

words,

corresponding Euler-Lagrange equations

ω1(ϕ*TN)

gSuperscipt jBaselin timesStarSetSartFactionparti l-difer ntialySupersciptalphaB selin Overparti l-difer ntialxSuperscipt Baselin parti l-difer ntialxSupersciptjBaselin EndFractionminusnormalup erGam aSubscript jSupersciptkBaselin StarFactionparti l-difer ntialySupersciptalphaB selin Overparti l-difer ntialxSupersciptkBaselin EndFractionplusnormalup erOmegaSubscriptbetag m aSupersciptalphaB selin StarFactionparti l-difer ntialySupersciptbetaB selin Overparti l-difer ntialxSuperscipt Baselin EndFractionStarFactionparti l-difer ntialySupersciptgam aB selin Overparti l-difer ntialxSupersciptjBaselin EndFractionE dSet

▾(dϕ)

=

0

induced from the Levi-Civita connections

in mind the elastic nature of M, trace

it becomes

TϕN,

only compactly supported

trace

Bearing

→

∂yα/∂xi∂yβ/∂xigijhαβ where g is the metric

ϕ is said to be harmonic iff it is on

norm

▾(dϕ)

on

M and N.

is called the tension of ϗ. In local coordinates

where

andf

are

g

symbols

Christoffel

mapping ϕ and

the second fundamental form of the

tension,

however, does not in

Perhaps

M and N

on

Without the trace,

behaves well with respect to

compose for the

general

respectively.

composition

of

▾(dϕ)

is

The

composition.

mappings.

neater way to write the energy is as

a

upper E left-parenthesis phi right-parenthesis equals integral Underscript upper M Endscripts one-half times trace d phi times normal upper Lamda times times asterisk times d phi

where

*

Ω1(ϕ*TN)

:

to h. The

—>

Ω.m-1(ϕ*TN)

Euler-Lagrange equations

is the

are

Hodge *-operator

is the

Ωm-1(ϕ*TN)

→

dual of the

Hodge

N, maintaining

a more

is the

Ωm(ϕ*TN)

more

M and the trace is with respect

then

▿(*dϕ) where ▿ :

on

0

=

pull-back of the

Levi-Civita connection

usual tension. In local coordinates

abstract notation

▿(*dϕ) notation) on

(or

N.

on

abstract index

M

on

d At this

point

an

obvious

with the

analogy

Yang-

familiar to many mathematicians and

analogy

maps under the

name

of ‘𝕔𝕡n model’,

of the gauge field F and

analogue

there is the Bianchi

identity

▿F

=

▿(dϕ)

ϕ*▿δ where

δ

=

▿(ϕ*δ)

=

0. The

on

N is the

▿δ is exactly the torsion of ▿. Now recall that the

(being conformally invariant)

and that twistor

It is natural to ask whether there is The

special

invariant when

conformally ±

self-dual,

*F

dimension 2, make

sense

specifically

acting

=

however, *2

▿ is

Kähler,

so

a

so

not ...

Yang-Mills

granite).

analogue

bundle

complex.

But now,

necessarily complex Kähler

the

complex

it is easiest to take N

than the strongest one,

—1

=

so

as a

▿(*dϕ)

=

illuminating

(Kronecker delta)

and

special in dimension 4

way of

looking

equations ▿(*dϕ)

now on

if N has

a

only

can

an

almost

=

be

special *dϕ

=

complex

N is made from

at them.

0

complex

complex

±i▿(dϕ)

=

0

linear is

conformally manifestly

are

solutions

namely

identity.

±idϕ. stone

i.e.

more

(more rigid

an

equivalent

In

For this to

manifold and

nicely compatible metric,

Now, if dϕ is ± self-dual,

▿(±idϕ)

are

=

consequence of the Bianchi

of ± self-dual

linear. Indeed ▿ being

stone from now on.

are

because

case

2 for then the *-operator is

N should be

From

even

section

in dimension 4 there

±F, which satisfy the equations

TN must be

=

1- forms. Thus, the harmonic field

invariant. For Euclidean

holds in the harmonic

of this for harmonic maps.

dimension for harmonic maps is dim M on

For harmonic maps dη is the

Yang-Mills equations an

(an

interested in harmonic

0. Recall that in gauge theories

tautological

theory gives

analogy

an

=

corresponding identity

Ω1(TN)

being

algebra’).

‘current

replaces ▿(*F)

0

=

or

to mind

equations springs

the latter

physicists,

‘π-model’,

▿(*dϕ)

Mills action and

Hermitian

to N

being

as

the Korn-Licktenstein theorem the *-operator

required. By

structure where be

becomes

*

i.e.

complex linear, Now to the

as a

way of

be clearly

dϕ is

in the

after

complex. there is

complex

becomes

a

The

rather

hope

that

by

photon

complexification will throw

ϕ 𝕔M :

our case

→ 𝕔N on some

equations.

The derivative

dϕ may be split

(1, 0)

and

(0,1)

To

going

on

of M. Then

these

equations

are

straightforward

easy to

M but with the If

ϕ

:

(ϕ(p), ϕ̄(q) . in 𝕔M A

so

In

identify: 𝕔M

holomorphic

general

a

complexify. =

▿(dϕ)

ℝ-analytic ϕ :

complexification, ∂ refers to differentiation

regarded

satisfy

one can a

to differentiation

along M̄with

M

=

along

on a

4-dimensional should

one

complexify to

▿(*dϕ)

dϕ  02;ϕ =

+

obtain

version of the =

0

as

follows.

 02;̄ε usually

called

0 it follows that ϕ is harmonic ⇔

=

▿( 02;̄ϕ)

=

are

0.

already complex, is the

diagonal (m

a

small

 02;ϕ +  02; 4;ϕ is

now

↦ (m,m̄)).

neighbourhood

=

of M

of M in M × M̄.

ψ satisfies

manifested

holding the

occurring parametrically.

complexiflcations

given by 𝕔ϕ(p, q̄)

neighbourhood

of ϕ : M → N iff

M whilst

their

where M̄ denotes the smooth manifold

complexify to only

complexification

as a

complexified

it is best to rewrite

notation for any

general

Gauss),

automatically ℝ-analytic (by

complexification 𝕔ϕ 𝕔M →Ǖ 4; N is

as a

→ 𝕔N is the

to

then take

from Riemann surfaces.

:

The splitting dϕ

p

Kähler,

ϕ will

example,

will

ℝ-analytic case

hope.

structure and g

M → N

then the

way that Maxwell’s

same

mappings

can

‘thickening’

into details it is clear that

Since M and N

M × M̄, for

then its

it is best to maintain 𝕔M

holomorphic ψ : 𝕔M

condition

to

conjugate holomorphic

M → N is

should be

This

the existence of

particular,

the

into its self-dual and anti-self-dual parts

parts. Bearing in mind that

be viewed

construction

ℝ-analytic category (due

exactly

Riemann surface and N

identify

In

Riemannian manifolds is

▿(∂ϕ)=equivalenty In this form it is

in the 2- dimensional

harmonic

further upon this

neighbourhood

harmonic field

the

a

equations

Without

after

dϕ

antiholomorphic).

ℝ-analytic

Cauchy-Riemann equations in

can

simple geometric

a

‘right-flat’).

holomorphicity

geometric light

explain

of M

to

The

Maxwell’s

—

mapping between ℝ-analytic

ellipticity). Thus, in

equally well

analogous

Cauchy-Riemann equations.

A harmonic

theory

4-fold is

original

becomes

example,

works

observation.

which leads to

The purpose of this article is to

for

means

conformally right-flat space).

conformal structure in the

a

simple geometric

lead to the twisted

version of the

procedure

condition

integrability

no

geometric interpretation equations

correspondence, same

structure induced

a

anti-self-dual

geometric significance, namely through a-planes,

on a

The Ward

(for

complex

a

the condition that

Twistor

theory?

More precisely, if the

complexification.

complexification.

(where the

ϕ is holomorphic (and

conformal geometry in dimension 4

doing

conformal structure takes into the

means

to

equivalent

idϕ is precisely

=

what has this to do with twistor

question:

seen

self-dual

Thus, *dϕ

i.

multiplication by

M is

on

the

‘reality’

geometrically:

after

M̄ variable fixed and  02;̄ refers

In other words

(1,0)-forms complexify

to the

holomorphic torsion holomorphic iff

to M̄. The connection ▿ on N

cotangent bundle

▿(∂ϕ)

=

free connection

on

torsion free connection ▿ then

0. To interpret this

can

let μ : 𝕔M

the

as

(plus some simple topological restrictions

reasons

a

as a

section of the bundle

M. In

on

ℝ-analytic)

holomorphic

(holomorphic) projection

=

manifold with

to be harmonic

onto the first factor.

f▿μs + ∂f ⊗ s.

the fibres of

particular,

to a

relative connection▿μ on μ*Ω1(ϕ*TL)

pull-back of a bundle

on

it is

M

Thus

μ*Ω1(ϕ*TL)

(since, by

dimension

μ)▿μis relatively flat).

Therefore

on

constant on the fibres of p or, in other

ϕ is harmonic iff ∂ϕ μ**Ω1(ϕ*TL) is covariant down to

M be

—>

satisfying →μ(fs)

canonically regarded

and hence ϕ*TL maybe

if L is any

generally,

interpreted

be

a

i.e.

complexifies (assuming

define ϕ : 𝕔M → L

we

geometrically

𝕔 : Ω1,0(ϕ*TL) → Ω2(ϕ*TL)

Then

𝕔N. More

words, pushes

this proves

PROPOSITION. The zero set of ∂ϕ consists of the fibres of μ : 𝕔M → M over a discrete set of points and, in particular, by rescaling, ∂ϕ defines a complex direction at ϕ(x) even if ∂ϕ(x) = 0 (unless ∂ϕ ≡ 0). □

This

proposition

is

one

of the

from Riemann surfaces into

key steps

complex projective

solutions in the 𝕔𝕡n-1 model, Nucl. of the

classical 𝕔𝕡n-1

general

& Wood,

in the recent classification of harmonic spaces

Phys.

95B, 419-422. D. Burns,

Lett.

model, Phys.

maps

General classical

(1980)

B174, 397-406. Din & Zakrzewski

Harmonic maps from surfaces to

(1983)

& Zakrzewski

(Din

isotropic

(1980) Properties

unpublished.

Eells

spaces, Advances in Math.

complex projective

49, 217-263). Actually, the whole classification theorem complexifies rather well. Because it would make this article rather too the

Firstly

complexification

Fubini-Study

metric

and

can

so

gives

𝕡(V)

rise to

 02;̄ϕ

are

0.

In the

(ðR

to

case

on

a

mapping ϕ : to

of L

some

on

V is the

same as an

ϕ (make local choices and

of the

𝕔M

Fubini-Study

L is said to be

→

derivatives

𝕡(V) × 𝕡(V*)

provided ϕ avoids

the

patch).

� F0;̄S

-

The

quadric {(R,S)

key points.

gives

rise to the

isomorphism

V̄ ≅ V*

=

it is

along

metric and there is

isotropic M̄.

iff ∂ϕ and

This is the

⟨R,ð̄S⟩/⟨R,S⟩) proposition

𝕡(V)

×

a

corresponding

higher

case

for

derivatives

example

if

essentially just algebra (albeit cunning algebra)

ϕ 𝕔M → 𝕡(V) × 𝕡(V*) is harmonic and isotropic then

⟨ð,S⟩R/⟨R,S⟩,

V

on

,

:

-

of the

𝕡(V) × 𝕡(V*). The natural pairing ⟨ ⟩ : V ⊗V* → Ǖ 4; 𝕔(V) × 𝕡(v*) induced by ⟨(a,b),(c,d)⟩ (⟨a,c⟩, ⟨b,d⟩) on

 02;̄ϕ and higher =

describe

just

space. An Hermitian form

The Hermitian form

complexification

orthogonal

to show that if =

𝕡(V).

to go into detail I will

complex projective

complexified

general,

M

D(R, S)

of

𝕔2-valued metric

a

connection. In

=

be

This is the

V ⊗ V*.

along

on

lengthy

where

(R,S)

:

so

is Dϕ defined

𝕔M → V × V* is

a

by

lift of

above is used to show that D is well-defined

𝕡(V*)

s.t.

⟨R,S⟩ 0} (D =

is

essentially ∂).

Hence

the classification. In the best of worlds there would be Witten ambitwistor

description

of

a

harmonic

Yang-Mills.

analogue

of the

Isenberg-Yasskin-Green

𝕔M is the ambitwistor space for M.

&

References Burns , D.

unpublished. General classical solutions in the 𝕔𝕡n-1 model , Nucl.

Din & Zakrzewski

( 1980 )

Din k, Zakrzewski

( 1980 ) Properties

Eells & Lemaire Eells & Wood ,

( 1978 )

( 1983 )

A report

of the

on

general

classical 𝕔𝕡n-1 model ,

Phys.

harmonic maps , Bull. L.M.S. 10 , 1 68

B174 , 397 406

.

Lett. 95B , 419 422

.

Phys.

-

-

-

Harmonic maps from surfaces to

.

complex projective spaces

Advances in Math

,

.

49 , 217 263 ). -

§II.1.16

Contact birational

correspondences

between twistor spaces of Wolf spaces

by

P.Z.Kobak

Introduction. can

The task of

be often

even

data which

define

harmonic maps

(cf.

in S4 from twistor

as

completely

projections

of

holomorphic

be constructed

as

l

horizontal distribution

Bryant’s (the

manifold of

defined

by

the

result

on

the

language of complex geometry

in

formulae

For

manifolds which

example

horizontal

one can

curves

in 𝕔𝕡3

the twistor space, i.e. is

follows: if

f, g

are

can

(a

curve

functions

a

a

complex

𝕔3).

correspondence

We shall call this the

produce

2-spheres

is horizontal iff it is the twistor

on a

fibres).

Riemann surface

(Bryant 1982).

contact structure and Lawson

contact birational

is

authors

some

be used to

orthogonal to

meromorphic

and

approaches

construct minimal

is horizontal in 𝕔𝕡3

by defining

following

on

𝕔ᵖ3 arises from

complete flags

complex

Rawnsley 1990).

Burstall and

and g ≠ const then the curve

curves can

problem to

characterise such maps. One of several

fibrations of almost

tangent to the horizontal distribution Such

in various classes of Riemannian manifolds

twistor methods have been successful to the extent that

theory. Indeed,

twistor spaces

2-spheres

the

accomplished by translating

specifying holomorphic to use twistor

minimal

classifying

between

(1983)

Ǖ 4;𝕡3

and

The

reinterpreted

F12(𝕔3)

Bryant-Lawson correspondence,

it is

(Gauduchon 1984):

b colon upper F Subscript 1 2 Baseline left-parenthesis double-struck upper C cubed right-parenthesis contains-as-member left-parenthesis x 0 colon x 1 colon x 2 semicolon xi 0 colon xi 1 colon xi 2 right-parenthesis right-ar ow from bar left-bracket x 2 xi 2 minus x 0 xi 0 colon 2 x 0 xi 2 colon x 0 xi 1 colon minus 2 x 1 xi 2 right-bracket element-of double-struck upper C double-struck upper P cubed

(1) where

i

The

flag manifold F12(ℂ3)

be identified with 𝕄T*ℂ𝕡2. Moreover, all non-vertical contact

curves

ℂ𝕡2*

𝔾r2(ℂ3)

≃

in 𝕡T"*ℂ𝕡2

are

is the twistor space of

and

can

Gauss lifts of

curves

in ℂ𝕡2. The birational map b

S4. This map is and

𰎾2

(here

=

conditions

on

in

F12(ℂ3):

on

the divisor S1 ∪ S2

the

Gauss lifts

Burstall

(1990)

of

F12(ℂ3)

ℂ𝕡2 and ℂ𝕡2*

to

used to construct horizontal

are

symmetric quaternion-Kähler

Wolf spaces, have many

have twistor spaces which

interesting generalisation

very

in

properties

common

fibrations of

are

they

manifolds

are

of the

These

endowed with

Burstall and Rawnsley

quaternionic

THEOREM 1 are

proof

to the added a

with

node;

of such

considered here

can

be

from

projectivised

singular

For

example the

a

Bruhat

divisors

decomposition

explicit formulae for which

(1)

is

a

special

S1, S2

For

more

F12(C3); Another

case.

nilpotent

details and

simple

Lie

can

between root

over

algebras.

curves

1983). to

spaces, known

of twistor spaces recruited

1990);

we

reserve

the term

structures

(see

as

with

use

nilpotent

corresponding

Kobak

through

algebras.)

Lie

the nodes linked

Our aim is to present

the fact that the twistor spaces

orbits.

quaternionic

Hasse

This makes it

are

decomposition.

the Schubert varieties in

and the

flag

manifold 𝕄T*ℂ𝕡n+1 of

correspondences

can

be derived from

(1972),

correspondences

(1993), Chapter

the Wolf spaces

to

diagram is shown below. We also derive

of Burstall

Wolf spaces

possible

twistor spaces of Wolf spaces

orbit to its tangent space. Moreover,

ℂ𝕡2n+1

of applications of the

example

dimension

in Baston and Eastwood 1989

crosses

Lie groups described in Warner

see

same

twistor spaces of Wolf spaces

l1, l2 in F12(ℂ3), defined above,

Wolf spaces.

For

symmetric

be described in terms of the Bruhat

interpretation

examples

to constructions of harmonic maps Twistor fibrations

highest

and lines the

We shall

contact birational maps between

the coordinatisations of

§1.1.4.

Dynkin diagrams

correspondences a

conventions

projectivised nilpotent

as

correspondences

of

(With

correspondences.

linear maps from

sets of these

the

Lawson

quaternion-Kähler

quaternionic

generalised Heisenberg

are

represented

construct contact birational

(cf.

twistor spaces of Wolf spaces of the

the fact that

to extended

their nilradicals

geometric description

on

nilradicals.

isomorphic

correspond

flag manifolds

holomorphic

This

contact manifolds.

complex

of Theorem 1 is based

flag manifolds

these

as

respectively).

quaternion-Kähler geometry).

on

(BURSTALL 1990). Quaternionic

birationally equivalent The

are

details

family

a

generalised flag more

0

ℂ𝕡2: being quaternion-Kähler they ℂᵔ ;1. In general,

twistor space for the twistor spaces which arise from

Salamon 1982 for

=

contact manifolds with fibre

from among

(cf.

x0

Bryant-Lawson correspondence

spaces.

with S4 and

complex

Riemannian symmetric spaces of compact type

equations

stay away from L1 ∪ L2 and

transversal to the kernel of the differential of b a

the

given by

and I

rise to minimal immersions in S4 if

give

twistor spaces of compact as

in ℂ𝕡2 which

curves

are

found

projections

to the twistor space of

curves

0 and is well defined away from the lines l

p1 and P2 denote the canonical

imposes

be used to transfer these

away from the surfaces S1 and S2

biholomorphism

a

can

are

in

arising

3 a

.

bijective correspondence

from

su(n)

and

sp(n)

are

with compact the Grass-

𝔾r2(ℂn) and H𝕡n-1 with twistor spaces given by the flag manifolds F1,n-1(ℂn) hyperplanes in ℂn) and ℂ𝕡2n-1 respectively. All flag manifolds have natural projective

mann

manifolds

(lines

in

realisations

Baston and Eastwood

(cf.

tivised

nilpotent

arising

from

orbits. More

precisely,

complex simple

a

system ▵ and root spaces a

and

negative

where N

Gℂo

=

and 0

≠ o

,

nilpotent

Kostant-Kirillov-Soriau

2-form),

with the radial vector field

gives

quaternionic

Lie(Gℂ). a

decomposition

gℂ. Adjoint

denoted here

▵

Cartan

=

by

a

a

nilpotent

projec-

Wolf space

∪ ▵_ into

▵+

natural

a

are

subalgebra

nilpotent orbit;

root

orbits carry

ω. For

G𝕡-equivariant complex

a

a

root then there is an identification Z

highest

orbit in

twistor space of

Let j ⊂ gℂ be

N is called the highest

∈ s

the smallest nontrivial

let Z denote the

α ∈ ▵. We choose

If ρ ∈ ▵+ is the

roots.

twistor spaces of Wolf spaces

particular

algebra gℂ =

Lie

root

In

1989).

positive

𝕄N

=

with

⊂ ᵔ ;gℂ,

this is in fact

symplectic

form

(the

orbit N the contraction of ω

contact structure θ on

𝕡N We have

omega left-parenthesis left-bracket e comma up er A right-bracket comma left-bracket e comma up er B right-bracket right-parenthesis equals left pointing angle e comma left-bracket up er A comma up er B right-bracket right pointing angle comma theta left-parenthesis left-bracket e comma up er X right-bracket right-parenthesis equals left pointing angle e comma up er X right pointing angle comma

(2) [e, gℂ]

where TeN is identified with references

where

Kobak

see

We shall

gℂ

=

LEMMA 2.

