Table Of ContentFurther a dvances
in t wistor t heory
Volume Ill: C urved t wistor
spaces
CHAPMAN & HALUCRC
Research N otes in Mathematics Series
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R. Aris, University of Minnesota S. Mori, Kyoto University
G.l. Barenblatt, University of Cambridge L.E. Payne, Comell U niversity
H. Begehr, Freie Universitiit Berlin D.B. Pearson, University o f Hull
P. Bullen, University of British Columbia I. Raeburn, University ofN ewcastle, Australia
RJ. E lliott, University ofA lberta G.F. Roach, University of Strathclyde
R.P. Gilbert, University of Delaware I. Stakgold, University of Delaware
D. Jerison, Massachusetts Institute of Technology W.A. Strauss, Brown University
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at S tony B rook
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Mason
St Peter's College and the Mathematical Institute, Oxford
LPHughston
King's College London
PZ Kobak
Instytut Matematyki, Uniwersytet Jagiellonski Krakow
K Pulverer
Center for Mathematical Sciences, Munich University of Technology,
Munich
(Editors)
Further advances
in twistor theory
Volume Ill: Curved twistor
spaces
Boca Raton London New York
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Preface
Twistor theory originated as an approach to the unification of q uantum theory
and general relativity by reformulating basic physics in terms of t he geometry
of twistor space. Although the basic physical aspirations remain in many re
spects unfulfilled, the t wistor c orrespondence and its many g eneralizations have
provided and continue to provide powerful mathematical tools for the study of
problems in such areas as differential geometry, nonlinear equations, and repre
sentation theory. At the same time, the theory continues to offer new insights
into the nature o f q uantum theory and gravitation. Some critics have suggested
that t wistor t heory has lost its w ay; our reply would be, yes, perhaps it has ... ,
but it has struck o ut i n many significant new and unexpected directions.
The p resent w ork, Volume Ill, is in fact the f ourth volume in a series of com
pilations o f articles f rom Twistor N ewsletter, a photocopied and p artly h andwrit
ten j ournal t hat h as been published from time t o t ime since 1976 by members of
Roger P enrose's r esearch group i n Oxford. Much of t he m aterial i n these articles
has not been published elsewhere. Those articles that have tend, in their earlier
incarnation reproduced here, to b e less formal and f ocus more o n the m otivation
and the key ideas, providing an easier entry into the subject. Thus it is hoped
that, to a certain extent, these volumes provide a review, albeit biased, patchy,
and preliminary, of t he advances in twistor theory over the l ast 20 years.
The zeroth volume, Advances in Twistor Theory, appeared in 1979 edited
by one of u s (LPH) and Richard Ward, and c overed all the m aterial available at
the t ime. The s ubsequent v olumes, Further Advances in T wistor Theory, divided
the m aterial i nto categories. The f irst volume, subtitled The Penrose Transform
and Its Applications, i s primarily concerned with linear aspects o f twistor t heory,
the Penrose transform of f ree fields and its various generalizations to fields on
homogeneous spaces and related topics concerning such fields on Minkowski
space. The second volume, subtitled Integrable Systems, Conformal Geometry
and Gravitation, is concerned with applications of f lat or homogeneous twistor
spaces to nonlinear problems; integrable systems of n onlinear equations on the
one hand, through conformal geometry, to quasi-local definitions of mass in
general relativity on the other. This third volume is concerned with deformed
twist or spaces and their applications.
Deformed twistor s pace constructions s tarted i n 1976 with Penrose's nonlin
ear graviton construction. This solved the anti-self-dual half of t he problem of
finding a twistor correspondence for curved vacuum (Ricci-flat) space-times. It
had the r emarkable f eature that s olutions to t he n on linear vacuum equations o n
space-time were encoded into the d eformed complex structure o n twistor s pace.
For l ocal solutions i n space-time, this d eformed complex s tructure o n the t wistor
space is specifiable in terms of f ree functions (see the introduction to Chapter
1 for more details). This volume is concerned with applications and extensions
of this construction motivated both by questions in differential geometry, and
the desire to f ind a twistor correspondence for general Ricci flat space-times in
pursuance of t he basic twist or programme.
Chapter one presents articles that give examples and development of the
theory of the original nonlinear graviton construction and extensions and ap
plications that a re motivated by questions in differential geometry. The o riginal
nonlinear graviton construction is briefly reviewed in the i ntroduction and fur
ther a rticles d evelop i ts t heory, provide e xamples a nd a re c oncerned with a pplica
tions to manifolds in Euclidean signature. The main extension of t he construc
tion considered in this chapter is to quaternionic manifolds in 4k-dimensions,
k > 1 , and various applications of t his construction are presented. This c hapter
is perhaps the m ost p atchy as the s ubject has been developed very s ubstantially
over the last 20 years by differential geometers who do not usually contribute
to Twistor Newsletter. To compensate for this deficit, the introduction to the
chapter recommends a number of s urvey articles.
Chapters 2 to 4 are c oncerned with different approaches to f inding a twistor
correspondence for space-times in four dimensions that a re not necessarily anti
self-dual.
Chapter 2 is devoted to articles on spaces of c omplex null geodesics in the
complexified space-times. LeBrun shows that these can be used to encode an
arbitrary conformal structure (in arbitrary dimension) into the deformation of
the c omplex structure o f t he s pace of n ull geodesics. The v arious articles in this
chapter i ntroduce the c onstruction and its various properties, and lead towards
the e ventual characterization of t hose that a rise from conformal structures c on
taining a vacuum (Ricci-flat) metric.