Killing

Finally,

(1)

The

denotes the

m_

of the Bruhat

nilpotent

form

on

gℂ(for

details and further

more

1994).

the

properties

root

is the

and

simply

connected. If

viewing a ω lies

decomposition

nilpotent algebra s

decomposition

orbits but the

G

We shall write

denotes the least number of factors in the

root reflexions.

highest

(•, •)

following conventions. Lie(Gℂ) and Gℂis connected

use

gℂthen l(ω) the basic

and

general

case

of

flag

projectivised highest root nilpotent

.

manifolds.

is similar,

see

of

(We

w

as a

in the

Weyl group

into

product

The

a

following

consider here

Baston and Eastwood

orbit 𝕄N ⊂ ᵔ ;gℂ has

of ᵔ ;gℂ,

point

a

Bruhat

of

W of

simple

lemma lists

projectivised 1989.)

decomposition

double-struck up er P equals quare-union Underscript alpha el ment-of up er L Endscripts normal e normal x normal p times Subscript minus Baseline times German g Subscript alpha Superscript double-struck up er C Baseline equals quare-union Underscript alpha el ment-of up er L Endscripts up er N Superscript alpha Baseline German g Subscript alpha Superscript double-struck up er C Baseline times left-parenthesi d i s j o times i n t u n i o n right-parenthesi comma

where L ⊂ ▵ denotes the set of

algebra

nα ⊂ m_ defined

by

long roots

and Nα = exp nα is the

nilpotent

Lie group with Lie

the formula SuperscriptalphaBaseline qualsStarLayout1stRow cir led-plusEndLayoutUnderscriptgam ael ment-ofnormalup erDeltaSubscriptminusBaselinecom aleft-parenthesi alphatimescom agam aright-parenthesi les -than0Endscripts imesGermangSubscriptgam aSuperscriptdouble-struckup erCBaselineperiod

(2)

If α ∈ L then the map

Bruhat cell A of the shortest Let

(1990).

us

a

complex diffeomorphism

length such

=

between

ℂk

and the

and ω ∈ W is the element

where k

consider the

Let ▵k

A is

that

ωα

=

ρ.

following special

∪ {0} : (α, ρv) {α ∈A▵

case =

k}

of

a

grading of gℂ,

defined

by

Burstall and

Rawnsley

and put

GermangSuperscriptleft-parenthesi kright-parenthesi Baseline qualsStartLayout1stRow circled-plusEndLayoutUnderscriptalphatimesel ment-ofnormalup erDeltaSuperscriptkBaselineEndscriptstimesGermangSubscriptgam aSuperscriptdouble-struckup erCBaselinetimes emicol nthentimesGermangSuperscriptdouble-struckup erCBaseline qualsStartLayout1stRow circled-plusEndLayoutUnderscriptkequalsnegative2Overscriptkequals2EndscriptstimesGermangSuperscriptleft-parenthesi kright-parenthesi Baselineandtimesleft-bracketGermangSuperscriptleft-parenthesi irght-parenthesi Baselinecom aGermangSuperscriptleft-parenthesi jright-parenthesi Baselineright-bracketsubset-ofGermangSuperscriptleft-parenthesi iplusjright-parenthesi Baselineperiod

(3)

Here g

is

by

definition the Cartan

algebra j.

We have the

following equalities:

German g Superscript left-parenthesi plus-or-minus 2 right-parenthesi Baseline equals German g Subscript plus-or-minus rho Superscript double-struck up er C Baseline times comma German g Superscript left-parenthesi 0 right-parenthesi Baseline equals German l comma and German g Superscript left-parenthesi plus-or-minus 1 right-parenthesi Baseline circled-plus German g Superscript left-parenthesi plus-or-minus 2 right-parenthesi Baseline equals Subscript plus-or-minus Baseline

where A

is the nilradical and I the reductive part of p.

n

geometric interpretation.

to Theorem 1 any

According

of dimension 2n +1 is birationally equivalent

ℂ𝕡2n+1

contact structure on

ℂ2n

on

comes

with the radial vector

description

of such

Let R for

⊂ ▵+.

gℂ.

The

This

grading (3)

simple

⟨H,Hα⟩

that

by

for

the

z

𢂇 N

we

symplectic

for

have

a

a

=

roots and choose

α(H)

contact

TZN

=

nilpotent

(the complex

geometric

and

ℂ𝕡2n+1

hand, namely 𝕡TZN

TZN.

at

[z, gℂ]

explicit

a more

In fact contact birational follows.

as

H

basis

⟨Xα,X-α) can

orbit

complex symplectic structure

gℂ→ TzN which are defined

for all H ∈ j and

shows that the tangent space

more

Weyl

a

minimal

manifold to ℂ𝕄2n+1

structure on

be constructed from linear maps

denote the set of means

might hope

One

field).

equivalences. Indeed,

𝕡TZN can

complex contact

from the contraction of the constant

with the contact structure induced maps from 𝕡N to

as a

projectivised

,

=

be written

(so [Xα,X-α]

1

=

Hα).

as

up er T Subscript Baseline Subscript z Baseline equals circled-plus double-struck up er C up er H Subscript rho Baseline equals German g Superscript left-parenthesi 2 right-parenthesi Baseline circled-plus German g Superscript left-parenthesi 1 right-parenthesi Baseline circled-plus double-struck up er C up er H Subscript rho Baseline

(4) (here

a

.

This

gives

a

decomposition gℂ

TzN ⊕ Vz

=

where

up erVSubscriptzBaseline quals eft-parenthesi GermangSuperscriptleft-parenthesi 0right-parenthesi Baselinecir led-minusdouble-struckup erCup erHSubscriptrhoBaselineright-parenthesi cir led-plusGermangSuperscriptleft-parenthesi negative1right-parenthesi Baselinecir led-plusGermangSuperscriptleft-parenthesi negative2right-parenthesi Baseline quals eft-parenthesi intersectionup erHSubscriptrhoSuperscriptup-tackBaselineright-parenthesi cir led-plusStartLayout1stRow cir led-plusEndLayoutUnderscriptalphael ment-ofnormalup erDeltacom aleft-parenthesi alphatimescom arhoright-parenthesi les -than-or-slanted-equals0Endscripts imesGermangSubscriptalphaSuperscriptdouble-struckup erCBaselineperiod

Let p

:

gℂ

=

denote the linear

TzN ⊕ Vz → TzN

projection

to the first summand. The map

pi colon g Superscript double-struck up er C Baseline contains-as-member x right-ar ow from bar p left-parenthesis x right-parenthesis plus left-parenthesis x comma up er X Subscript negative rho Baseline right-parenthesis z ampersand up er T Subscript z Baseline times

(5) is

simply

a

of p with

composition

of 2 and is constant

on

the

a

linear

remaining

that

=

on

exp n be the

and let π :

z

of

TZN

components in the direct

between the canonical contact structures LEMMA 3. Let C

endomorphism

gℂ → TZN

which stretchesgℂ by the factor

sum

(4)

.

There is

a

simple

relation

𝕡N and 𝕡TZN:

big cell

be defined

in the Bruhat

decomposition

by Formula (5)

.

of 𝕡N. We

assume

If x ∈ n_ then

pi left-parenthesis e Superscript x Baseline z right-parenthesis equals 2 z plus left-bracket x comma z right-bracket period

(6) Moreover, π induces

Proof.

We

begin

a

1

:

1 contact map from C to 𝕡TzN.

with the

proof

of Formula

(6)

.

If

x

is

an

element of

n_

then

e Superscript x Baseline z equals normal up er A normal d Subscript Baseline Subscript e Sub Superscript x Subscript Baseline times z equals e Superscript normal a normal d Super Subscript x Superscript Baseline times left-parenthesi z right-parenthesi equals normal up er Sigma Underscript n equals 0 Overscript 4 Endscripts times StartFraction 1 Over n factorial EndFraction left-parenthesi normal a normal d Subscript x Baseline right-parenthesi Superscript n Baseline left-parenthesi z right-parenthesi equals z plus left-bracket x comma z right-bracket plus one-half eft-bracket x comma left-bracket x comma z right-bracket right-bracket plus period period period

(7)

where the omitted terms

belong

We

to n

can

write

x

n_

⊂ Vz

we can

where xρ,xα ∈ ℂ. Since X

and

[x,[X-ρ,z]] lie

in

xρX-ρ when calculating π([x, [x, z]]).

The

remaining summands give

the summand

ignore

normalup erSigmaUnderscriptalphael ment-ofnormalup erDeltaSuperscript1BaselineEndscripts imesup erNSubscriptnegativealphatimescom arhoBaselinetimesxSubscriptalphaBaselinel ft-bracketxcom aup erXSubscriptrhominusalphatimesBaselineright-bracketequalsnormalup erSigmaUnderscriptalphacom abetael ment-ofnormalup erDeltaSuperscript1BaselineEndscripts imesup erNSubscriptnegativealphatimescom arhoBaselinetimesxSubscriptalphaBaselinetimesxSubscriptbetaBaselinel ft-bracketup erXSubscriptminustimesbetaBaselinetimescom aup erXSubscriptrhominusalphatimesBaselineright-bracketperiod

Nα,β α + β ࣔ 0,

The constants with

α +

β ≠ ρ are either

defined

are

by

the formula

lie in S

zero or

[Xα, Xβ]

The summands

NαβXα+β.

=

and the summands with

α +

β

ρ give

=

minusone-halfnormalup erSigmaUnderscriptalphacom abetael ment-ofnormalup erDeltaSuperscript1BaselineUnderUnderscriptalphaplusbetaequalsrhoEndscriptsxSubscriptalphaBaselinetimesxSubscriptbetaBaselineleft-parenthesi up erNSubscriptnegativealphacom a lphaplusbetaBaselinetimesup erHSubscriptbetaBaselineplusup erNSubscriptnegativebetacom a lphaplusbetaBaselinetimesup erHSubscriptalphaBaselineright-parenthesi equalsone-halfnormalup erSigmaUnderscriptalphacom abetael ment-ofnormalup erDeltaSuperscript1BaselineUnderUnderscriptalphaplusbetaequalsrhoEndscriptsup erNSubscriptalphacom abetaBaselinetimesxSubscriptalphaBaselinetimesxSubscriptbetaBaselineleft-parenthesi up erHSubscriptalphaBaselineminusup erHSubscriptbetaBaselineright-parenthesi

since

N-α,α+β

since

⟨Hα Hβ,Hα+β⟩

=

—

N-β,α+β

—

(X-p,Xp)

Nα,β (cf.

⟨α β, α+ β⟩ and | α|

to 𝕡N

x,y ∈ n_ and let if and

only

(adx,)ky

=

if

a

=

⟨y,az⟩

as

projections

(Formula (2) ),

0

0 if k > 1. As

a

But

Hα

—

Hβ is perpendicular

This shows that

(6)

and

contact map from C to

a

ex ∈ Gℂ. The vector =

=

=

𝕡TZN

and to

3).

p.

| β| . (7) implies that ⟨exz,X-p⟩z z,

It remains to show that π induces

tangent

1972,

Warner

-

=

1, Formula

=

=

p(exz)

=

i.e.

∈ TazN

𝕡TZN.

⟨a-1y, z⟩

=

projects

[x, z].

+

Hρ

Since

follows. We shall represent vectors

of vectors tangent to N and to TzN

[y,az]

z

to

Let

respectively.

to a contact vector in Tℂ(az)𝕡N

0. But a-1 y

=

Ade-x y

=

y

—

adx

y since

result

left-bracket y comma a z right-bracket element-of upper T Subscript a z Baseline times times p r o j ects to a contact vector in times upper T Subscript double-struck upper C left-parenthesis a z right-parenthesis Baseline times double-struck upper P left right double arrow left pointing angle y minus left-bracket x comma y right-bracket comma z right pointing angle equals 0 period

(8) We shall

now

calculate

We have

π*[y,az].

Baker-Campbell-Hausdorff formula,

so

s

the

by

the differential of π is

given by

the formula

pi Subscript asterisk Baseline times colon up er T Subscript a z Baseline contains-as-member left-bracket y times comma a z right-bracket imes right-ar ow from bar left-bracket y plus one-half times left-bracket y x right-bracket comma z right-bracket imes element-of up er T Subscript left-parenthesi 2 z plus left-bracket x comma z right-bracket right-parenthesi Baseline times left-parenthesi up er T Subscript z Baseline up er N right-parenthesi period

(2)

It follows from

Consequently,

[A, z) ∈TZN projects to π*[y, az] is contact in 𝕄TzN

that X

the vector

=

a

contact vector in

iff

iff

T(ℂy)𝕄TzN

a

⟨A, Y⟩

=

0.

We have:

.

leftpointingangle2zplusleft-bracketxcom azright-bracket imescom ayplusone-halftimesleft-bracketycom axright-bracketrightpointingangletimesequalsleftpointingangleleft-bracketxcom azright-bracket imescom ayrightpointingangletimesplus2leftpointinganglezcom ayrightpointingangletimesplusleftpointinganglezcom aleft-bracketycom axright-bracketrightpointingangletimesplusone-half eftpointingangleleft-bracketxcom azright-bracketcom aleft-bracketycom axright-bracketrightpointingangle2leftpointinganglezcom ayplusleft-bracketycom axright-bracketrightpointingangletimesplusone-halftimesleftpointingangleleft-bracketxcom azright-bracketcom aleft-bracketycom axright-bracketrightpointingangle quals0

(the

last

equality

follows from

The map 𝕄π : 𝕄N → 𝕄TzN a

is

[x, z]

and the fact that

(8) a

candidate for

a

contact birational

map is birational if it is rational and has rational

if it on

can

be written

as z

M. Birational maps

↦ are

r(z)

=

[1

:

f1:...:

∈ g(1) ⊕g(10) and

inverse;

fn]

generically biholomorphic

where

a

map

fi

are

[x, y]

□ ∈ g(-2)

correspondence. r :

M

→

We recall that

N ⊂ ℂ𝕄n is rational

global meromorphic

and well defined away from

a

functions

subvariety

of

codimension 2. It is clear that ᵔ ;π is well defined cell C. It turns out that

𝕄(N∩Vz)

on

is

a

the set

𝕄N𝕄(N∩Vz).

union of Bruhat cells.

This set

can

be

bigger

than the

big

1. Rank two root systems

Figure

Arrows denote negative roots, the roots in ▵1 are marked by ∂, the highest root ρ is marked by ▵ (and -ρ in Dynkin diagrams by ☉ ). Thick lines represent

simple roots. The numbers next to the long roots indicate the dimensions of the corresponding Bruhat

LEMMA 4. With the above notation

cells.

have:

we

double-struck up er P times left-parenthesi intersection up er V Subscript z Baseline Subscript Baseline right-parenthesi equals quare-union Underscript gam a el ment-of up er L com a left pointing angle gam a com a rho right pointing angle les -than-or-slanted-equals 0 Endscripts up er N Superscript gam a Baseline times German g Subscript gam a Superscript double-struck up er C Baseline times left-parenthesi d i s j o times i n t u n i o n right-parenthesi period

Moreover, if γ ∈ ▵ 29; L

First note that if y ∈ g

Proof. Let τ be

have an

m_-module,

a nonzero

It remains to show that 𝕄 is

no

components ing(2) so =

Let in

Fig.

root.

0. This ends the

us

have

proof

look at the

a

1 show that

are

precisely

Now

σα(ρ)

cannot be

diagram

—

To

see

in the

gℂ sp(n,ℂ).

and,

N

on

C

cases, when

gℂ has

variety (codimension 3)

For Ci the set the

on

this note that, are

disjoint

configuration might

seem

from

a

result

This is as we

All

for G2

the roots

If

to

a

is such

precisely

a

(Schubert

2-dimensional

root

2, the cells of

σα(ρ) a

when

(α, ρ) ∩ ϕ.

simple

Diagrams

In fact this situation

α). 𝕄(N∩Vz) precisely

σα(ρ) ≠.

surprising

As

have

(codimension 2)

to Lemma

parametrised by

does not lie in ▵1 .

cell.

big

consider the rank 2 root system span

s

.

rank 2.

Vz) is 𝕄(N∩

according

corresponding

of type A1 ⊕ A1) since

This

since Vz

Elements of N

.

lie in

simplest

cells).

of 𝕄N"

Bruhat cell is

σα(ρ)

to y.

□

𝕄πis defined only

decomposition

⟨ρ, αv)α and

g࠶contains the =

so

equal

𝕄VZ.

[X-ρ,y]

3-dimensional Schubert

1 and it is clear that

for

i.e. when

a

is

each Bruhat cell of codimension at least 1. Let τ ∈ ▵1

≠ ρ (σα∈W is the reflexion

semisimple (i.e. Fig.

shown in

in

since Φ is not radial.

for root systems Cn.

corresponding

ρ

=

entirely

components of

(codimension 1),

σα(ρ)

Lemma 4, the

y in g

all elements of the Bruhat cell

is the union of two 1-dimensional Schubert varieties

𝕄(N∩ Vz)

codimension 1 in the Bruhat α ∈ R and

on

by definition the closures of Bruhat

Schubert variety occurs

lies

variety 𝕄(N∩Vz)

for the root system A2 and is varieties

so

m

singular

nonzero

.

∈ m_ then the component of Adex

x

Consider the vector field Φ : y ↦ [X-ρ,y]

a

π*(Φ)

the Bruhat cell N

component in TzN. If δ ∈ ▵1 ∪ {ρ} then g

the Bruhat cell exp

be

long

and

singular on

root. If τ ∈ ▵1 ∪ {ρ} then g

long

a

N is

then the map π is

for which

root

σα(ρ)

▵1.

This root system

systems of rank 2

when the extended

possible only

then, by

are

Dynkin

for the root systems

Cn,

expect ᵔ ;π to be birational. Moreover,

the

twistor space

quaternionic

however, that in this

case

PROPOSITION 5. If

gℂis

then the linear map π :

𝕄Nis

the map

a

in such

the

projective

extends to

ᵔ ;πtrivially

complex simple

gℂ→

case

Lie

algebra

space

ℂ𝕄2n+1

𝕄N.To summarise,

itself. We shall see,

we

have the

and N ⊂ gℂ is the minimal

TzN, defined in Formula (5) induces ,

from 𝕄N to 𝕄TzN. Moreover, the map 𝕄π is well defined

on

a

following orbit

nilpotent

contact birational map ᵔ ;π

the set

StarLayout1stRow cir led-plusEndLayoutUndersciptalphael ment-ofnormalup erDeltaSuperscipt1Baselineinters ctionup erLunio StarSetrhoEndSetEndscriptsup erNSupersciptalphaBaselinetimesGermangSubscriptalphaSupersciptdouble-struckup erCBaselinetimescom a

and if

gℂis

not of

The cell structure of

REMARK 6.

Eastwood

type Cn then 𝕄πis singular beyond the big cell N

1989);

𝕄(N∩Vz).

For

example

following diagram for the

flag

one

𝕄(N

"*" the

Vz)

one

can

diagrams (see

Baston and

find which Bruhat cells lie in the

singular

set

gets the

the group F4. In

diagram weights corresponding

cells in

manifolds is encoded in Hasse

from these diagrams

to

marked with

are

a

𝕄(N'∩VzZ) is a 10-dimensinal variety (codimension 5 in𝕄N). set

Highest π :

sp(n,(ℂ)

(ei)=i...2n

(ℂ2n)‘.

nilpotent orbit in sp(n,ℂ).

root →

TzN. Let ω be

in which

We have

the Cartan

a

We shall find

explicit

nondegenerate skew-symmetric

2-form

where

a

formulae for the on

ℂ2n. We choose

(ei)i=1...2n

to define a

basis

a

is the dual basis for and

s

algebra consisting of the diagonal matrices

projection

decomposition sp(n, ℂ)

we can use =

TZN⊗VZ

and the associated map π : N → TzN The map

φ :

parametrises 𝕡N

=

the minimal

𝕔𝕡2n-1).

Take

We have i

z

ℂ2n \{0}∋x→x  97;ω(x,.)&#Nx2 08; orbit in

nilpotent =

and put

φ(e1)

w

=

x

and TzN —

sp(n,𝕔) (this φ*(Te1𝕔2n)

—

shows that N =

{e1 ⊗ω(v, •)

=

+

v

(ℍn \ {0})̸ ⊗ω(e1,

·): v

2

and

𝕔2n}.

x1e1. We get:

phi left-parenthesis x right-parenthesis equals left-parenthesis x 1 e 1 plus w right-parenthesis circled-times omega left-parenthesis x 1 e 1 plus w comma dot right-parenthesis equals x 1 squared e 1 circled-times omega left-parenthesis e 1 comma dot right-parenthesis plus x 1 left-parenthesis e 1 circled-times omega left-parenthesis w comma dot right-parenthesis plus w circled-times omega left-parenthesis e 1 comma dot right-parenthesis right-parenthesis plus w circled-times omega left-parenthesis w comma dot right-parenthesis period

Now

(we

w

w ⊗ ω(w,

can

use

•)

Vz and, since

the scalar

z

=

e1

product ⟨A, B⟩

ω(ei, •), =

we

have

trace AB since it is

x

proportional

to the

Killing form).

(φ*(ei))i=1... 2n

In the basis

so

x

C)

the map π o

for TzN

π vanishes if

x1

0

=

(this

φ is given by

the formula

g

condition defines the complement of the

big

cell

but 𝕔π obviously extends to 𝕔𝕡2n-1.

Highest

root

birational

φ

where D

𝕔n+1

{(x,ξ)

have

surjective

a

:𝕔n+1

×

similar way

a

is the

1,𝕔)

one can

twistor spaces of

quaternionic

where N ⊂ sl((n +

case we

In

sl(n + 1, 𝕔).

between the

and 𝕡N,

≥ 2). In this

n

orbit in

correspondence

𝕔𝕡2n-1

between that

nilpotent

highest

root

find formulae for

𝔿r2(𝕔n+1) nilpotent

and

orbit

a

contact

ℍ𝕡n-1, (we

i.e.

assume

map

(𝕔n+1)* N ξ ⊗ x → (x,ξ) ∋ D ⊃

0, x ≠ 0, ξ ≠ 0} (note that there is a bijection (𝕔n+1)* : ξ(x) 𝕡N ∋ [A] → (im A ⊂ ker A) F1,n(𝕔n+1)). We adopt the same notation as in the previous example (but basis elements in 𝕔n+1 are indexed now by the set {0.n}) and take z φ(e0,en) e0⊗en. =

×

=

=

=

Then

TZN

η(e0)

+

φ(T(e0,en)D)

=

en(v)

=

en,

e0⊗

The Cartan

TZN⊗VZ

consists of the tensors e0 ⊗ η +

0. We choose the

e1 ⊗ en,..., en-1 ⊗en, e0 ⊗

algebra j consisting

where Vz is

perpendicular such that

following

to

of the

en-1,

Hρ

e0 ⊗ e0

—

en ⊗ en.

⊗ en where η

⊗en-2

e0

diagonal matrices

spanned by matrices ei⊗ej =

v

(𝕔n)*,

𝕔n and

v

basis for TzN:

with i

Now take

sl(n, Ǖ 4;) gives

in

≠ j,

e0 e0 ⊗

e1,

e0 ⊗

i≠ 0 and

a

—

en ⊗ en.

decomposition sl(n, Ǖ 4;)

j ≠ n,

and

=

by diagonal matrices g and

x

We have

g

xcircled-timesxiequalsnormalup erSigmaUnderscriptiequals1OverscriptnEndscriptstimesx0xiSubscriptiBaselinetimese0circled-timeseSuperscriptiBaselineplusnormalup erSigmaUnderscriptiequals1Overscriptnminus1EndscriptstimesxSubscriptiBaselinetimesxiSubscriptnBaselinetimeseSubscriptiBaselinetimescircled-timeseSuperscriptnBaselineplusnormalup erSigmaUnderscriptiequals0OverscriptnEndscriptstimesxSubscriptiBaselinetimesxiSubscriptiBaselinetimeseSuperscriptiBaselineplusnormalup erSigmaUnderUnderscriptiequals1com aelipsi com anUnderscriptinot-equalsjEndscriptsEndscriptstimesnormalup erSigmaUnderscriptjequals0Overscriptnminus1EndscriptstimesxSubscriptiBaselinetimesxiSubscriptjBaselinetimeseSubscriptiBaselinetimescircled-timeseSuperscriptjBaselineperiod

The last summand in the formula above

projects orthogonally

in j to the matrix

into account the summand

(x ®⊗Xξ-Pρz

belongs

to

Vz and the diagonal matrix

½(x0ξ0— x£ξ)(eo0®⊗e°0— =

xo0ξe00®⊗en

we

en

i

®⊗e"n GC𝕔PρC⊂TZN■.Taking

get

PROPOSITION 7. The formula left-parenthesis x 0 colon ellipsis colon x Subscript n Baseline times semicolon xi 0 colon ellipsis times colon xi Subscript n Baseline right-parenthesis right-ar ow from bar left-bracket 2 x 0 xi Subscript n Baseline colon x 1 xi Subscript n Baseline times colon ellipsis colon x Subscript n minus 1 Baseline xi Subscript n Baseline times colon x 0 xi Subscript n minus 1 Baseline times colon ellipsis colon x 0 xi 1 times colon one-half left-parenthesis x 0 xi 0 minus x Subscript n Baseline xi Subscript n Baseline right-parenthesis right-bracket

defines

a

contact birational map from

the set

{x0

ξn =

=

0}

REMARK 8. In the basis

and

case

F1,n(𝕔n+1)

to

biholomorphic if and only

when

n

=

2

one recovers

𝕔𝕡2n-1. This map is well defined if x0ξn

≠ 0.

Gauduchon’s Formulae e

away from

(1)

.

To

see

this

use

the

References Baston , R. J. and Eastwood , M. G. ( 1989 ) The Penrose transform, its interaction with representation Clarendon Press , Oxford Mathematical Monographs.

theory,

Bryant

R. L

,

( 1982 ) Conformal

DifF. Geom. 17 , 455 473

and minimal immersions

of compact surfaces

into the

4-sphere

,

J.

-

.

Burstall , F. E.

( 1990 ) Minimal surfaces in quaternionic symmetric manifolds, Cambridge University Press 231 235

spaces ,

Geometry

of low-dimensional

-

.

,

Burstall , F. E. and

Rawnsley , J. H. Lect. Notes Math. 1424

( 1990 )

Twistor

theory for

Riemannian symmetric spaces ,

Springer-Verlag,

.

Gauduchon , P. Kobak , P. Z.

( 1987 )

( 1993 )

La correspondance de Bryant ,

Astérisque

181 208 -

,

.

Quaternionic geometry and harmonic maps , D. Phil. Thesis,

University of Oxford

Kobak , P. Z.

Systems

( 1994 ) Twistors, nilpotent orbits and harmonic maps , Harmonic Maps and A.P. and Wood , J.C. , eds.), Vieweg , Braunschweig/Wiesbaden Fordy , (

Astérisque

Salamon , S. M. Swann , A. F. Warner , G.

Integrable

.

Lawson , H. B. , Jr , 624 ,

.

( 1983 ) Surfaces minimales (Soc. Math. France,

121 122 -

( 1982 ) Quaternionic

( 1991 ) HyperKähler ( 1972 )

Harmonic

Kahler

et la construction de Calabi-Penrose , Sém. Bourbaki 1985

197 211 ). -

,

manifolds

Invent. Math. 67 143 171 -

,

.

and quaternionic Kähler geometry , Math. Ann. 289 , 421 450

analysis

-

on

semi-simple

Lie groups , vol. I ,

Springer-Verlag

.

.

2

Chapter

to conformal

Applications

geometry

M&LHEbyM.I§natursog.dhwP2GJcot.i1nd, There

several

are

significant

geometry. The links the

isomorphism

are

of Lie

ways in which

spinors

particularly strong

and the fact that the irreducible

simple (it

SO(4) into

self-dual and

a

irreducible). dual

only

one

of

su(2)

(whereas

of two

important

𝕔2).

on

As

reason

for this is

and the

easy to describe a

Weyl

Weyl

the

curvature

Weyl

decomposes curvature is

curvature and bundles with self-

achievements of twistor

equations

(as

consequence the group

in other dimensions, the

to define metrics with self-dual

subjects

Yang-Mills

underlying

especially

are

orthogonal group),

construction for self-dual solutions of Einstein’s

for self-dual

conformal differential

ࣅ su(2) × su(2)

spinor representations

such

anti self-dual part

This allows

connections,

graviton

is the

an

the

in four dimensions. The

representations

powers of the fundamental

is not

impinge on

algebras

so(4)

symmetric

and twistors

theory,

Penrose’s non-linear

and Ward’s twistor construction

fields.

A similar coincidence of low-dimensional Lie

algebras

can

be

regarded

as

the basis of twistor

theory, namely so(6, Ǖ 4;) and its various real forms, such

as

so(4,2) The

spin representation

isomorphism. six

These

dimensions, just

of

so(6,
D554;)

is

do

spinors

a

in any

n-form is

higher

totally null,

The second the

dimension. cf. the

isomorphism

point being

that

the

in four dimensions

(A spinor

appendix also

su(2, 2).

spinors

are


D554;4 induced by the above

tool for differential geometry in

(cf. §§11.2.2 5 ). -

on

Part of the

automatically

special utility

‘pure’

when the associated

of Penrose & Rindler 1986, and S.B.Petrack in the

use

here

pure, whereas this is not the

in 2n dimensions is said to be

underpins

SO(4,2)

representation

powerful computational

arises from the fact that in six dimensions case

ࣅ

precisely

isomorphisms give as

≅ sl(4),

§1.3.9.)

of twistors in four-dimensional conformal geometry,

is the group of conformal motions of

space. This is familiar in that natural constructions

on

flat twistor space

compactified (such

as

Minkowski

forming

various

DOI: 10.1201/9780429332548-2

2. sheaf

to conformal

Applications

geometry

rise to

cohomology groups) give

Minkowski space. In this context

on

conformally invariant objects (such

‘conformally

differential

as

invariant’ may be taken to

mean

operators)

‘invariant under

conformal motions’. There is also On

compactified

much wider

a

in which twistors enter into conformal differential geometry.

sense

Minkowski space, there is

an

exact sequence of vector

OB1;OA࢐ 92; 0.

0 → OA → The bundles at either end twistor space manifold

as

fibre.

are

& MacCallum

with

1972).

bundles and the bundle in the middle is the trivial bundle with this situation

Surprisingly,

(if spin). Moreover,

naturally equipped

spin

a

bundles, usually denoted

persists

the bundle in the middle

conformally

This is the

on a

(called

four-dimensional conformal

general

the bundle of ‘local

twistors’)

invariant connection called local twistor transport connection associated with Cartan’s

spin

SO(4,2)

comes

(Penrose

conformal

connection. The local twistor connection may be defined of

a

metric in the conformal class

factor induces

a

Penrose & Rindler 1986 for full

(see

A choice of conformal

details).

splitting Oα

of the above exact sequence. Under a

quite explicitly in terms of the Levi-Civita connection

twistor Zα represented

a

by (wA, πA')

=

OA &#c2295; OA,

conformal

rescaling

ĝab

with respect to gab is

→

Ω2gab

the

splitting changes

so

that

represented by

left-parenthesi ModifyngAboveomegaWithcaretSuperscriptup erABaselinecom aModifyngAbovepiWithcaretSubscriptup erAprimeBaselineright-parenthesi equals eft-parenthesi omegaSuperscriptup erABaselinecom apiSubscriptup erAprimeBaselineplusnormalinormalϒSubscriptup erAup erAprimeBaselineomegaSuperscriptup erABaselineright-parenthesi

with respect to ĝab where Υa

=

▿a log Ω. The covariant derivative

can

be

expressed

as

nabla left-parenthesis omega Superscript up er A Baseline comma pi Subscript up er A prime Baseline right-parenthesis equals left-parenthesis nabla omega Superscript up er A Baseline plus i epsilon Subscript up er B Superscript up er A Baseline pi Subscript up er B prime Baseline comma nabla pi Subscript up er A prime Baseline plus i up er P Subscript b up er A up er A prime Baseline omega Superscript up er A Baseline right-parenthesis

where normal up er P Subscript a b Baseline equals normal up er Phi Subscript a b Baseline minus normal up er Lamda g Subscript a b Baseline equals StartFraction 1 Over 1 2 EndFraction up er R g Subscript a b Baseline minus one-half up er R Subscript a b Baseline period

Here Rab, is the Ricci tensor and R is the scalar curvature. It as

defined is invariant under conformal

rescalings.

can

be checked that this connection

Local twistors at

a

point

can

be

thought

of

as

the flat space twistors associated to the flat conformal Minkowski space that best approximates the

space-time

to second order at that

point.

Many of the articles in the chapter

use

this local twistor construction, for example in the

of conformally invariant differential operators concerned with the More

more

generally,

that is anti-self-dual

geometric in 2n

(cf. §§11.2.13

realization of twistors

dimensions,

(resp. self-dual).

an

α-surface

as

-

construction

19 ). Several other articles

‘α-surfaces’

(resp. β-surface)

(and is

a

dual twistors

totally

as

are

β-surfaces).

null n-surface

§11.2.1

Introduction

The interest in foliations of space-times by such surfaces goes back in effect decades to the

1961,

when Ivor Robinson

subsequent development

of

null

a

space-time, a

rays

real curved space-time with

solution of Maxwell’s

naturally

with twistor

in

theory

chapter or

touch

foliations

the articles

the Kerr theorem and the

recurrent theme in the

important

an

as

on

a

this theme and

by Hughston

Summary of chapter. as an

aid to

as

Jeffreyes

interest in its

signature

own

(including generalizations

of twistor

theory.

consequence

as a

and related

has from the outset been

Several of the articles in this

various aspects of the geometry of shearfree ray congruences

explore

the

play

do the familiar two-component

regards

the

complexifies

in

§§11.3.4

-

space-time

a

admits

a

Killing spinor (cf.

6 ).

§§11.2.2-4 by Hughston

and

Jeffryes apply flat

twistor space

the differential geometry of curved six-dimensional spaces. Here the twistors

analysing

the

on

foliation by α-surfaces and

Goldberg-Sachs theorem)

development

The articles in

live in the tangent space, and as

to a

one

of non-trivial ways, and

variety

by α-surfaces. Such foliations also arise when and

influence

hyperbolic signature

If

equations.

equivalent

the illumination of various aspects of this result

constructions, such

significant

of β-surfaces.

consisting

This result ties in

to have a

was

than three

Robinson’s theorem showed that associated with

on a

discovers that the shearfree congruence is

foliation

conjugate

geodesic

(algebraically degenerate)

one

result that

a

general relativity.

any shearfree congruence of null there is

published

more

right,

same

spinors

role relative to the

to

space-time.

of the associated Riemannian and may

ultimately

be of

The

resulting formalism,

(or pseudo-Riemannian)

in the

use

underlying six-dimensional

study

of certain

manifold

which is flexible

metric, is of

global problems

some

in the

geometry of six dimensional manifolds (e.g. in connection with the existence of complex structures). See

§1.4.17 (Minimal

surfaces and

strings

in six

dimensions, by Hughston k. Shaw)

for

a

related

application. In

employed to

eight In

the

§11.2.5

same

to prove

an

dimensions

§11.2.6

theorem. In

a

analogue

(and

was

(see

(using

a

In

these ideas

is used to are

zero

In

and

a

§11.2.9

10

a

p-fold spinor

of

unless p are

studies

a

§7.3)

a zero =

Hughston &

3p +

rest

dimensions)

are

Mason

1988). of Robinson’s

mass

a

on zero

spinor field satisfies

field of valence p + q, then it is

q.

provided

holomorphic functions

conformally

which says that if

characterized in

natural invariant formulation is

Bailey

for geometry in six

extended to embrace the Sommers-Bell-Szekeres theorem

rest mass fields in terms of -

spinors

elementary (i.e. algebraic) proof

an

Sommers 1976, Penrose & Rindler

null self-dual Maxwell fields

alpha-planes, null

produce

repeated spinor of the Weyl spinor, §11.2.8

the

as

of the Kerr theorem in six dimensions. The result is then extended

the shearfree ray condition and is also

twistors

later extended to 2n dimensions in

spinor method

§11.2.7

rest mass fields

methods

terms of

integrable

for Penrose’s

with

original

distributions of identification of

simple poles (Penrose 1968, 1969).

invariant connection obtained

by decomposing

the

local twistor connection in the presence of

conformally Bailey goes

case

naturally sit inside twistor

‘Goldberg Sachs’

also. In

condition is

relative

§11.2.11

relationship

explored

in

§11.2.12

The articles

cohomology

on

an

§§11.2.13

17

-

and many of them

goals A

in

Fefferman & Graham

curved space. This is

to a

Differential on

up to more

(1985) using can

pursued

spinors,

some

flat

alternative

be

generated

that,

applied

of

In

.

are

obtained

tensors

complete

list

in which all the

using representation

connection).

to the

In

§11.2.13

Weyl spinor in

originally

discovered by

is that

§11.2.13

four

conformally

conformally invariant operators in

§11.2.14 it

of the bundles

weights

operators

A subtext of

theory

§11.2.15

In

are

functions

§ 11.2.16

on

is shown that the

assumption

which the invariant operators act

by allowing non-integral weights

§11.2.17

on

projective

a

a

simpler

space

case

in

even

were

representation theory

case one

them

derivatives,

and

by Weyl).

polynomial invariants

with all derivatives

is studied in which

by regarding

classified

(which

conformal

point together

In the flat

corresponding non-projective space, taking tensors

thought of as

the structures of bundles of infinite

of the

corresponding

be

studied in terms of their

their

can

one

of functions

jets and

compared

considers

a

(and

to their

projective

many invariants

generate

homogeneous

as

perhaps

functions

on

the

constructing all the affine invariants In

§11.2.17

it is shown that certain

invariants do not arise in this way.

‘exceptional’ in

admits

degenerates

classified

when

machinery.

structure rather than a conformal structure.

Finally

are

a

(1987)

(using

from the

the bundle of sections at

order).

generally bundles)

weighted

multipole-like

space).

(that is,

finite

a

in conformal manifolds is

non-trivial scalar conformal invariant

in further detail in

counterparts in flat space. In

for

on curves

the local twistor

such

polynomial conformal invariants can

bundles

jet

Then In flat

neighbourhood.

study of conformally invariant operators,

to curved space on

highly

not essential—no new

(and

by α-planes.

formal

rise to

give

is contained in Eastwood & Rice

key step

in Eastwood &; Rice that the conformal

integral is

a

conformal invariants is to construct

studying

generalized

invariant tensors and scalars

dimensions

foliation

in terms of these structures.

interpreted

concerned with the

are

operator is constructed that acts

dimensions, it gives rise

are

Penrose’s

study Killing spinors.