Chapter 3 is concerned with articles on hypersurface twistor s paces, twistor
spaces that can be defined in terms of t he first and second fundamental forms
of a hypersurface in a general 4-dimensional space-time. In the first instance
the construction is presented in the context of a complexified initial data s ur
face, but it also has a real version appropriate to space-times of Lorentzian
signature f or which the t wistor space is a 5-dimensional Cauchy-Riemann man
ifold rather than a complex 3-manifold. The articles focus on the structure of
hypersurface twistor spaces (for example, the Chern-Moser connection in the
Cauchy-Riemann case) and the formulation of t he vacuum Einstein constraint
and evolution equations in terms of s tructures o n the twistor spaces.
Although the c haracterization of t he c onformal vacuum equations f or spaces
of c omplex null geodesics is mathematically neat, i t e nds up being too u nwieldy
to b e a suitable candidate f or a full twistor description of a vacuum space-times
or for useful applications. Similarly, the fact that hypersurface twistor spaces
are tied to a hypersurface in space-time is a disadvantage if one is seeking a
fundamental twistor correspondence for a general vacuum space-time, although
it is perhaps an advantage if one wishes to address questions associated with
initial data. Chapter 4 is concerned with the various attempts to find a more
fundamental twistor c orrespondence for vacuum space-times. There are a num
ber of i ntriguing different approaches, and many of t hese relate to o ther areas.
However, we still await the d efinitive resolution of t his problem.
Each chapter has an introduction that s ets the scene for the topics in that
chapter, reviews the necessary background material and sets each article in its
context. Thus w e hope that t his book will be a suitable introduction and survey
of t he t opics of t he articles contained herein.
We would like to t hank R oger Penrose, for his continued encouragement and
support in this project.
-L.J. M ason, L.P. Hughston, P.Z. Kobak and K. Pulverer, October 2000.
Note o n c ross-referencing: W e refer to t he o riginal Advances i n Twistor The
ory (L.P. Hughston & R.S. Ward editors, Pitman R esearch Notes in Mathemat
ics, number 37, 1979) as volume 0, a nd by §0.5.1 w e mean Article 1 i n Chapter
5 of that volume. In the current Further Advances in Twistor Theory series,
the volume preceding the present book is Volume 11: Integmble Systems, Con
formal Geometry and Gmvitation (L.J. Mason, L.P. Hughston and P.Z. Kobak,
Longman, Pitman Research Notes in Mathematics Series, number 23), 1995).
By §1I.2.3 we mean Article 3 of Chapter 2 of that book. Similarly §L2.1 de
notes A rticle 1 o f Chapter 2 of Volume I: Applications of t he Penrose Transform
and §III.2.1 the c orresponding article in this volume. The c ontents o f t he e arlier
volumes appear after the contributor list.
Contents
Chapter 1: The nonlinear graviton and related constructions
111.1.1 The Nonlinear Graviton and Related Constructions by L.J. Mason 1
111.1.2 The Good Cut Equation Revisited by K.P. Too 9
III.1.3 Sparling-Tod Metric = Eguchi-Hanson by G. Bumett-Stuart 14
III.1.4 The W ave Equation Transfigured by C.R. LeBrun 17
111.1.5 Conformal Killing Vectors and Reduced Twistor Spaces
by P.E. Jones 20
111.1.6 An Alternative Interpretation of S ome Nonlinear Gravitons
by P.E. Jones 25
111.1.7 .n"-Space from a Different Direction
by C.N. Kozameh and E.T. Newman 29
III.1.8 Complex Quaternionic KKhler Manifolds by M.G. Eastwood 31
111.1.9 A.L.E. Gravitational Instantons and the Icosahedron
by P.B. Kronheimer 34
111.1.10 The Einstein Bundle of a Nonlinear Graviton
by M. G. Eastwood 36
111.1.11 Examples of A nti-Self-Dual Metrics by C.R. LeBrun 39
111.1.12 Some Quaternionically Equivalent Einstein Metrics
by A.F. Swann 45
111.1.13 On the T opology of Q uaternionic Manifolds by C.R. LeBrun 48
111.1.14 Homogeneity of T wistor S paces by A.F. Swann 50
111.1.15 The T opology of A nti-Self-Dual 4-Manifolds by C.R. LeBrun 53
111.1.16 Metrics with S.D. Weyl Tensor from Painleve-VI by K.P. Tod 59
111.1.17 Indefinite Conformally-A.S.D. Metrics on S2 X S2 by K.P. Tod 63
111.1.18 Cohomology of a Quaternionic Complex by R. Horan 66
III.1.19 Conform ally Invariant Differential Operators o n Spin Bundles
by M. G . Eastwood 72
111.1.20 A Twistorial Construction of { I, l}-Geodesic Maps by P.Z. Kobak 75
111.1.21 Exceptional Hyper-Kahler Reductions
by P.Z. Kobak and A .F. Swann 81
111.1.22 A Nonlinear Graviton from the Sine-Gordon Equation
by M. Dunajski 85
111.1.23 A Recursion Operator f or A.S.D. Vacuums and ZRM Fields
on A.S.D. Backgrounds by M. Dunajski and L.J. Mason 88
Chapter 2: Spaces of c omplex null geodesics
III.2.1 Introduction to S paces of C omplex Null Geodesics by L.J. Mason 97
111.2.2 Null Geodesics and Conformal Structures by C.R. LeBrun 102