S is studied and shown to

invariant differential operators in Minkowski space

conformally

a

essentially yields

.

invariants.

polynomial

theory,

This

space and therefore have

between conformal circles and parameters

One of the

and scalars. of

spinor fields.

it is shown that this structure survives in the curved

satisfied,

structure. Robinson’s theorem in flat space is

The

of

the structure of the space S of leaves of

investigate

to

on

space the space S would

When the

pair

a

invariant eth and thorn operators. This is first used to

§11.2.18 a

-

19

some

applications

are

given.

conformal factor with vanishing

and fails when the

Weyl

In

§11.2.18

trace free

tensor is self-dual

Ricci

(the

condition that

a

tensor is

result

requires

given. a

a

conformal manifold This condition

nondegeneracy

con-

dition

the

on

distinguishes

in

Weyl tensor). However,

§11.2.19

a

conformally invariant

Weyl tensors

between space-times with self-dual

that have

tensor is

vanishing

presented that

Ricci tensors, and

those that cannot.

References Eastwood , M. G. & Rice , J. and their curved

( 1992 ), 213

144

analogues

( 1987 ) Conformally ,

Comm. Math.

Phys

invariant differential operators on Minkowski space 109 , 207 228 , and Erratum, Comm. Math. Phys., -

.

.

Fefferman , C. & Graham , C. R.

d’Aujourdui Astérisque ,

95 116

( 1985 )

Conformal invariants , in Élie Cartan et les

.

Hughston

,

L. P. k, Ward , R.S.

( 1979 )

Advances in twistor

Hughston

,

L. P. & Mason , L. J.

( 1988 )

A

275 285 -

generalised

theory

Pitman

,

.

Kerr-Robinson theorem , Class.

Quant. Grav

.

5

,

.

Hughston transform

,

,

L. P. & Mason , L. J. ( 1990 ) Further advances in twistor theory, Volume I: The Penrose Pitman research notes in mathematics series, 231 , Longmans .

Penrose , R.

( 1968 ) Twistor quantisation

Penrose , R.

( 1969 )

Solutions of the

Penrose , R. & MacCallum , M. A. H. and

Mathématiques

-

,

space-time Phys. Repts. ,

Penrose , R. & Rindler , W. Robinson , I.

( 1961 )

Null

and curved rest-mass

zero

space-time

equations

( 1972 ) Twistor theory:

6C , 241 315

,

,

Int. J. Theor.

J. Math.

an

Phys.

1 , 61 99 -

,

10 , 38 39 -

,

to the

approach

Phys.

.

.

quantization of fields

-

,

( 1986 ) Spinors

electromagnetic

.

and

space-time

fields , J. Math.

,

Vol. 2 , CUP

Phys.

2

.

290 291 -

,

.

Sommers P.D. ( 1976 ) Properties of shearfree congruences of null geodesics Proc. Roy. Soc. A349 .

309 318

,

-

.

§II.2.2HL.ughPsto.nDifferential Geometry Twistors

are

useful and

illuminating

account of the fact that twistors the

geometry

spinors

in Six Dimensions

are

in the

the

analysis

spinors

by

(TN 19, January 1985)

of manifolds of dimension six.

for the group

O(6, 𝕄).

Thus twistors

of six dimensional spaces similar in many respects to the role

in the geometry of four-manifolds. Whether these considerations

remains to be seen—my purposes here of results in outline form.

are

primarily geometrical,

This is

play

a

on

role in

played by two-component

are

of any

physical

and I shall summarise

a

interest number

Conventions: Point:

Xi

Metric.

i,j, k

0,1, 3,&2,5;4,#x03B123;,

=

Xαβ(skew,

=

gij

etc.

abstract index

Vi(X)

Vector field:

Vαβ(X) (skew)

=

Vαβ P[αQβ],

Null vector field:

=

where

normal up er Omega Subscript i j Baseline times up er V Superscript i Baseline times up er V Superscript j Baseline quals 0 left right double ar ow up er V Superscript alpha beta Baseline quals up er P left-bracket imes Superscript alpha Baseline up er Q Baseline Superscript beta Baseline right-bracket

Two-forms:

Fijk

Anti-self-dual 3-forms: Curvature tensor:

Qα(X)

are

‘spinor fields’.

ϕαβ

Fijk

~

ψαβ

R

spinor:

spinor:

~

(105 components)

a

(84 components)

R

(⊞symmetry,

Vacuum Bianchi identities: Ricci identities: Define

and

d

Self-dual 3-forms:

Ricci

Pα(X)

F

Three-forms:

Conformal

convention)

εαβγδ(εαβγ�=ε0B410B2)

=

20

components)

s

b

a

Then

.

we

find the

following

relations:

whitesquareSubscriptbetaSuperscriptalphaBaselinexiSubscriptgam aBaseline qualsnormalup erPsiSubscriptbetagam aSuperscriptalphadeltaBaselinetimesxiSubscriptdeltaBaselineplusnormalup erPhiSubscriptbetagam aSuperscriptalphadeltaBaselinetimesxiSubscriptdeltaBaselineplusdeltaSubscriptgam aSuperscriptalphaBaselinetimesnormalup erLamdaxiSubscriptbetaBaseline

whitesquareSubscriptbetaSuperscriptalphaBaseline taSuperscriptdeltaBaseline qualsminusnormalup erPsiSubscriptbetagam aSuperscriptalphadeltaBaselinetimestimesetaSuperscriptgam aBaselineminusnormalup erPhiSubscriptbetagam aSuperscriptalphadeltaBaselinetimestimesetaSuperscriptgam aBaselineminusdeltaSubscriptbetaSuperscriptdeltaBaselinenormalup erLamdaetaSuperscriptalphaBaseline

where normalup erPhiSubscriptbetagam aSuperscriptalphadeltaBaseline qualsnormalup erPhiSubscriptleft-bracketbetagam aright-bracketSuperscriptleft-bracketalphadeltaright-bracketBaseline qualsnormalup erPhiSubscriptleft-bracketbetagam aright-bracketleft-bracketrhosigmaright-bracketBaseline psilonSuperscriptalphadeltarhosigmaBaselineperiod

Note that these formulae

are

actually simpler in form

‘Maxwellian’ equations: Fijk =F[ijk] are

equivalent

□ψβγ

=

to:

0 where □

General solution of

▿αβϕα; =

=

0

and

than their four-dimensional

analogues!

&&##x03C8x;x03B1;&9#x053B;2 Set ▿[iFjkl0 ] and ▿iFijk 0. &3x03D5;αβ 0. In flat ▿αβψβγ 6-space these imply □ϕβγ

~

=

=

=

=

These 0 and

▿i▿i.

▿αβϕβγ

=

0 in flat space:

phi Superscript alpha beta Baseline left-parenthesi up er X Superscript rho sigma Baseline right-parenthesi equals contour-integral up er Z Superscript alpha Baseline up er Z Superscript beta Baseline f left-parenthesi up er X Subscript rho sigma Baseline times up er Z Superscript sigma Baseline comma up er Z Superscript sigma Baseline right-parenthesi cubed up er Z times comma

where D3Z a

suitable

O(8, 𝕔),

εαβγδZα dZβ ∧ dZγ ∧ dZδ and F(Wρ, Zσ) is homogeneous region of the space WαZα = 0. Note that the pair {Wα, Zα] =

i.e. is in effect

Comment

(1994).

I. Robinson Festschrift.

a

of is

degree —6, a

spinor

defined

on

for the group

‘twistor’ for the flat six space.

For further discussion of this formula

(Cf.

also Penrose & Rindler

1986,

see

pp.

my article

462-464.)

(Hughston 1986)

in the

Algebraic classification in four dimensions.

symmetric spinor fields

of

Reality

distinct types. The most

Pαpβ for

some

conditions aside,

a

of these

degenerate

in six dimensions is

field

ϕαβ

(which

at each

can

I shall call

intricate matter than

a more

point

‘null’)

be

of four

one

essentially

ϕαβis of the form

is when

spinor field Pα(X).

1. LEMMA. If ▿αρϕαβ = 0 and

ϕαβ

=

PαPα then Pα(X) satisfies

left-parenthesi up er P Superscript alpha Baseline nabla up er P Superscript left-bracket gamma Baseline right-parenthesi up er P Superscript delta right-bracket Baseline equals 0 times period

(*) 2. REMARK. This condition is

to the

analogous

shear free condition

geodesic

left-parenthesi o Superscript up er A Baseline nabla Subscript up er A prime up er A Baseline o Superscript left-bracket up er B Baseline right-parenthesi o Superscript up er C right-bracket Baseline quals 0

for

a

spinor field

in four dimensions.

3. PROBLEM. such that Solutions of

(*)

spinor

degree,

field

satisfies

▿αβϕαγ

a

can

4. THEOREM. of some

Suppose Pα(X)

be

0 for

=

regions

Pα(X) according

above. Does there

consideration of

Suppose Fr(Wα, Zα), on

as

suitable choice of the scalar

generated by

defined

(*)

r

of the

1, 2, 3 is

=

a

analytic triple

quadric WαZα

=

Pα(X)

satisfies

(Pα▿αβP[r)Pδ]

5. PROBLEM. Does every 6. LEMMA.

spinor

Suppose

a

exist

a

field

ϕαβϕPαPβ =

ϕ(X)?

varieties of

appropriate

of holomorphic

0. Then the

codimension:

functions, homogeneous

variety Fr

=

0 determines

a

to the scheme

Fr(XαβPβ(X),Pα(X)) and

necessarily

=0,

0.

=

analytic Pα(X) satisfying (*)

arise in this

curved six-dimensional space satisfies

Rij

=

way?

0 and has

a

degenerate Weyl

to the extent that normal up er Psi Subscript gam a delta Superscript alpha beta Baseline quals up er P Superscript alpha Baseline up er P Superscript beta Baseline up er Q Subscript gam a delta

(**) for

some

Pα, Qαβ.

Then

(Pα▿αβP[γ)Pδ]

7. PROBLEM. Does the

Weyl spinor necessarily 8. DEFINITION. let

converse

of the form

&

(Robinson

ka(x) (a

=

1...n)

metric of M then for suitable

=

hold,

0.

in the

Ω(x)

that if Pα satisfies

(*)

and

Rij

=

0 is the

(**) ?

Trautman)

be

sense

a

In

a

manifold

vector field which is

we

have gab

=

Ω2ĝab

Mof n

dimensions

conformally geodesic,

such that ka▿akb

=

(signature

unimportant)

i.e. if ĝab is the

0, where ▿a is the

connection associated with gab and indices

Lkak[agb][ckd]

=

ϕk[agb][ckd] for some

9. REMARK. If M is or

space-time

ϕ,

or

raised and lowered with gab. Then ka is

are

equivalently ▿(akd)

+

ξ(akb)

for

then this definition reduces to the ‘standard’

shear-free

if

some 𻳨a,

ones

if ka is null

timelike.

10. LEMMA.

the

Suppose

spinor

fields Aα and Bα each

Kαβ A[α Bβ] is geodesic and shearfree in the =

11. REMARK. To show we

ψgab

=

have

Kαβis geodesic

Aα▿αβAρ = λβAρ and

is

sense

Bα▿αβBρ

=

=

Then the vector field

.

noted above.

straightforward enough:

AαBβ▿αβBρ μBρfor suitable λ, μ.

λAρ and

satisfy (*)

μβBρ for

Since

; ;β,

some

μβ.

Aα and Bα satisfy (*) Thus

AαBβ▿αβAρ

=

Whence

up er A Superscript alpha Baseline up er B Superscript beta Baseline nabla up er A Superscript left-bracket rho Baseline up er B Superscript sigma right-bracket Baseline equals left-parenthesis lamda plus mu right-parenthesis up er A Superscript left-bracket rho Baseline up er B Superscript sigma right-bracket Baseline times period times white square

To show

Bβ] is A[α

shearfree is

more

12. PROBLEM. Show that the

13. REMARK.

conjugation rules, +-----

we

Reality a

i.e.

impose

14. PROBLEM. In

a

all ‘null’ solutions of the

intricate.

converse

conditions: for with

standard

a

to Lemma 10 does not hold.

signature

Hermitian correlation of

conjugation,

equations,

impose

signature

the ‘usual’ twistor

+ +--. For

signature

component by component.

i.e. s

real six-dimensional curved vacuum

we + +----

space-time

of

f

i.e. for which

determine

signature +---.

References

Hughston

,

Festschrift

L. P.

( 1986 ) Applications

volume),

of

SO(8) spinors

,

eds. W. Rindler and A. Trautman

Penrose , R. & Rindler , W.

( 1986 ) Spinors

and

in Gravitation and

Geometry (I.

( Bibliopolis Naples )

pp. 253 287

,

Space-Time Vol. ,

2

,

Robinson

-

.

Cambridge University

Press

.

§II.2.3

A Theorem

Null Fields in Six Dimensions

on

Hby ughston L.P.

(TN 20, September

1985) In what follows I shall outline

regarded By

a

as a

generalization

rather

a

striking result, holding

of Robinson’s theorem

‘massless field’ in six dimensions I

▿δαϕαβ.;

0;

=

LEMMA.

to be a

‘totally

mean

null’ field it must

Suppose ϕαβ. γsatisfies

on

null

in six

electromagnetic fields ϕαβ. [&#xP0x0B3B34;;]

satisfy

0 for

=

these conditions; then Pα must

can

be

in four dimensions. ϕαβ. γ which satisfies

symmetric spinor field

a

which

dimensions,

some

spinor

Pα.

satisfy

left-parenthesi up er P Superscript alpha Baseline nabla up er P Superscript left-bracket gamma Baseline right-parenthesi up er P Superscript delta right-bracket Baseline equals 0 period

(1) Proof.

It will be

easily

seen

that

ϕαβ. γis totally

null iff there exists

a

scalar ψ such that

phi Superscript alpha beta el ipsi gamma Baseline equals e Superscript psi Baseline up er P Superscript alpha Baseline up er P Superscript beta Baseline period period period up er P Superscript gamma Baseline times period times

(2) The

zero

rest mass condition then

and indeed is

implies,

equivalent

to:

up er P Superscript beta Baseline up er P Superscript gamma Baseline times el ipsis up er P Superscript delta Baseline nabla psi plus up er P Superscript gamma el ipsis Baseline times up er P Superscript delta Baseline nabla Subscript alpha beta Baseline up er P Superscript beta Baseline plus left-parenthesis n minus 1 right-parenthesis up er P Superscript beta Baseline left-parenthesis nabla up er P Superscript left-parenthesis gamma Baseline right-parenthesis times el ipsis up er P Superscript delta right-parenthesis Baseline equals 0 comma

(3) where

(1)

n

is the valence of

follows at

THEOREM. Let Pα be

over

γ and

e

the condition □

locally, a

a

result that is

essentially

a converse

holomorphic spinor field satisfying (1)

manifold endowed with

holomorphic with

Pε and skews

(n ⩾2).

once

Now I shall establish,

complex

ϕαβ. γIf one multiplies (3) by

a

non-degenerate holomorphic

connection. Then

locally

there exists

a

to this lemma.

on a

region

of a six-dimensional

metric tensor and

totally

a

Riemann-compaatible

null massless field of valence n,

principal spinor Pα, providing that left-parenthesi n minus 2 right-parenthesi up er P Superscript alpha Baseline up er P Superscript beta Baseline times normal up er Psi Subscript alpha beta Superscript rho left-bracket sigma Baseline times up er P Superscript au right-bracket Baseline quals 0 com a

(4) where

s

is the

Proof. (1)

is

Weyl spinor (conformal

equivalent

curvature

to the existence of a

spinor).

spinor ∧α such

that

up er P Superscript alpha Baseline nabla up er P Superscript gamma Baseline equals normal up er Lamda Subscript beta Baseline up er P Superscript gamma Baseline times comma

(5) whence up er P Superscript beta Baseline nabla Subscript alpha beta Baseline psi plus nabla up er P Superscript beta Baseline minus left-parenthesis n minus 1 right-parenthesis normal up er Lamda Subscript alpha Baseline equals 0

(6)

as follows from (3). Now consider an equation of the form upper P Superscript beta Baseline nabla Subscript alpha beta Baseline psi plus upper A Subscript alpha Baseline equals 0

(7) with

ψ unknown, Aα specified,

and

Pβsatisfying (1)

Such

.

an

equation admits solutions, locally,

the Frobenius theorem, iff Aα satisfies

by

up erPSuperscriptalphaBaselinenablaSubscriptalphaleft-bracketbetaBaselineup erASubscriptgam aright-bracketBaseline qualsnormalup erLamdaSubscriptleft-bracketbetaBaselineup erASubscriptgam aright-bracketBaselineperiod

(8) To

see

use

of

the necessity of

(5)

(8)

operate

(7)

on

with

P� 3C1;▿σρap,

and skew

over

α and σ; (8)

then follows

by

.

We wish to

whether there exists

see

a

scalar

ψ such

(6) holds;

that

thus

we

examine the

expression

up er P Superscript alpha Baseline nabla Subscript alpha left-bracket beta Baseline up er A Subscript gam a right-bracket Baseline minus normal up er Lamda Subscript left-bracket beta Baseline up er A Subscript gam a right-bracket Baseline qual-col n up er I Subscript beta gam a

(9) with up er A Subscript alpha Baseline equals nabla up er P Superscript beta Baseline minus left-parenthesis n minus 1 right-parenthesis normal up er Lamda Subscript alpha Baseline times period

(10) A

straightforward

calculation

gives up er I Subscript beta gamma Baseline equals minus left-parenthesi n minus 2 right-parenthesi up er P Superscript alpha Baseline nabla Subscript alpha left-bracket beta Baseline normal up er Lamda Subscript gamma right-bracket Baseline times period

(11) To arrive at

(11)

use

is made of the Ricci

identity

(12) where R is the scalar curvature; furthermore

we

require

the

simple identity

left-parenthesi nablaSubscriptdeltaleft-bracketbetaBaselineup erPSuperscriptalphaBaselineright-parenthesi timesleft-parenthesi nablaSubscriptgam aright-bracketalphaBaselineup erPSuperscriptdeltaBaselineright-parenthesi equals0period

(13) Now

we

skewing

wish to examine the over

ηand β.

expression appearing

A short calculation

in

(11) Suppose .

we

operate

on

(5)

with Pξ▿ξη

gives

(14) but the

vanishing

of the left side of this

equation is, by

another Ricci

identity, equivalent

to

up er P Superscript alpha Baseline up er P Superscript beta Baseline normal up er Psi Subscript alpha beta Superscript rho left-bracket sigma Baseline up er P Superscript au right-bracket Baseline quals 0 period

(15) Therefore the vanishing of Iβα, the desired integrability condition, is equivalent Note that for no

restrictions

n

are

=

2, the

imposed

case on

corresponding

the curvature

to

(4)

□

.

to the classical Robinson theorem in dimension

beyond

those

already implied by (1) ;

these

four,

conditions,

incidentally,

are

p

as

Pε over γ and ε. In flat these may be

given

space,

generated

via

appropriately simple pole

a

solution of

contour

a

follows from

integral

(1)

,

(14) directly by skew-symmetrization

null fields of any valence

formula with

a

holomorphic

can

with

be constructed:

function

showing

an

structure.

Gratitude is expressed

to Lionel Mason and Ben

Jeffryes,

both of whom in discussion and correspondence

made contributions to these results.

(1994).

Comment

For

a

generalization

of these results to

higher

dimensions

see

Hughston,

L.P.

k. Mason, L.J. (1988) A generalized Kerr-Robinson theorem, Class. Quant. Grav. 5, 275-285.

§II.2.4 A

Six Dimensional

One of the in the

areas

algebraic

approaches).

'Penrose

in which the

use

of

classification of the

Rather than

looking

Jefryes diagram' by

(TN

B.P.

21, February 1986)

spinors greatly simplifies four-dimensional general relativity Weyl

for

tensor

(see

eigenspinors

Q

is

Penrose & Rindler 1986 for details of various and

eigenvalues λ of the Weyl spinor ΨABCD

such that phiSubscriptup erCSuperscriptup erABaselinetimesnormalup erPsiSubscriptup erAup erBSuperscriptup erCup erDBaseline quals amdaphiSubscriptup erBSuperscriptup erD

(1) (classification space

then

being

with

regard

spanned by the eigenspinors),

(pnd’s)

is used. oA is

a

pnd

of

a

to the

multiplicity

classification

of the

and the dimension of the

by the multiplicity of the principal

oAoBoCoD ΨABCD

ΨABCD

eigenvalues

0,

=

or

written

null directions

alternatively

oSuperscriptup erABaselineoSuperscriptup erB aselineoSuperscriptleft-bracketup erCBaselinenormalup erPsiSubscriptup erAtimesup erBSuperscriptup erDright-bracketleft-bracketup erEBaselineoSuperscriptup erFright-bracketBaseline quals0semicol n

(2) oA is

a

double pnd if oSuperscriptup erABaselineoSuperscriptup erB aselineoSuperscriptleft-bracketup erCBaselinenormalup erPsiSubscriptup erAup erBSuperscriptup erDright-bracketup erEBaseline quals0

(3) and

so on.

The

reason

for the curious

case; here of course

position of the

anti-symmetrisation

indices is for easier

with

an

upstairs

comparison

index is

with the six-dimensional

equivalent

to contraction with a

downstairs index. We

might hope

Hughston’s

for

a

similar

article II.2.2 for the

simplification

in

spinor notation),

studying

curved six-dimensional spaces

within which the

Weyl tensor

is

(see

L.P.

represented by

the

totally trace-free Weyl spinor to

classify

by

the

The

than in four

multiplicity

use

of the

d which has 84

dimensions;

of its

as an

analogous

example

This

eigenvalues.

components! Everything consider the

gives ({partitions

of

use

is much

of equation

2} +1

—

3)

(1)

more

to

different

complicated

classify ΨABCD

algebraic

classes.

scheme in six dimensions phiSubscriptgam aSuperscriptalphaBaselinetimesnormalup erPsiSubscriptalphabetaSuperscriptgam adeltaBaseline quals amdaphiSubscriptbetaSuperscriptdelta

(4) leads to

({partitions

directions

By

way of

analogy the

on

does not of

imply

+ 1

=

with

different classes.

136)

Weyl spinord

and

Luckily

a

concept similar

Pαs with the

a

(3)

wish to consider

we

number of copies of a

to

principal

null

indices ofs

upstairs

then there will be

s

and Pα. Unlike in four dimensions the

operations

of

or

that

possible

contracting

some

algebraic operation

an

spinor Pα such

Pα = 0. It is clear that all that is

or

Should

d

and

equations (2)

that either i

anti-symmetrising

indices of

14}

still be used.

can

O

of

is

combination

Pαs with the downstairs

algebraic relationship

anti-symmetrisation

g

some

between d

and contraction

are

not

equivalent. Now for the

diagram. Given

d and Pα we abbreviate up er P Superscript left-bracket epsilon Baseline normal up er Psi Subscript gam a delta Superscript alpha right-bracket beta Baseline up er P Superscript delta Baseline quals 0 times a s times normal up er Psi Subscript bulet star Superscript right-bracket bulet

and normal up er Psi Subscript gamma delta Superscript alpha beta Baseline equals 0 times a s times normal up er Psi Subscript bul et bul et Superscript bul et bul et

and

so on.

Then

(5)

The

diagonal relationships

are

obvious;

involves

This classification scheme

just

carried out inverted with

downstairs

a

an

the vertical

ones

arise because t is

upstairs spinor Pα; clearly

spinor instead.

The

the whole

corresponding

totally

trace-free.

procedure

number of freely

can

be

specifiable

components for each class is

(6)

Reality

conditions may restrict the

then a

is

impossible. Unlike

algebraic types possible,

in four dimensions

a

for instance if the space is Riemannian

generic Weyl spinor

will not have

pnd’s.

For

a

a in

given

a

chosen basis the condition

four components of Pa. The question arises “What out to be connected to the existence of

is six

’JiJJ

use

special spinor

homogeneous quartic equations

on

the

is all this!”. This classification scheme turns

fields. If Pa satisfies

left-parenthesi up erPSuperscriptalphaBaselinetimesnablaSubscriptalphabetaBaselineup erPSuperscriptleft-bracketgam aBaselineright-parenthesi timesup erPSuperscriptdeltaright-bracketBaseline quals0

(7) then the

Weyl spinor

this

equation).

of

Pα such that

a

is of type

If the space is

a with

respect

to Paα (see II.2.2 and II.2.3 for the

Kähler manifold

a

it

(not necessarily vacuum)

of

significance

implies

the existence

up er P Superscript left-bracket gamma Baseline nabla up er P Superscript delta right-bracket Baseline equals 0 times

(8) forcing .

Weyl spinor

the

to be of

is a If in addition there

fashionable Calabi-Yau

a

type awith respect

constant

spaces),

holomorphic

to Pα. If the space is vacuum it is of

3-form

there would be in the

(as

Weyl spinor

this PB1; is constant and the

case

of

type

currently

is of typea .

References

Hughston

,

L. P. ,

§11.2.2

and

§11.2.3.

Penrose , R. & Rindler , W.

(1986 ) Spinors and Space-Time, Space-Time Geometry Cambridge University Press

§II.2.5

vol. II:

Spinor

and Twistor Methods in

.

,

Null Surfaces in Six and

Eight

(TN 22, September 1986)

Dimensions Hby ughston L.P.

1. This note is concerned with the construction of null surfaces of dimension of dimension 2n. For the construction of null 2-surfaces in four dimensions ‘Kerr theorem’

(see

Penrose 1967,

general analytic spinor field ξA

§8;

Penrose & Rindler 1986,

which satisfies ξAξBBF;A'A£ψB

essentially arbitrarily specified analytic of this construction exist in six and In dimension six the

ξαV▿αβξγ

=

Λβξγ for

‘twistor

quadric).

surface in

eight

some

Λβ is

space’ (which in

given this

in terms of

case

projective

dimensions

general analytic spinor

=

as

field an

§7.4)

n

we

in

complex flat

which shows how to

0. The solution is twistor space.

given

specify

the

in terms of

an

Remarkably, analogues

well.

ξα(Xi) (α

1...4i

=

=

analytic variety of dimension

is the space of

space

have the well-known

‘pure’ spinors

for

1...6) satisfying three in the associated

SO(8)—a six-dimensional

In dimension

ξαξβΩαβ= 0 is

and

ξαΓiαα1iξβ BF; Λα1ξβ =

the associated twistor

(of

general analytic spinor

the

eight

a

given helicity)

The pattern of

=

question

1

...

8, i

=

1

...

8) satisfying

variety of dimension four in is the space of pure

spinors

‘purity’

conditions

on

the relevant

essential way.

We wish to solve the

2. Dimension Six.

α

arbitrary analytic

the twistor space in

case

similar in each case, and involves the

proof is an

given by

an

ξα(Xi) (here

SO(10).

for the group

and twistors in

spinors

space—in

this

field

equation

left-parenthesi xiSuperscriptalphaBaselinenablaSubscriptalphabetaBaselinexiSuperscriptleft-bracketgam aBaselineright-parenthesi timesxiSuperscriptdeltaright-bracketBaseline quals0com a

(1) where

α

=

1...4. Here %VF;αβ

equation (1)

is that it is the

arbitrary,

ηβ], ηβξ[α

=

A twistor for

ᵔ ;6

integrability

be

can

(1) corresponds (locally)

an

Let

to

ZαWα

pair

a

analytic variety =

coordinatises 1D5546. The

—Xβα

0

(r

0, and suppose

=

a

significance

of

of null vector fields of the form

family

commutation, hence providing for

a

family

of null 3-surfaces.

Zα, Wα satisfying ZαWα = 0: the

projective

in 𝕄1D561;7. It will be shown that each solution of

projective quadric Q

Fr(Zα ,Wα) =

=

condition for the

represented by a

projective quadric

where Xαβ

to be closed under

twistor space for ᵔ ;C6 is thus

THEOREM 1.

∂/∂Xαβ,

of dimension 3 in

1, 2, 3) be

an

Q

.

analytic variety

spinor held ξα(Xi)

defined

of dimension 3 in the

on an

open

region

U of

𝕄6

satisfies up er F Superscript r Baseline left-parenthesi xi Superscript alpha Baseline comma up er X Subscript alpha beta Baseline times xi Superscript beta Baseline right-parenthesi equals 0

(2) for each value of Xi

⊂ U. Then ξα(Xi) satisfies

Proof. Since Fr(Zα, Wα) we

have

whence

is

(ZαẐα + WαŴα)Fr

ξα(Ẑα

—

XaαβŴα)Fr

—

(1)

.

by hypothesis homogeneous nrFr

—

0

by

=

0

on

the

of

degree (say)

variety (where Ẑα

the substitution

(Zα, Wα)

=

=

nr in

Zα and Wα jointly

∂/∂Zα,Ŵα

=

∂/∂Wα);

(ξα, Xαβξβ). Writing

up erRSubscriptalphaSuperscriptrBaselinecol n-equal eft-parenthesi ModifyngAboveup erZWithcaretSubscriptalphaBaselineminusup erXSubscriptalphabetaBaselinetimesModifyngAboveup erWWithcaretSuperscriptalphaBaselineright-parenthesi up erFSuperscriptr

(3) we

have xi Superscript alpha Baseline upper R Subscript alpha Superscript r Baseline equals 0 comma times left-parenthesis r equals 1 comma 2 comma 3 right-parenthesis times period

(4) Furthermore, by

differentiation of

(2)

we

get

left-parenthesi nablaSubscriptrhosigmaBaselinexiSuperscriptalphaBaselineright-parenthesi timesModifyngAboveup erZWithcaretSubscriptalphaBaselineup erFSuperscriptrBaselineSuperscriptBaselineplusleft-parenthesi nablaSubscriptrhosigmaBaselineup erXSubscriptalphabetaBaselinetimesxiSuperscriptbetaBaselineright-parenthesi timesModifyngAboveup erWWithcaretSuperscriptalphaBaselineup erFSuperscriptrBaseline quals0times emicol n

whence left-parenthesi nablaSubscriptrhosigmaBaselinexiSuperscriptalphaBaselineright-parenthesi timesModifyingAboveup erZWithcaretSubscriptalphaBaselineup erFSuperscriptrBaselineplusepsilonSubscriptrhosigma lphabetaBaselinetimesxiSuperscriptbetaBaselineModifyingAboveup erWWithcaretSuperscriptalphaBaselineup erFSuperscriptrBaselineplusleft-parenthesi nablaSubscriptrhotimes igmaBaselinexiSuperscriptbetaBaselineright-parenthesi timesup erXSubscriptalphabetaBaselinetimesModifyingAboveup erWWithcaretSuperscriptalphaBaselineup erFSuperscriptrBaseline quals0com a

so left-parenthesi nabla Subscript rho sigma Baseline xi Superscript alpha Baseline right-parenthesi times up er R Subscript alpha Superscript r Baseline plus epsilon Subscript rho sigma lpha beta Baseline times xi Superscript beta Baseline ModifyingAbove up er W With caret Superscript alpha Baseline up er F Superscript r Baseline quals 0 semicol n

whence

V▿ρσFr(ξα, Xαβξβ)

=

0,

and

by

transvection of this relation with ξρ we get: left-parenthesis xi Superscript rho Baseline nabla Subscript rho sigma Baseline xi Superscript alpha Baseline right-parenthesis times up er R Subscript alpha Superscript r Baseline equals 0 comma left-parenthesis r equals 1 comma 2 comma 3 right-parenthesis period

(5) Now since R

(r

R

),

3. Dimension

In this

to

case

1...8, i

—

1

..

1,2,3)

—

linearly independent

are

it follows from Here

Eight. generate

(4)

we use

(5)

and

.8) satisfying

ξαξβΩαβ

space in that dimension—thus

0

=

(where

is taken to be

ξα

some

of the

variety (i.e.

λσ.

□

the notation set out in my article for the I. Robinson Festschrift.

family of null four-surfaces in 𝕄8

a

generic points

vectors at

that ξρ▿ρσξα = λσξαfor

we

require

‘pure’ spinor field),

a

spinor

a

field ξα(Xi)

Ωαβ is the natural ‘metric’ induced

(α

=

the spin

on

and

xi Superscript alpha Baseline nabla Subscript alpha lpha times prime Baseline xi Superscript beta Baseline quals normal up er Lamda Subscript alpha times prime Baseline xi Superscript beta Baseline period

(6) where

▿αα1

=

(▿i

Λiαα' ▿i

∂/∂Xi).

=

One

can

that

verify

(6) together

withξαξα = 0

and sufficient conditions for all vector fields of the form V and closed under commutation with A ‘twistor’ for dimension W

satisfying

‘purity’

relations for the

Hughston §1.2.8

of dimension ten, which In what follows a

set of

equations

homogeneous a

of

suitable set of

we

an

degree jointly

with

ξα(Xi)

(α

=

value of Xi ⊂ U. Then

(r

projective

(Cf.

These conditions

Cartan 1937,

pure twistors is

of dimension four in

1...6. For each value of

a

0 and

are

the

Petrack 1982,

complex manifold

S10, given locally by

r we

require

Thus F

.

that Fr be for

=

1..

.6)

define

an

analytic variety

1..

), defined

.8)

ξαsatisfies

,

=

Z

Proof. By homogeneity a

W

=

nr.

(given by

1...8, i

=

r

in Zα and W

THEOREM 2. Let F in the space S10

by Zα,

.

with ZαZβΛαβ

) .

analytic variety V4 ,

integers

arbitrary)

necessary to be null

shall denote S10.

F

some

defined

The space of such

§1.3.9).

consider

we

spinors ( Z

the incidence relation T

SO(10) spinor

and Petrack

of pure

pair

a

are

another.

one

is

eight

(

a

(1)

have

we

,

on a

i.e.

region

U of ᵔ ;8 satisfies F

(ξα▽α 'ξ[β)ξγ]

=

of dimension four

and suppose

a

spinor field for each

0.

Z

where V4, on Z

Z

; whence xi Superscript alpha Baseline up er R Subscript alpha Superscript r Baseline equals 0

(7) where R of F

,

we

get:

nabla up er F Superscript r Baseline left-parenthesis xi Superscript beta Baseline comma up er X Superscript beta prime beta Baseline xi Subscript beta Baseline right-parenthesis equals 0 comma

left-parenthesi nabla xi Superscript beta Baseline right-parenthesi times ModifyingAbove up er Z With caret Subscript beta Baseline up er F Superscript r Baseline plus nabla left-parenthesi up er X Superscript beta prime beta Baseline xi Subscript beta Baseline right-parenthesi times ModifyingAbove up er W With caret Subscript beta prime Baseline up er F Superscript r Baseline equals 0 comma

left-parenthesi nablaxiSuperscriptbetaBaselineright-parenthesi timesModifyingAboveup erZWithcaretSubscriptbetaBaselineup erFSuperscriptrBaselineplusnormalup erGam aSubscriptalpha lphatimesprimeiBaselinetimesnormalup erGam aSubscriptbetabetaprimeSuperscriptiBaselinexiSuperscriptbetaBaselineModifyingAboveup erWWithcaretSuperscriptbetatimesprimeBaselineup erFSuperscriptrBaselineplusleft-parenthesi nablaxiSubscriptbetaBaselineright-parenthesi up erXSuperscriptbetaprimebetaBaselinetimesModifyingAboveup erWWithcaretSubscriptbetatimesprimeBaselineup erFSuperscriptrBaseline quals0period

(r

=

1..

.6)

with Xαα'

=

XiΓiαα'. Furthermore by differentiation

Thus

V

,

and therefore

left-parenthesi xiSuperscriptalphaBaselinenablaSubscriptalpha lphatimesprimeBaselinexiSuperscriptbetaBaselineright-parenthesi up erRSubscriptbetaSuperscriptrBaselineplusleft-parenthesi xiSuperscriptalphaBaselinexiSuperscriptbetaBaselinenormalup erGam aSubscriptalpha lphatimesprimeiBaselinetimesnormalup erGam aSubscriptbetabetaprimeSuperscriptiBaselineright-parenthesi ModifyngAboveup erWWithcaretSuperscriptbetaprimeBaselineup erFSuperscriptrBaseline quals0period

But in

eight dimensions

there is the remarkable

identity

=

Q

; thus

xiSuperscriptalphaBaselinexiSuperscriptbetaBaselinenormalup erGam aSubscriptalpha lphatimesprimeiBaselinetimesnormalup erGam aSubscriptbetabetaprimeSuperscriptiBaseline qualsleft-parenthesi xiSuperscriptalphaBaselinexiSuperscriptbetaBaselinenormalup erOmegaSubscriptalphabetaBaselinetimesright-parenthesi normalup erOmegaSubscriptalphatimesprimebetaprimeBaseline quals0period

Therefore left-parenthesi xi Superscript alpha Baseline nabla xi Superscript beta Baseline right-parenthesi times up er R Subscript beta Superscript r Baseline equals 0 period

(8) Thus ξα and ξα▽α '

are

each

orthogonal by equations (7)

vectors, i.e. the vectors R and

are

But

.

therefore each orthogonal to

seven

they

also each

are

independent

Comment work. See also

(1994).

equation (1)

to six

orthogonal are

in

briefly

(1988)

for the

linearly independent to the vector

therefore

§11.2.2

to ‘null’ fields in six dimensions

See Hughston k. Mason

Hughston (1979)

(8)

vectors; and

The result outlined here for six dimensions is mentioned discussion of the relation of

and

see

Ωβγξ;

proportional.

without

proof.

For

□ a

§11.2.3.

higher-dimensional analogues

of this

page 146.

References

Cartan E. ( 1937 ) The Theory of Spinors ,

Hughston

,

Hughston

,

L. P.

,

Twistors and Particles ,

L. P.

( 1986 ) Applications

volume),

Hughston

L. P. & Mason , L. J.

,

Springer

Lecture Notes

on

Equation

Physics

and

97

.

Spinors

in

Higher Dimensio,ns

.

Festschrift

275 285

Dover 1966 ).

L. P. A Remarkable Connection between the Wave

§1.2.8 Hughston

( 1979 )

(reprinted by

of

SO(8) spinors

,

eds. W. Rindler and A. Trautman

( 1988 ) A generalized

in Gravitation and

Geometry (I.

( Bibliopolis Naples )

pp. 253 287

Robinson

-

,

Kerr-Robinson theorem , Class.

.

Quant.

Grav. 5 ,

-

.

Penrose , R.

( 1967 ) Twistor Algebra

,

Penrose , R. & Rindler , W.

( 1986 )

Petrack , S. B. An Inductive

Approach

J. Math.

Phys.

8 , pp. 345 366 -

and

Spinors Space-Time, Space-Time Geometry (Cambridge University Press ). Petrack , S. B.

( 1982 ) Spinors (Oxford University ).

and

to

Higher

vol. II:

Dimensional

Complex Geometry

in

.

Spinor

and Twistor Methods in

Spinors §1.3.9

Arbitrary

,

.

Dimensions ,

Qualifying

Thesis

§II.2.6

A Proof of Robinson's Theorem

In 1976 Paul Sommers

published

an

Hby ughston L.P.

a new

proof

aspects of his argument and of the development of the

some

& Rindler

(1986), §7.3.

THEOREM

The methods

a

of

are

(Robinson). Suppose M

metric gab. Let kA be

is

some

that

same

interest in their

tightens

material

own

1985)

on an

open set U ⊂

up and

as

on

shear-free

improves

on

outlined in Penrose

right.

complex manifold of dimension

a

field defined

spinor

20, September

elegant simplified proof of Robinson’s theorem (1961)

In what follows I shall outline

congruences.

(TN

four with

a

holomorphic

Msatisfying

kap a Superscript up er A Baseline kap a Superscript up er B Baseline nabla kap a Subscript up er B Baseline equals 0 period

(1) Then for each

p ∈ U there exists

point

V with V

neighbourhood

a

V ⊂ U such that there exists

a

scalar ψ

on

where

,

phi Superscript up er A up er B Baseline equals e Superscript psi Baseline times kap a Superscript up er A Baseline kap a Superscript up er B Baseline period

(2) Note that

Proof.

𥯺'AϕAB

=

0 is

equivalent, by (2)

,

to

kap a Superscript up er B Baseline times kap a Superscript up er A Baseline times nabla Subscript up er A prime up er A Baseline psi plus kap a Superscript up er B Baseline nabla kap a Superscript up er A Baseline plus kap a Superscript up er A Baseline nabla kap a Superscript up er B Baseline equals 0 period

(3) Since

(1)

is

to the existence of a

equivalent

spinor λA,

such that

kap a Superscript up er A Baseline nabla Subscript up er A up er A prime Baseline kap a Superscript up er B times Baseline quals lamda Subscript up er A prime Baseline kap a Superscript up er B times

(4) it follows

by

insertion of

(4)

in

(3)

that

we

seek

a

scalar

ψ such

that

kappa Superscript upper A Baseline nabla Subscript upper A upper A prime Baseline psi plus nabla kappa Superscript upper A Baseline plus lamda Subscript upper A prime Baseline equals 0 period

(5) Now consider

an

equation

of the form kap a Superscript up er A Baseline nabla Subscript up er A up er A prime Baseline psi plus alpha Subscript up er A prime Baseline equals 0 comma

(6) where

kA satisfies

solutions of

(4)

(6) locally

As

.

a

if and

lemma

only

we

require the

fact that if αA, is

specified

then there exist

if αA, satisfies kap a Superscript up er A Baseline nabla Subscript up er A up er A prime Baseline alpha Superscript up er A prime Baseline quals lamda Subscript up er A prime Baseline alpha Superscript up er A prime Baseline period

(7) The

proof

of this lemma follows

as a

necessary condition

we

as an

transvect

application of the (6)

kap a Superscript up er B Baseline nabla Subscript up er B Superscript up er A prime Baseline left-parenthesi kap a Superscript up er A Baseline nabla Subscript up er A up er A prime Baseline psi rght-parenthesi plus kap a Superscript up er B Baseline nabla Subscript up er B Superscript up er A prime Baseline alpha Subscript up er A prime Baseline quals 0 semicol n

with

K

Frobenius theorem.

to obtain

(To

see

how

(7)

arises

whence, left-parenthesi kap a Superscript up er B Baseline nabla Subscript up er B Superscript up er A prime Baseline kap a Superscript up er A Baseline right-parenthesi nabla Subscript up er A up er A prime Baseline psi plus kap a Superscript up er B Baseline kap a Superscript up er A Baseline nabla Subscript up er B Superscript up er A prime Baseline nabla psi equals kap a Superscript up er B Baseline nabla Subscript up er B up er A prime Baseline alpha Superscript up er A prime Baseline com a

which

by (4) gives

shows that

(7)

A

,

is also

by

use

of

(6) gives (7)

.

Frobenius’ theorem

sufficient.)

We wish to determine whether there exists examine the

which

a

scalar

I

expression

with

ψ such

that

is satisfied. Thus

(5)

a

.

we

must

We have:

uperI qualskpaSupersciptuperABaselin abl SubscriptuperAuperApimeBaselin eft-parenthsi nabl kap SupersciptuperB aselin plusamd SupersciptuperApimeBaselin rght-parenthsi mnuslamd SubscriptuperApimeBaselin eft-parenthsi nabl kap SupersciptuperABaselin plusamd SupersciptuperApimeBaselin rght-parenthsi equalskpaSupersciptuperABaselin abl SubscriptuperAuperApimeBaselin abl kap SupersciptuperB aselin pluseft-parenthsi kap SupersciptuperABaselin abl SubscriptuperAuperApimeBaselin amd SupersciptuperApimeBaselin plusamd SupersciptuperApimeBaselin abl SubscriptuperApimeuperABaselin kap SupersciptuperABaselin rght-parenthsi equalsminuskap SupersciptuperABaselin abl SubscriptuperBuperApimeBaselin abl kap SupersciptuperB aselin plus2kap SupersciptuperABaselin abl SubscriptuperApimeBaselin abl kap SupersciptuperB aselin pluseft-parenthsi nabl kap SupersciptuperABaselin amd SupersciptuperApimeBaselin rght-parenthsi equalsminusleft-bracketnabl eft-parenthsi kap SupersciptuperABaselin abl SubscriptuperASupersciptuperApimeBaselin kap SupersciptuperB aselin rght-parenthsi mnusleft-parenthsi nabl kap SupersciptuperABaselin rght-parenthsi left-parenthsi nabl kap SupersciptuperB aselin rght-parenthsi rght-bracketplus2kap SupersciptuperABaselin whitesquarekap SupersciptuperB aselin plusnabl kap SupersciptuperABaselin amd SupersciptuperApimeBaselin period

But kA

A1;ABkB

vanishes for any kA since □ AB kB

=

-3▿kA. Furthermore

we

have

left-parenthesi nabla Subscript up er A prime up er B Baseline kap a Superscript up er A Baseline right-parenthesi left-parenthesi nabla Subscript up er A Superscript up er A prime Baseline kap a Superscript up er B Baseline right-parenthesi equals 0

for any kA. Thus: up erItimesequalsminusnabla eft-parenthesi kap aSuperscriptup erABaselinenablaSubscriptup erASuperscriptup erAprimeBaselinekap aSuperscriptup erB aselineright-parenthesi plusnablakap aSuperscriptup erABaselinelamdaSuperscriptup erAprimeBaseline qualsminusnabla eft-parenthesi lamdaSuperscriptup erAprimeBaselinekap aSuperscriptup erB aselineright-parenthesi plusnablakap aSuperscriptup erABaselinelamdaSuperscriptup erAprimeBaseline quals0period

Since I vanishes the THEOREM as

above,

condition for ψ is satisfied, and the theorem is

integrability

(Sommers-Bell-Szekeres generalization (1)

kA satisfies

.

Furthermore let

kA

be

of Robinson’s

proved.□

result). Suppose,

ap-fold principal spinor (p ⩾ 1)

in the

venue

of a massless field

of valence p + q. Then:

ABCDkAkBkC 0, = (p-2-3q)ψ where

ψ

The The

ABCD is

proof

the

Weyl spinor.

follows

Goldberg-Sachs

essentially

the

same

theorem follows if

principal spinor of the Weyl spinor,

we

line of

reasoning

note that a

and that in

in the first theorem

as

spinor satisfying (1)

a vacuum

the

is

(see 11.2.7).

automatically

a

1-fold

Weyl spinor

satisfies the zero-rest-mass

to formulate a

purely ‘covariant’ spinorial

equations. Comment

me

The idea that it should be

to Robinson’s theorem and the

approach and

(1994).

in conversation in the

early

possible

Goldberg-Sachs

1970’s

Martin

by

theorem

Walker,

was

suggested

who in turn attributed the idea to

Robert Geroch.

References Robinson I. ( 1961 ) Null electromagnetic fields J. Math. Phys. 2 p. 290 291 -

,

,

to Paul Sommers

,

.

Sommers , P. D.

of shear-free congruences of null

( 1976 ) Properties

A349 , p. 309 318

geodesics

,

Proc. Roy. Soc. Lond. ,

-

§II.2.7

A

.

Simplified

Proof of

a

Theorem of Sommers

Hby ughston L.P.

(TN

22, September

1986) 1. Introduction. In 1976

published by

THEOREM 1. If

(p ⩾ 1)

of

a

Weyl spinor,

interesting

an

theorem

P. D. Sommers. His main result is a

spinor

on

zero-rest-mass fields in curved

zero-rest-mass field of valence p + q, then it is also unless p

=

the

space-time

was

follows:

field ξA which satisfies ξAξB^▿A'AξB a

=

0 is

a

p-fold principal spinor

repeated princpal spinor

of the

3q + 2.

The purpose of this note is to present

respects

on

as

original argument

devised

condensed

a

by

of this theorem,

Sommers. In what follows I

the notation and conventions of Penrose k. Rindler 2. The SFR Condition.

proof

(1986) (cf.

The shear-free ray condition

on

in

improving

employ

as

far

as

in

some

possible

particular §7.3).

ξA, given by

xi Superscript upper A Baseline xi Superscript upper B Baseline nabla xi Subscript upper B Baseline equals 0

(2.1) be

can

expressed alternatively

in either of the forms nablaSubscriptleft-parenthesi up erABaselinexiBaselineSubscriptup erBright-parenthesi Baseline qualsnormalup erLamdaSubscriptup erAprimeBaselineSubscriptleft-parenthesi up erABaselinexiBaselineSubscriptup erBright-parenthesi

(2.2) or xi Superscript up er A Baseline nabla Subscript up er A up er A prime Baseline xi Superscript up er B Baseline equals eta Subscript up er A prime Baseline xi Superscript up er B times Baseline period

(2.3) It is

helpful

to be able to use both of these

shall find it useful to have at LEMMA 1.

our

▿A,AξA + ΛA,AξA

disposal =

a

expressions

in

relationship

between ηA, and

2ηA, (where ξA

Proof. nabl Subscriptu erApimeuperABaselin xSubscriptu erB aselin equalsnblaSubscriptlef-parenthsi uperABaselin xBaselin Subscriptu erBight-parenthsi Baselin plusnabl Subscriptlef-bracketuperABaselin xBaselin Subscriptu erBight-bracketBaselin equalsnormaluperLamd Subscriptu erApimeBaselin Subscriptlef-parenthsi uperABaselin xBaselin Subscriptu erBight-parenthsi Baselin plusone-halfepsilonSubscriptu erAuperB aselin abl xiSupersciptu erCBaselin equalsnormaluperLamd Subscriptu erApimeuperABaselin tmesxiSubscriptu erB aselin minus ormaluperLamd Subscriptu erApimeBaselin Subscriptlef-bracketuperABaselin xBaselin Subscriptu erBight-bracketBaselin plusone-halfepsilonSubscriptu erAuperB aselin abl xiSupersciptu erCBaselin equalsnormaluperLamd Subscriptu erApimeuperABaselin tmesxiSubscriptu erB aselin plusone-halfepsilonSubscriptu erAuperB aselin tmeslft-parenthsi nabl Subscriptu erApimeuperCBaselin xSupersciptu erCBaselin minus abl xiSupersciptu erCBaselin rght-parenthsi period

∈ 0).

computations,

and

as a

ΛA,A:

consequence

we

Contraction of each side with 3.

ξA,

followed

by

use

of

(2.3)

,

then

It will be useful to have another lemma at

Principality.

our

yields

the desired result.

disposal

which relates

ξA

□ to the

Weyl spinor: LEMMA 2. A of the the

spinor

Weyl spinor.

Weyl spinor is Proof.

Held

ξA

which satisfies the SFR condition is

A necessary and sufficient condition for ξA to be V

necessarily a

a

principal spinor

repeated principal spinor

of

.

We have a

; whence

xiSupersciptuperB aselin abl SubscriptuperBuperBpimeBaselin eft-parenthsi xSupersciptuperCBaselin abl SubscriptuperCSupersciptuperBpimeBaselin xiSubscriptuperABaselin rght-parenthsi equalsxiSupersciptuperB aselin abl SubscriptuperBuperBpimeBaselin eft-parenthsi etaSupersciptuperBpimeBaselin xiSubscriptuperABaselin rght-parenthsi com aleft-parenthsi xSupersciptuperB aselin abl SubscriptuperBuperBpimeBaselin xiSupersciptuperCBaselin rght-parenthsi nabl SubscriptuperCSupersciptuperBpimeBaselin xiSubscriptuperABaselin plusxiSupersciptuperCBaselin xiSupersciptuperB aselin abl SubscriptuperBuperBpimeBaselin abl xiSubscriptuperABaselin equalseft-parenthsi xSupersciptuperB aselin abl SubscriptuperBuperBpimeBaselin etaSupersciptuperBpimeBaselin rght-parenthsi xSubscriptuperABaselin plusetaSupersciptuperBpimeBaselin eft-parenthsi xSupersciptuperB aselin abl SubscriptuperBuperBpimeBaselin xiSubscriptuperABaselin rght-parenthsi com aleft-parenthsi etaSubscriptuperBpimeBaselin tmesxiSupersciptuperCBaselin rght-parenthsi tmesnabl xiSubscriptuperABaselin plusxiSupersciptuperB aselin xiSupersciptuperCBaselin whitesquarexiSubscriptuperABaselin equalseft-parenthsi xSupersciptuperB aselin abl SubscriptuperBuperBpimeBaselin etaSupersciptuperBpimeBaselin rght-parenthsi xSubscriptuperABaselin plusetaSupersciptuperBpimeBaselin eft-parenthsi etaSubscriptuperBpimeBaselin xiSubscriptuperABaselin rght-parenthsi emicoln

whence

from which it follows at

a

ψABCDξAξBξCξD 4.

0, and that a

Proof of Theorem 1.

field

for

=

ϕA. .E.

some

If

we

scalar ψ.

transvect

Let

□

ξA

ϕA. .E

that

.

be

a

with q

Therefore, given

V

xi Superscript up er A Baseline period period period times xi Superscript up er B Baseline nabla phi Subscript up er A period period period times up er B up er C up er D period period period times times up er E Baseline equals 0 comma

nabla left-parenthesi xi Superscript up er A Baseline period period period times xi Superscript up er B Baseline phi Subscript up er A times period period period times up er B times up er C up er D times period period period times up er E Baseline right-parenthesi minus phi Subscript up er A times period period period times up er B times up er C up er D times period period period times up er E Baseline times nabla Superscript up er C prime up er C Baseline left-parenthesi xi Superscript up er A Baseline period period period times xi Superscript up er B Baseline right-parenthesi equals 0 comma

nabla left-parenthesi e Superscript psi Baseline times xi Subscript up er C Baseline times xi Subscript up er D Baseline times period period period times xi Subscript up er E Baseline right-parenthesi minus q phi Subscript up er A times period period period times up er B times up er C up er D times period period period times up er E Baseline times normal up er Lamda Superscript up er C prime up er C Baseline xi Superscript up er A Baseline period period period times xi Superscript up er B Baseline equals 0 comma

eSuperscriptpsiBaselinel ft-parenthesi xiSubscriptup erCBaselinetimesnablapsirght-parenthesi xiSubscriptup erDBaselinetimesperiodperiodperiodtimesxiSubscriptup erEBaselinepluseSuperscriptpsiBaselinel ft-parenthesi nablaxiSubscriptup erCBaselineright-parenthesi xiSubscriptup erDBaselinetimesperiodperiodperiodtimesxiSubscriptup erEBaselinepluseSuperscriptpsiBaselinetimesxiSubscriptup erCBaselinenablaSuperscriptup erCtimesprimeup erCprimeBaselinel ft-parenthesi xiSubscriptup erDBaselinetimesperiodperiodperiodtimesxiSubscriptup erEBaselineright-parenthesi minusqeSuperscriptpsiBaselinenormalup erLamdaSuperscriptup erCprimeup erCBaselinexiSubscriptup erCBaselinexiSubscriptup erDBaselinetimesperiodperiodperiodtimesxiSubscriptup erEBaseline quals0semicol n

which

once

gives e Superscript psi Baseline times xi Subscript up er D Baseline times period period period times xi Subscript up er E Baseline left-bracket xi Subscript up er C Baseline times nabla psi times plus nabla xi Subscript up er C Baseline minus left-parenthesis p minus 1 right-parenthesis eta Superscript up er C times prime Baseline minus q normal up er Lamda Superscript up er C prime up er C Baseline xi Subscript up er C Baseline right-bracket equals 0 comma

whence xi Superscript up er C Baseline times nabla psi plus nabla Subscript imes up er C times prime up er C Baseline xi Superscript up er C Baseline minus q times normal up er Lamda Subscript up er C prime up er C Baseline times xi Superscript up er C Baseline plus left-parenthesis p minus 1 right-parenthesis eta Subscript up er C prime Baseline equals 0 period

p-fold principal spinor of

ξs we get

,

we

have:

a

valence p + q zero-rest-mass

By

lemma 1

from the

have

we

equation

—Λc'cξc

above to

—2ηc'cξc,

=

give

so

the term

involving qΛ.c'cξc

be eliminated

can

us

xi Superscript upper C Baseline nabla psi plus left-parenthesis q plus 1 right-parenthesis times nabla Subscript upper C prime upper C Baseline xi Superscript upper C Baseline plus left-parenthesis p minus 2 q minus 1 right-parenthesis times eta Subscript upper C prime Baseline equals 0 period

Now this is

of the form

equation

an

with respect toa Therefore r

=

q + 1,

we

investigate

s

p

=

—

2q

and

use

+ αc'

ξcΛcc'ψ

=

0 from which it follows at once,

of the SFR condition, that

αc'

must

the consequences of this relation with

αA,

satisfy

r▿A,AξA

=

by differentiation a

+

.

sηA,

where

,

1. We have:

—

xi Superscript up er A Baseline times nabla Subscript up er A up er A prime Baseline alpha Superscript up er A prime Baseline minus eta Subscript up er A prime Baseline times alpha Superscript up er A prime Baseline equals 0 comma

r xi Superscript up er A Baseline nabla nabla xi Superscript up er B Baseline plus s xi Superscript up er A Baseline nabla eta Superscript up er A prime Baseline minus r eta Subscript up er A prime Baseline nabla Subscript up er A Superscript up er A prime Baseline xi Superscript up er A Baseline equals 0 comma

2 r xi Superscript up er A Baseline nabla Subscript left-parenthesis up er A Baseline nabla Baseline Subscript up er B right-parenthesis Superscript up er A prime Baseline xi Superscript up er B Baseline minus r xi Superscript up er A Baseline nabla nabla xi Superscript up er B Baseline plus s xi Superscript up er A Baseline nabla eta Superscript up er A prime Baseline plus r eta Superscript up er A prime Baseline nabla Subscript up er A prime up er A Baseline xi Superscript up er A Baseline equals 0 comma

2rxiSuperscriptup erABaselinewhitesquarexiSuperscriptup erB aselineminusrleft-bracketnablaleft-parenthesi xiSuperscriptup erABaselinenablaSubscriptup erASuperscriptup erAprimeBaselinexiSuperscriptup erB aselineright-parenthesi minusleft-parenthesi nablaxiSuperscriptup erABaselineright-parenthesi left-parenthesi nablaxiSuperscriptup erB aselineright-parenthesi right-bracketplusleft-parenthesi sminusr ight-parenthesi xiSuperscriptup erABaselinetimesnablaetaSuperscriptup erAprimeBaselineplusrleft-bracketxiSuperscriptup erABaselinenablaSubscriptup erAup erAprimeBaseline taSuperscriptup erAprimeBaselineplusetaSuperscriptup erAprimeBaselinenablaxiSuperscriptup erABaselineright-bracketequals0com a

from which it follows that

algebraic identity thus

we see

that if

to lemma 2 that

As

was

,

V, s

∈ (i.e.

ξA is

pointed

a

p∈

out

(1994).

Comment would be

an

by Sommers, to Kundt

and

the Ricci relation

2)

we

have a

□AB ξB = —3ΛξA,

the

,

of the

peculiar

a

Weyl spinor.

□

theorem

its

Goldberg-Sachs

.

from which it follows

(and

& Thompson) follows immediately

as a

the And

according

generalization

due to

consequence of Theorem

§7.3).

to

interesting challenge

refractory

use

These results have in the meanwhile

whereas Robinson’s theorem more

3g

+

repeated principal spinor

Penrose & Rindler 1986,

(cf.

by

and the SFR condition r

Robinson &i Schild and 1

s

can

come

easily

be

up with yet

a

appeared

shorter

in

proof.

geometrized (cf. §11.2.8)

the

Hughston (1987).

Note in

particular

Goldberg-Sachs

It

that

result is

to four dimensions.

References

Goldberg p. 13 23

J. N. & Sachs , R. K.

,

( 1962 )

A theorem

on

Petrov types , Acta.

Phys. Polonica, Suppl.

22 ,

-

.

Hughston

,

L. P.

( 1987 )

Remarks

on

Sommers theorem Class. Quant. Grav. 4 1809 1811 -

,

,

Kundt W. & Thompson A. H. ( 1962 ) Le tenseur de Weyl et ,

isotropes

sans

,

une congruence associée de , distorsion , C.R. Acad. Sci. Paris 254 , p. 4257 4259

Penrose , R. & Rindler , W.

( 1986 ) Spinors and Space-Time, Space-Time Geometry ( Cambridge University Press ).

.

géodésiques

-

.

vol. II:

Spinor

and Twistor Methods in

Robinson , I. & Schild , A.

Phys.

4 , p. 484 489

( 1963 )

Generalisation of

a

theorem

by Goldberg

and Sachs , J. Math.

-

.

Sommers , P. D. ( 1976 ) Properties of shear-free congruences of null A349 , p. 309 318

geodesics

Proc. Roy. Soc. Lond. ,

,

-

.

§II.2.8 A

Twistor

Description

of Null Self-dual Maxwell Fields

Eby astwo d M.G.

(TN 20,

September 1985) Robinson’s theorem: After complexification, Robinson’s theorem (Robinson, 1961, motivation in the birth of is

an

i.e.

a

integrable

self-dual Maxwell field.

E may be defined or,

equivalently,

a

defining

a

integrable 2-forms a

field is

to

This

can

field

be

proved

and

a

locally

null

locally

integrability

(≡simple)

on

M

2-form defines a

(locally).

defining a

plane

congruence of

E

closed self-dual 2-form,

a

then shows that as

fact,

more a

(given by pull-back

1-1

a

a

integrability

precisely,

of E

suppose E is

correspondence

a

Without

simple 2-form

under θ : M →

distribution it follows that and

a

required.

criterion:

self-dual and the

Then there is

a-surfaces,

important

spinors (e.g. Sommers 1976):

integrability

able to choose this form to be closed. In

exactly specified by E,

a

,

α-planes is, by definition, necessarily

S and closed 2-forms

by

is the condition

computation

the theorem is immediate from Frobenius’

being

defined

solved for ψ whence V

and let S denote the space of leaves

on

complex

an

conformal manifold and E ⊂ TM

with the aid of

A curvature

distribution of

equivalent

α-planes,

spinor

may be

spinors, however,

since

by

then E may be

a

a

is

twistors)

distribution of

states that if M is a

between

S). Finally,

null self-dual Maxwell

2-form ω on

S,

the 2-dimensional

parameter space for the congruence. A twistor

description:

twistor space T

above,

a

Suppose

now

parameterizing

submanifold of T:

that M is

conformally right

the a-surfaces. Then

a

flat

so

that it has

self-dual null Maxwell field

a

3-dimensional

gives

S

as

As noted above, the Maxwell field

specifies

2-form ω on S. The usual Penrose

a

identifies the space of all self-dual Maxwell fields with the canonical bundle Ω3. Hence there should be

natural

a

transform, however,

cohomology H1(T, k)

where

k

is the

homomorphism

normal up er Gamma left-parenthesi up er S comma normal up er Omega squared right-parenthesi right-ar ow up er H Superscript 1 Baseline left-parenthesi up er T comma kap a right-parenthesi period

(*) It is somewhat easier to ‘Real’ twistors: the

namely of

see

what is

The usual twistor

(+,+,-,-) and

“α -planes”.

Victor Guillemin has and it

analogue.

Gr2(R4)

=

RP is the space of

twistor

theory

in ‘real’ twistor

on

and

one

been

recently

theory:

Gr2(C4)

between

correspondence

between RM

correspondence

signature

going

=

RP

family

RP3.

of

investigating

that most twistor constructions have

seems

and CP3 has

RM has

totally

null

a

a

real form,

conformal metric

2-planes

in RM, the

this ‘black-and-white’ version of

an

(often simpler)

black-and-white

The Penrose transform is replaced by the Gelfand-Radon transform e.g. normal up er Gam a left-parenthesi double-struck up er R times double-struck up er P com a kap a right-parenthesi ModifyingAbove right-ar ow With asympto icaly-equals StartSet omega times el ment-of normal up er Gam a left-parenthesi double-struck up er R times times com a normal up er Omega Subscript plus Superscript 2 Baseline right-parenthesi s period times t period times times d omega equals 0 EndSet

which is

simpler in in

locally (i.e.

a

that functions have

neighbourhood

shown, however,

has

that

of

globally

a

replaced cohomology. Although line in

it is

an

RP),

Suppose such on

that ω is

a

it, which

a

as

for the

may also be denoted

by

holomorphic ω.

The

on

an

RM

(it

(all

its null

will have

of

(✶)

so

on)

geodesics S

↪RP

be defined

V. Guillemin and suggests

closed).

are

singularities

case, one obtains a surface

analogue

can

isomorphism.

all helicities and

Zoll manifold

null self-dual Maxwell field

technicalities). Then,

never

isomorphism (for

that the crucial property of RM is that it is now

it is then

the transform

but

EF and

a

ignore 2-form

is thus

normalup erGam aleft-parenthesi up erScom anormalup erOmegasquaredright-parenthesi right-ar ownormalup erGam aleft-parenthesi double-struckup erRtimesdouble-struckup erPcom akap aright-parenthesi period

R(✶) A function

f

on

RP may be

integrated

over

S

f right-arrow integral Underscript s Endscripts f omega

against ω.

The linear functional

therefore defines

a

distribution-valued 3-form

supported

on

S. This is Guillemin’s identification of

S determines the singularities of ω.

R(✶).

Ordinary

twistors:

As

defining S,

and therefore

a

an

divisor,

S

rise to

gives

a

line bundle

ξ over

T with

a

section

s

∈ Γ(T, ξ)

exact sequence: 0 right-ar ow xi asterisk right-ar ow Overscript times s Endscripts Subscript upper T Baseline right-ar ow Subscript s Baseline right-ar ow 0 period

(✶✶)

ξ|s

Moreover,

=

N the normal bundle of S in T and the sequence may therefore be rewritten: ξ →N →0. 0 → OT →

that N ⊗

Finally, noting

k|S

=

Ω2

on

it may be rewritten

S,

a

and

is

(✶)

given by

connecting homomorphism

the

of the

corresponding long exact

sequence:

normal up er Gamma left-parenthesi up er S comma normal up er Omega squared right-parenthesi right-ar ow up er H Superscript 1 Baseline left-parenthesi up er T comma kap a right-parenthesi right-ar ow Overscript imes s Endscripts up er H Superscript 1 Baseline left-parenthesi up er T comma kap a circled-times xi right-parenthesi period

Note also that the null fields obtained in this way way of

that the

saying

identification of null

Googly photon?

class has

a

improved way

to

cohomology

:

The usual

covariant constant sections

its

connection

original

This is

own.

to do about the

the

surely

along

the leaves

(normal

googly photon

than null and

how

a

α-n-folds

S, and

(by

an

regard

an

rise to

a

leaves) equips for

a

a

a

along

an

invariant

[Penrose’s (1969) original

integrable

line-bundle

along

a

simple

as a

an

on a

line-bundle.

on

S. The residual information on

S with

give

a

connection of

clues

as

to what

to combine this construction with

trying

attempt

connection

distribution of α-planes and the

this line bundle

and also suggests in

Maxwell field is

null self dual field. This may

to describe ‘half

algebraically special’

(i.e. Yang-Mills bundles) gives

congruence of α-surfaces. These

The Frobenius

approach

Hughston

are

closed self-dual n-form

on a

definition orthogonal to self-dual

n-form ω on S.

Conversely,

to the Robinson theorem

has also shown

method works in dimension 6 and

spinor

In any case,

S

‘simple pole’ along

by ×s:

very

special

indeed

self-dual).

dimensions. L.P.

higher (even)

those annihilated

A similar construction for the non-linear version

dimensions:

Higher

give

photon construction

vector bundles with connection flat

(stronger

to the

general googly photon

the usual Ward twisted Maxwell fields.

exactly

fields].

A null self-dual Maxwell field is then flat

of the

are

(II.2.3

presumably

evidently

extends to

and lecture, Oxford 30

April 1985)

general spinor proof

is available.

a

conformal 2n-fold

gives

a

congruence of

an

n-dimensional space

Higher

dimensional twistors

n-forms) parameterized by

every such arises in this way.

rise to

only

exist in

This has

obvious way for

twistor space Zn

a

dimension

an

n(n

+

consisting

of

M denote

1)/2. Letting

flat space

conformally

open subset of

Zn, then (subject to mild topological restrictions

1983)

November

the

complex quadric

system of Pn’s lying therein (the

one an

Q2n,

on

M)

M.F.

and T the

Qn

Atiyah

of dimenion 2n.

α-Pn’s).

Zn has

corresponding

has shown

subset

(lecture, Oxford,

7

that up er H Superscript n left-parenthesi n minus 1 right-parenthesi slash 2 Baseline left-parenthesi up er T comma kap a right-parenthesi ModifyingAbove right-ar ow With asymptoticaly-equals StartSet omega element-of normal up er Gamma times left-parenthesi up er M comma normal up er Omega Subscript plus Superscript n Baseline right-parenthesi EndSet s period t period times d omega times equals 0 right-brace period

The

homomorphism Hn(n-1)/2(T, k) Γ(S, Ωn) →

is

given by composing

induced

by

the

a

series of

appropriate

version for real

connecting homomorphisms or by

Koszul

split quadrics.

complex

instead of

Note that

(✶✶).

n(n - 1)/2

a

spectral

There is

sequence construction

corresponding black-and-white

a

is the codimension of S in T

as one

would

expect. Many thanks

to Victor Guillemin for much

interesting

conversation.

References Penrose , R.

( 1969 )

Robinson I. ,

Solutions of the zero-rest-mass

( 1961 )

Null

Sommers , P. D. ( 1976 ) A349 , p. 309 318

electromagnetic

equations

fields , J. Math.

,

J. Math.

Phys.

2 , p. 290 291

10 , p. 38 39 -

.

-

of shear-free congruences of null

Properties

Phys.

geodesics

,

.

Roy. Soc.

Proc.

Lond. ,

-

.

§II.2.9

A

conformally

gruence

by

invariant connection and the space of leaves of

T.N. Bailey

Introduction.

This is

(TN 26,

a

report

which is the space of leaves of conformal a

complex

vacuum

space-times,

three manifold

a

March

on

shear free

con-

1988)

work in progress,

(complexified)

studying

the structure of the

shear free congruence.

the surface has the first formal

(which

a

I will show below that in

neighbourhood

in the flat space would be dual

complex surface

projective

of

an

twistor

embedding

space).

in

In order to describe this structure, I will first show that natural

spinor fields

has

‘conformally

invariant edth and thorn

interest in its It is

hoped

the Kerr The

a

own

conformally

conformal

invariant connection, which is

operators’.

This construction

complex space-time

essentially given by seems

to have some

with two Penrose’s

geometric

right.

that these these structures will

metric,

a

and there may be other

help

to

explain the separation

fields oA and

independent spinor

iA,

equations

in

applications.

invariant connection. Let M be

conformally

of various

a

defined up to scale.

complex

conformal

Equivalently

we

with two

space-time,

have

a

splitting

Superscript upper A Baseline equals upper O circled-plus upper I

(1) of the

spin

bundle. Assume also that

we are

given

an

identification of the

primed

and

unprimed

conformal weights left-bracket negative 1 right-bracket equals Overscript d e f Endscripts Subscript left-bracket up er A up er B right-bracket Baseline times ap roximately-equals Subscript left-bracket up er A prime up er B prime right-bracket Baseline period

This is

equivalent

to

allowing

conformal transformations epsilon Subscript up er A up er B Baseline right-ar ow from bar normal up er Omega times times epsilon Subscript up er A up er B Baseline

which is

a

splitting

in

of the form

epsilon Subscript up er A prime up er B prime Baseline right-ar ow from bar normal up er Omega times times epsilon Subscript up er A prime up er B prime Baseline

complexification of a real space-time.

natural condition if M is the

conformal class, the

only

equation (1)

allows

us

Given

a

metric in the

to define a one-form

up er Q Subscript a Baseline colon-equal negative 2 times o Superscript left-parenthesi up er B Baseline iota Superscript up er C right-parenthesi Baseline times partial-dif erential eft-parenthesi o Subscript left-parenthesi up er A Baseline iota Subscript up er C right-parenthesi Baseline right-parenthesi equals rho prime l Subscript a Baseline plus rho n Subscript a Baseline minus tau prime m Subscript a Baseline Subscript Baseline minus tau m overbar Subscript a Baseline

where ∂a is the metric

connection,

and

we

adopt the

convention that

oAiA

=

1 whenever

a

particular

metric has been chosen. Under conformal transformation up er Q Subscript a Baseline right-ar ow from bar up er Q Subscript a Baseline minus normal ϒ Subscript a Baseline times w h e r e times normal ϒ Subscript a Baseline equals normal up er Omega Superscript negative 1 Baseline partial-dif erential Subscript a Baseline normal up er Omega period

(2) The

significance

of

Qa

is that it enables

us

to

Recall the local twistor exact sequence 0 right-ar ow Subscript up er A prime Baseline times right-ar ow Superscript alpha Baseline right-ar ow Superscript up er A Baseline right-ar ow 0 pi Subscript up er A prime Baseline times right-ar ow from bar left-parenthesi 0 comma pi Subscript up er A prime Baseline right-parenthesi left-parenthesi omega Superscript up er A Baseline comma pi Subscript up er A prime Baseline right-parenthesi right-ar ow from bar omega Superscript up er A

and the conformal transformation rule omega Superscript up er A Baseline right-ar ow from bar omega Superscript up er A Baseline times pi Subscript up er A prime Baseline right-ar ow from bar pi Subscript up er A prime Baseline plus i normal ϒ Subscript a Baseline times omega Superscript up er A Baseline period

If

we

set alpha Subscript up er A prime Baseline times equals pi Subscript up er A prime Baseline plus i up er Q Subscript a Baseline times omega Superscript up er A

split

the local twistor bundle

as a

direct

sum.

then from

equation (2)

there is

conformally

a

invariant

splitting

SuperscriptalphaBaselinetimesModifyngAboveright-arowWithasymptoicaly-equals Superscriptup erABaselinecir led-plus Subscriptup erAprimeBaselinel ft-parenthesi omegaSuperscriptup erABaselinecom apiSubscriptup erAprimeBaselineright-parenthesi right-arowfrombaromegaSuperscriptup erABaselinecir led-plusalphaSubscriptup erAprimeBaselinetimes

(3) of Oα and I will

the ‘split co-ordinates’

use

The local twistor connection splits to

spin

(ωA, αA,)

henceforth.

give connections,

which I will denote

by ▽a,

on

the various

bundles. A brief calculation shows these to be Superscript up er A Baseline times colon nabla mu Superscript up er A Baseline equals partial-dif erential Subscript b Baseline mu Superscript up er A Baseline plus epsilon Subscript up er B Baseline times Superscript up er A Baseline up er Q Baseline Subscript up er C up er B prime Baseline mu Superscript up er C

Superscript up er A prime Baseline times colon nabla mu Superscript up er A prime Baseline equals partial-dif erential Subscript b Baseline mu Superscript up er A prime Baseline plus epsilon Subscript up er B prime Baseline times Superscript up er A prime Baseline up er Q Baseline Subscript up er B up er C prime Baseline mu Superscript up er C prime

Subscript upper A Baseline times times colon nabla mu Subscript upper A Baseline equals partial-dif erential mu Subscript upper A Baseline minus upper Q Subscript upper A upper B prime Baseline mu Subscript upper B Baseline

Subscript upper A prime Baseline times times colon nabla mu Subscript upper A prime Baseline equals partial-dif erential mu Subscript upper A prime Baseline minus upper Q Subscript upper B upper A prime Baseline times mu Subscript upper B prime Baseline

Subscript left-bracket up er A up er C right-bracket Baseline times times colon nabla Subscript b Baseline nu Subscript up er A up er C Baseline equals partial-dif erential Subscript b Baseline nu Subscript up er A up er C Baseline minus up er Q Subscript b Baseline times nu Subscript up er A up er C Baseline

(ωA, αA,)

If Zα=

is

a

local

twistor,

we can

write the local twistor connection

as

nabla Subscript b Baseline up er Z Superscript alpha Baseline quals left-parenthesi nabla Subscript b Baseline omega Superscript up er A Baseline plus i epsilon Subscript up er B Baseline times Superscript up er A Baseline alpha Subscript up er B prime Baseline com a nabla Subscript b Baseline alpha Subscript up er A prime Baseline plus i up er D Subscript a b Baseline times omega Superscript up er A Baseline right-parenthesi

(4) where Dab, is

a

conformally

invariant modification of Pab defined

by

up er D Subscript a b Baseline equals up er P Subscript a b Baseline minus partial-dif erential up er Q Subscript a Baseline plus up er Q Subscript up er A up er B prime Baseline times up er Q Subscript up er B up er A prime Baseline period

(For

a

definition and discussion of the modified curvature

(1986) §6.8 The

or

PAA'BB'see Penrose & Rindler

spinor

§11.2.1.) in

splitting

equation (1)

allows

us

to define the bundles

left pointing angle negative r prime comma negative r right pointing angle colon-equal up er O Superscript r prime Baseline circled-times up er I Superscript r

(note

that

section of

⟨1,1⟩

=

⟨—1, 0⟩,

[1]).

so

The connection ▿a

that λAoA

=

can

be

projected

on

to these.

For

example,

if λA

is

a

0, lamda Superscript up er A Baseline right-ar ow from bar minus o Superscript up er A Baseline times iota Subscript up er C Baseline times nabla Subscript b Baseline lamda Superscript up er C Baseline

is

a

connection, and its components

same can

way as the same

be

computed

with

expression ordinary

are

given by ‘conformally

with the metric connection

invariant edth and thorn’, in

 02;b, replacing

just

the

▿b has components that

edth and thorn.

Since ▿a agrees with &3x2202;a if you form any of the well known conformally invariant parts of the metric connection, there is scope here for The

expressions

which arise

components of Dab. The section.

as

producing

a

complete ‘conformally invariant GHP

curvatures when one commutes

geometrical significance

formalism'.

conformal edths and thorns

of these connections will be discussed in

a

are

later

Shear free congruences in Minkowski space. Before

starting

the situation in flat

a

(hereafter SFR)

In real Minkowski space,

space-time.

given by

is

spinor

a

field

on

the

general

case, I will review

shear free congruence of null

geodesics

satisfying o Superscript up er A Baseline o Superscript up er B Baseline partial-dif erential o Subscript up er B Baseline equals 0

(5) If oA is

it

analytic,

distribution is

be

can

integrable,

complexified,

and

gives

so

and it then determines

foliation of Minkowski space

a

when oA is shear free. The space of leaves S of this foliation is the twistor space P*, which describes the congruence, The surface S inherits

by complex surfaces, precisely

hypersurface

in dual

in

embedding,

in the

congruence

are

accompanying article, §11.2.11, isomorphic

projective

there is the tangent

particular

bundle of P*, the normal bundle sequence, and the restrictions of the line bundles

analysis

I will

now

describe how this

The

O(n).

shows how massless fields of various orders

to sections of sheaves on S.

This

β-planes.

to Kerr’s Theorem.

according

structure from its

some

distribution of

a

along

the

generalises

to

curved space. SFRs in curved

equation (5) surface

S,

,

and

gives

a

but there is in

The SFR defines bundle defined

In

space-times.

foliation in the

general

Maxwell

a

a

by S considered

to the existence of a one-form

no

complexification.

an

SFR is still

given by

solution of

a

The space of leaves still defines

a

complex

twistor space in which it is embedded. which in Minkowski space is the Ward transform of the line

field,

as a

general space-time,

divisor. This follows from the fact that

equation (5)

is

equivalent

φa with partial-difer ntialSubscriptup erAprimeleft-parenthesi up erASubSuperscriptoSubscriptup erBright-parenthesi Baseline qualsnormalup erPhiSubscriptup erAprimeleft-parenthesi up erASubSuperscriptoSubscriptup erBright-parenthesi Baselinetimescom a

and it is easy to

see

that φa has

satisfies

left-handed part a a

the freedom to be the

precisely

conformally flat space-time.

a

Maxwell field. The

vanishes

-ϕ(ABoC),

=

for

as

in

expected

1

The structures I shall describe that the SFR oA in the

ΨABCDoD

potential

on

S only exist under certain conditions. In particular, I will say

space-time M

satisfies the

Goldberg-Sachs

condition

(hereafter GS)

if

oAoBocΨABCD=0. We

assume

the GS condition holds

to exist otherwise.

The

henceforth,

Goldberg-Sachs

since

Theorem

no

implies

and it is therefore satisfied

o

construct bundles 1

An

SFR is thus

a

on

S,

we

make

use

of ▿a, the

charged twistor coupled

to

its

own

significant part

by

of the structure

that the GS condition is

all

conformally

conformally

vacuum

on

S

seems

equivalent

space-times.

to

To

invariant connection. First choose

canonically defined Maxwell field.

a

spinor direction iA all the bundles

on

to

complement

⟨r', r⟩)

of the foliation is both and

⟨r', r⟩s

the SFR oA, and deduce from the SFR and GS conditions that

bundle

corresponding

on

M with

O(S)A

vanishing

spinors proportional

S, whose sections

over

a

vector bundle on S.

the

quotient

is well defined

E. The part

oA▿a

We have

an

sections of the

injection

of the

→E → 0

of the local twistor connection preserves

E. Furthermore, it is flat

on

by definition

Oa 0 → ⟨0, 1⟩ → Oa

defining

thus define line bundles

can

are

conformal derivitive up the foliation.

The dual local twistor bundle also defines to oA into

of the connection that differentiates up the leaves

of the choice of iA and flat. 2 We

independent

rank two vector bundle

a

oA▿a

and OA', the part

on

the leaves and

so

defines

a

⟨0, 1⟩

and hence

rank three vector bundle

ε on S. Sections of ε can be realised

as

spinor fields

ξA'

satisfying

a

tangential twistor equation

3

oSuperscriptup erABaselinetimesnablaSubscriptup erABaselineSuperscriptleft-parenthesi up erAprimeSuperSubscriptxiSuperscriptup erBprimeSuperscriptright-parenthesi Baseline quals0

and

given

that sections of

O(S)A' are spinor fields satisfying o Superscript upper A Baseline times nabla xi Superscript upper B prime Baseline equals 0

we

get

an

injection

O(S)A'→ξwhich extends to

give

a

short exact sequence

O(S)A'0 →→ ε→⟨1,0⟩ → 0

given,

in terms of

equations, by

oSuperscriptup erABaselinenablaSubscriptup erAup erAprimeBaselinexiSuperscriptup erBprimeBaseline quals0timesright-ar owfrombaroSuperscriptup erABaselinenablaSubscriptup erABaselineSuperscriptleft-parenthesi up erAprimeSuperSubscriptxiSuperscriptup erBprimeSuperscriptright-parenthesi Baseline quals0timesStartBinomialOrMatrixoSuperscriptup erABaselineoSuperscriptup erB aselinenablaetaSubscriptup erABaseline quals0Cho seiotaSuperscriptup erABaseline taSubscriptup erABaseline quals0EndBinomialOrMatrix iSuperscriptup erAprimeBaselineright-ar owfrombariotaSubscriptup erABaselinetimesoSuperscriptup erB aselinetimesnablaSubscriptup erBup erBprimeBaselinexiSuperscriptup erBprimeBaseline

If

μA'

required can

4

is

a

section of

to make

OA' ⟨0, —1⟩),

a

calculation reveals that the condition

iAμA' a connecting vector

be identified with the tangent bundle

through by ⟨0,

—

1E9;s

to

give

to note that

T(S)

nearby of S.

leaf of the foliation. Thus,

The exact sequence above

= 0 is what is

O(S)A‘ ⟨0, can

—

1⟩s

be tensored

what in flat space would be the normal bundle sequence of S

0 →

2 It is helpful 3 To see this,

to a

oA▿aμB'

oAQa

T(S)

→

ξE8;0,-1⟩s → ⟨1, -l⟩s

→ 0.

is independent of iA

that GS and SFR imply oA oB

if oA is SFR. Dab = 0 and use

the conjugate version of equation (4). When writing down the splitting and connection on the dual local twistors, simply write down the conjugate pretending that note

Dab and Qa are real. 4 The connection here is

the tensor product of the conformally invariant

ones

on

the factors.

If is

is

one

given

equivalent

how

The

realise the first formal

spinor field oA defines

a

independent In

a

neighbourhood

SFR, there is

an

f

so

the

two-plane distribution,

When oA is

embedding

defines

of the

a

integral

A calculation shows that, worth of functions g

naturally

a

0

=

obey πA∂ag

of the lift of M

can

XAB

spinor field

now

describe

briefly

directly.

M in the projective spin bundle

on

=

neighbourhood (if it

lift of M

the

on

of are

order in

embedding

is

S.

on

any)

lifts of β-surfaces.

are

β-surfaces parametrised by S, functions

precisely

on

complex variables the

neighbourhood of

a

S.

lift of

S.

function

a

has

two functions of two

are

first

0 to

connections,

of this

sheaf O(1)

a

on

the first formal

neighbourhood

be written

g(x, πA) If the

invariant

conformally

I will

defined differential operator πA∂a which

surfaces of which

M. These form the formal neighbourhood sheaf O(1) In terms of the

of S

formal neighbourhood

given the GS condition, there

POA that

on

space-time

complex parameter family

two

POA which obey πA∂af

on

the normal bundle sequence

embedding.

embedding

an

first formal

a

detail; recall that POA has

more

and functions

of the

knowing

two-surface S̃ transverse to the foliation, and note that S̃ has

a

of the choice of S̃, and

slightly

defines

of

in the restriction of POA. The first

embedding

then

complex manifold,

natural

POA. Now realise S by choosing natural

a

knowing the first formal neighbourhood

to

one can

in

hypersurface

a

=

f(x)

+

iAXABπB

oAXAB 0

where

=

=

XABoB

satisfies

▿BA'XAB = ▿AA'f then it defines

a

section of O(1).

One result of this analysis is

Massless fields.

SFR, then,

minor

there is

an

of two

worth of left handed massless fields null

then to

remembering

one

that it has conformal

correspondence

In my

with sections

accompanying

correspond

article

§11.2.11

to sections of sheaves over

out that sections of the formal

fields which have functions of two

Apart

a

principal

complex

S of

sheaf

along

variables worth of such

neighbourhood

more severe

sheaf.

along

one

holomorphic function

it. If the field has

indicees,

n

are

in

one

⟨1, n + 1⟩)S.

S. Provided,

one

precisely

—1, it is easy to check that these fields

I show how in

neighbourhood

from that case, however,

second formal

weight

null direction

worth of order three Maxwell fields a

over

helicity,

Theorem,

of Robinson’s

generalisation

which states that if oA is

complex variables

for each

a

as

flat

space fields of various orders

usual,

that the GS condition

O(1) 97;⟨1,

l⟩s

on

S do

give

the congruence. Thus there

things, just

as

in the flat

along

holds,

oA

it turns

left handed Maxwell are

two

holomorphic

case.

curvature restrictions appear. To get three functions

requires

oAoBΨABCD

=

0 in which

case

it

seems

that S has

Killing spinors. Suppose null directions. The

M admits

a

and choose oA and iA to be

Killing spinor,

along its principal

Killing spinor equation partial-difer ntialSubscriptup erAprimeBaselineSuperscriptleft-parenthesi SuperSuperscriptup erASuperscriptomegaSuperscriptSuperSuperscriptup erBup erCSuperscriptright-parenthesi Baseline quals0

then

implies that

▿aω

=

on

1⟩is

⟨1,

both oA and iA

0 where ω is

a

section of

which

flat,

are

SFRs. The

⟨1, 1⟩.

This is

remaining parts

only possible

carries closed

by

a

a

number of consequences.

to

over

means

S, thereby giving

that

locally

reduce to

conformally invariant

solving

connection

In the

equation (2)

Further work is in progress

ideas in the next section, it will be

possible

on

to

an

isomorphism ⟨1, 1⟩)

shows that it

⟨1,

0⟩ and its

all

this,

≅ ⟨0,

0⟩)

the fact that

⟨1, l⟩s. Secondly,

metric thus

special

line bundle

single

0

it provides

Firstly,

it is exact, and

is contained in the

=

natural trivialisation of

a

conformal transformation.

Jeffryes (1984).

equation

implies

∂[aQb] This has

if the

of the

which

Qa

is

can

thus be made to vanish

constructed,

all the curvature information

(conformally invariant) connection;

since it

seems

explain the separation

likely that,

cf.

combined with the

of various differential

equations

in the Kerr solution.

Geometrical significance. To finish, I will mention which I have

just

started to follow up in collaboration with

connection constructed above is

an

structure, and the structure

has

obtained a

on

the

one

example (in

the

of

by

unique

oA and iA

so

The

M.A.Singer.

conformally

connection determined

complex space-time)

complexification of a real four-manifold X

compatible conformal Hermitian metric. The

defined

a

ideas due to R.Penrose and K.P.Tod

some

with

an

seems

invariant

geometrical

to be that which would be

almost

complex structure Jab

of Jab are the

eigenspaces

by

a

two-plane

and

distributions

that up erJSubscriptaBaselineSuperscriptbBaseline qualsileft-parenthesi oSubscriptup erABaselinetimesiotaSuperscriptup erB aselineplusiotaSubscriptup erABaselinetimesoSuperscriptup erB aselineright-parenthesi epsilonSubscriptup erAprimeBaselineup erBprime

The almost

suggestion

complex

is that the existence of a

metric. This whose

structure will be

view-point

seems

likely

Killing spinor since

is

something

when both oA and iA

equivalent

to the

are

SFRs.

Kähler condition

very similar has been

is somewhat different. I would like to thank

for discussions and Comment

very

integrable

Further, on

the

the Hermitian

given by Flaherty (1976),

M.A.Singer, R.Penrose,

and K.P.Tod

suggestions.

(1994).

This work

eventually

became

Bailey (1991a)

and

Bailey (1991b).

References T. N. ( 1991 ) Complexified conformal almost Hermitian structures and the , edth and thorn operators , Class. Quantum Grav. 8 , 1 4

Bailey

-

.

conformally invariant

Bailey

,

T. N.

( 1991 ) The

Theorem , J. Math.

space of leaves of a shear-free congruence, Phys. 32 , 1465 1469

,

and Robinson's

.

Flaherty E. J. Jr. ( 1976 ) Hermitian (Springer Verlag Berlin ). ,

46

multipole expansions

-

and Kählerian geometry in

relativity

,

Lecture Notes in

Physics

,

Jeffryes

B. P.

,

p. 323 341

( 1984 ) Space-times

with two-index

Killing spinors

Proc. Roy. Soc. London A392 ,

,

-

.

Penrose , R. & Rindler , W.

( 1986 ) Spinors and Space-Time, Space-Time Geometry ( Cambridge University Press ).

§II.2.10

A

conformally

This note is

invariant connection Bailey by T.N. the last section of my article

postscript to

a

vol. II:

associated to a direct sum

decomposition

behind the observations in that

article,

of

one

Spinor

(TN

§11.2.9

of the

the

on

and Twistor Methods in

27, December

conformally invariant connection

bundles. The

spin

1988)

stated for convenience in the

general

result

lying

holomorphic category,

is

as

follows: THEOREM held Jac with

.

Let M be

J(ab)

satisfying ▿aJba

=

conformal metric is If

one

is

given

a

=

complex conformal manifold with

a

0 and

JacJcb

0 and ▿agab

=

=

—δab.

Then there exists

for

Xagbc

conformal metric gab and

some

a

unique

given

tensor

torsion-free connection

The second condition is

Xa.

a

simply

▿a

that the

preserved. direct

sum

decomposition

of OA in

a

complex space-time

then

up erJSubscriptaBaselineSuperscriptbBaseline qualsileft-parenthesi oSubscriptup erABaselinetimesiotaSuperscriptup erBBaselineplusiotaSubscriptup erABaselinetimesoSuperscriptup erBBaselineright-parenthesi epsilonSubscriptup erAprimeBaselineup erBprimecom a

where oA, iA constitute the

resulting

a

thorn’ operators.

The

space-times

given

unclear,

and related

the

in components

significance

defines the connection is conformal

spin-frame defining

connection is

decomposition, by

satisfies the above conditions and

R. Penrose’s

‘conformally

of the rather strange condition and work continues areas.

on

the

use

on

invariant edth and

the derivitive of

Jab

which

of this connection in type D

Thanks to M.G.Eastwood and

M.A.Singer.

§II.2.11 sions

Relative

by

cohomology

T.N.

(TN March

26,Bailey

Introduction.

In my

(see §1.6.6

and

original and

§1.6.8

expanded

articles

The power series also The relative and let F be can

cohomology

a

locally

be described

open of F

cover over

gives

a

by

Ui of

a

a

a

which

precise

by

fi

as a

cohomology class,

first

seems

power series.

Let S be

relative Čech

a

on

cocycle,

but

of S in X; then

defining functions

expan-

a

hypersurface

X. The relative intuitive

good a

for

S,

a

one

—

is

fj

is

worldline

based

hypersurface,

in

cohomology is

be

multipole expansion.

complex

a

on a

can

fields.

algebraically special

picture

representative

as

given by

manifold X,

group H

follows: Choose a

holomorphic

set

fi of sections all of Ui ∩

on

an

Uj.

section of F.

holomorphic

then

of

sources on a

‘multipoles’

relative to

description

S, with the restriction that fi a

for

to be the twistor version of the

on

neighbourhood

multipole

of fields with

expressions

some

free sheaf of Ox modules

Ui that ‘blow up’

Now let gi be

I gave

description

version of the twistor

The freedom in each fi is the addition of

the

the twistor

on

Bailey 1985)

of‘power series’,

sort

a

Theorem and

1988)

world-line. In this note, I will show how in

series, Robinson's

power

might try

and

expand

the relative class defined

power series fSubscriptiBaseline qualsStarFractionfSubscriptiSuperscriptleft-parenthesi 1right-parenthesi BaselineOvergSubscriptiBaselineEndFractionplusStarFractionfSubscriptiSuperscriptleft-parenthesi 2right-parenthesi BaselineOvergSubscriptiSuperscript2BaselineEndFractionpluselipsi StarFractionfSubscriptiSuperscriptleft-parenthesi nright-parenthesi BaselineOvergSubscriptiSuperscriptnBaselineEndFractionpluselipsi

(1) To understand this

we

with transition functions which has

a

simple

which induces

a

gi/gj

zero on

map

on

need the divisor bundle L of on

Ui 29;

Uj.

which is defined to be the line bundle

The functions gi then

S. The section

s

sk

:

the relative

S,

gives F

→

us a

map

F

Lk

ࣹ

give

a

section

distinguished

s

of L

cohomology.

DEFINITION 1. The k-th order relative

cohomology

is defined

H

by

the exactness of

or

less

0 right-ar ow up er H Subscript up er S Superscript 1 Baseline left-parenthesi up er X com a script up er F times emicol n k right-parenthesi right-ar ow up er H Subscript up er S Superscript 1 Baseline times left-parenthesi up er X com a script up er F times right-parenthesi right-ar ow Overscript imes Superscript k Baseline Endscripts up er H Subscript up er S Superscript 1 Baseline left-parenthesi up er X com a script up er F times circled-plus up er L Superscript k Baseline right-parenthesi

The k-th order

cohomology

is thus the part which has

therefore corresponds to the first k terms in equation If ξ is can

a

sheaf

define the

on

X, and

(1)

a

pole

of order k

(ξ)(p) by

S,

and it

above.

I(p)ξ is the ideal of sections of ξ which vanish to

p-th formal neighbourhood sheaf

on

p-th

order

on

S

we

the short exact sequence

0 right-ar ow script upper I Superscript left-parenthesis p plus 1 right-parenthesis Baseline epsilon right-ar ow epsilon right-ar ow left-parenthesis epsilon right-parenthesis Superscript left-parenthesis p right-parenthesis Baseline right-ar ow 0

(2)

so

(ξ)(0)

that

is

ξ restricted to S.

just

LEMMA 1. There is

natural

a

isomorphism

up erHSubscriptup erS uperscript1Baselinetimesleft-parenthesi up erXcom ascriptup erFtimes emicol nkright-parenthesi ap roximately-equalsnormalup erGam aleft-parenthesi up erScom aleft-parenthesi scriptup erFtimescircled-plusup erLSuperscriptkBaselineright-parenthesi Superscriptleft-parenthesi kminus1right-parenthesi Baselineright-parenthesi period

The

proof is simply

with the freedom Thus

we

have

to observe that in

equation (1) above,

given by equation (2)

as

strictly

filtration

a

with the quotient at each stage

of the relative

given by

theLkofF&a smustgi#exc2vtio9e7n;

.

than

cohomology (rather

an

infinite direct

sum),

the exact sequence

0 right-ar ow normal up er Gamma times left-parenthesi up er S comma left-parenthesi script up er F times circled-plus up er L Superscript k minus 1 Baseline right-parenthesi Superscript left-parenthesi k minus 2 right-parenthesi Baseline right-parenthesi right-ar ow Overscript imes Endscripts normal up er Gamma times left-parenthesi up er S comma left-parenthesi script up er F times circled-plus up er L Superscript k Baseline right-parenthesi Superscript left-parenthesi k minus 1 right-parenthesi Baseline right-ar ow normal up er Gamma left-parenthesi up er S comma script up er F times circled-plus up er L Superscript k Baseline right-parenthesi right-ar ow 0 period

fields.

Algebraically special a

X in

region

cohomology

twistor space,

projective

of order k

orders

higher If

means

one

null,

on

order

correspond

an

are

analysis

can

corresponding

be

n means

the field has

a

a

hypersurface

to a shear free congruence.

for the relative case, and

as

when S is

applied

we

will say that

the congruence if its twistor function is in

principal

We a

H1(X, O(—n

null direction,

along

can

right

—

in

define

handed

2); k). Thus,

the congruence, and

to certain differential relations between the field and the congruence.

writes down the commutative

and whose columns there is

S just

on

massless field is of order k order 1

The above

induced

by sk

diagram

whose

F → F ⊗

:

Lk,

rows are

the relative

it is easy to

see

cohomology

that if

sequences,

H1(X, F)

=

0 then

exact sequence 0 right-ar ow StartFraction normal up er Gamma left-parenthesi up er X comma script up er F times circled-plus up er L Superscript k Baseline right-parenthesi Over normal up er Gamma left-parenthesi up er X comma script up er F times right-parenthesi EndFraction right-ar ow up er H Subscript up er S Superscript 1 Baseline left-parenthesi up er X comma script up er F times emicol n k right-parenthesi right-ar ow up er H Superscript 1 Baseline left-parenthesi up er X comma script up er F times emicol n k right-parenthesi right-ar ow 0

Since L has the section once, we can write L

=

s

which has

M(1)

n

simple

where M is

bundle of the ‘Maxwell field of the k