Dunajski M. Solitons, Instantons, and Twistors
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Solitons, Instantons, and
Twistors
Maciej Dunajski
Department of Applied Mathematics and Theoretical
Physics, University of Cambridge
1
3
Great Clarendon Street, Oxford ox2 6DP
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Preface
This book grew out of lecture courses I have given to mathematics Part II, Part
III, and graduate students in Cambridge between the years 2003 and 2008.
The first four chapters could form a basis of a one-term lecture course on
integrable systems covering the Arnold–Liouville theorem, inverse scattering
transform, Hamiltonian methods in soliton theory, and Lie point symmetries.
The additional, more advanced topics are covered in Chapters 7 and 8. They
include the anti-self-dual Yang–Mills equations, their symmetry reductions,
and twistor methods. Chapters 5, 6, and 9 provide material for an advanced
course on field theory where particular emphasis is paid to non-perturbative
solutions to classical field equations. We shall discuss scalar kinks, sigma-
model lumps, non-abelian magnetic monopoles in R
3
, instanton solutions
to pure Yang–Mills equations in R
4
, and finally gravitational instantons.
Although the material is entirely ‘classical’, the motivation comes from quan-
tum field theories, including gauge theories, where it is necessary to consider
solutions of the non-linear field equations that are topologically distinct from
the vacuum. Chapter 10 contains a discussion of the anti-self-dual conformal
structures, some of which have not been presented in literature before. This
chapter links with the rest of the book as the anti-self-dual conformal struc-
tures provide a unifying framework for studying the dispersionless integrable
systems. There are three appendices. The first two provide a mathematical
background in topology of manifolds and Lie groups (Appendix A), and
complex analysis required by twistor theory (Appendix B). Appendix C is self-
contained and can form a basis of a one-term lecture course on overdetermined
partial differential equations. This appendix gives an elementary introduction
to the subject known as the exterior differential system.
Although the term soliton – a localized non-singular lump of energy –
plays a central role in the whole book, its precise meaning changes as the
reader progresses through the chapters. In Chapters 1, 2, 3, and 8 solitons
arise as solutions to completely integrable equations. They are time-dependent
localized waves which scatter without emitting radiation, and owe their sta-
bility to the existence of infinitely many dynamically conserved currents. In
Chapters 5 and 6 the solitons are topological – they are characterized by
vi Preface
discrete homotopy invariants (usually in the form of Chern numbers) which
are conserved as the continuous field configurations have finite energy. Finally
in Chapter 9 the gravitational solitons arise as lifts of Riemannian gravitational
instantons to higher dimensions.
The book should be of use to advanced undergraduate and research stu-
dents, as well as experts in soliton theory who want to broaden their tech-
niques. It is aimed at both mathematicians and those physicists who are willing
to go beyond perturbation theory. The revived interest in twistor theory in
recent years can be largely attributed to Witten’s twistor-string theory [185]. It
is hoped that those researchers who come to twistor strings with the string
theory or quantum field theory (QFT) background will find this book an
accessible introduction to twistor theory.
There are some excellent text books which treat the material presented
here in great depth. Readers should consult [122] for inverse scattering trans-
form, [124] for the symmetry methods, and [114] for topological solitons.
The twistor approach to integrability is a subject of the monograph [118],
while [83, 132, 175] are books on twistors which concentrate on aspects of
the theory other than integrability. The full treatment of exterior differential
systems can be found in [23].
The twistor approach to integrability used in the second half of the book
has been developed over the last thirty years by the Oxford school of Sir Roger
Penrose with a particular input from Richard Ward. I am grateful to Sir Roger
for sharing his inspirational ideas with the rest of us. His original motivation
was to unify general relativity and quantum mechanics in a non-local theory
based on complex numbers. The application of twistor theory to integrability
has been an unexpected spin-off from the twistor programme.
While preparing the manuscript I have benefited from many valuable discus-
sions with my colleagues, collaborators, and research students. In particular I
would like to thank Robert Bryant, David Calderbank, Mike Eastwood, Jenya
Ferapontov, Gary Gibbons, Sean Hartnoll, Nigel Hitchin, Marcin Ka
´
zmier-
czak, Piotr Kosinski, Nick Manton, Lionel Mason, Vladimir Matveev, Pawe l
Nurowski, Roger Penrose, Prim Plansangkate, Maciej Przanowski, George
Sparling, David Stuart, Paul Tod, Simon West, and Nick Woodhouse. I am
especially grateful to Paul Tod for carefully reading the manuscript.
Finally, I thank my wife Asia and my sons Adam and Nico not least for
making me miss several submission deadlines. I dedicate this book to the three
of them with gratitude.
Cambridge Maciej Dunajski
January 2009
Contents
List of Figures xii
List of Abbreviations xiii
1 Integrability in classical mechanics 1
1.1 Hamiltonian formalism 1
1.2 Integrability and action–angle variables 4
1.3 Poisson structures 14
2 Soliton equations and the inverse scattering transform 20
2.1 The history of two examples 20
2.1.1 A physical derivation of KdV 21
2.1.2 Bäcklund transformations for the Sine-Gordon equation 24
2.2 Inverse scattering transform for KdV 25
2.2.1 Direct scattering 28
2.2.2 Properties of the scattering data 29
2.2.3 Inverse scattering 30
2.2.4 Lax formulation 31
2.2.5 Evolution of the scattering data 32
2.3 Reflectionless potentials and solitons 33
2.3.1 One-soliton solution 34
2.3.2 N-soliton solution 35
2.3.3 Two-soliton asymptotics 36
3 Hamiltonian formalism and zero-curvature representation 43
3.1 First integrals 43
3.2 Hamiltonian formalism 46
3.2.1 Bi-Hamiltonian systems 46
viii Contents
3.3 Zero-curvature representation 48
3.3.1 Riemann–Hilbert problem 50
3.3.2 Dressing method 52
3.3.3 From Lax representation to zero curvature 54
3.4 Hierarchies and finite-gap solutions 56
4 Lie symmetries and reductions 64
4.1 Lie groups and Lie algebras 64
4.2 Vector fields and one-parameter groups of
transformations 67
4.3 Symmetries of differential equations 71
4.3.1 How to find symmetries 74
4.3.2 Prolongation formulae 75
4.4 Painlevé equations 78
4.4.1 Painlevé test 82
5 Lagrangian formalism and field theory 85
5.1 A variational principle 85
5.1.1 Legendre transform 87
5.1.2 Symplectic structures 88
5.1.3 Solution space 89
5.2 Field theory 90
5.2.1 Solution space and the geodesic
approximation 92
5.3 Scalar kinks 93
5.3.1 Topology and Bogomolny equations 96
5.3.2 Higher dimensions and a scaling argument 98
5.3.3 Homotopy in field theory 99
5.4 Sigma model lumps 100
6 Gauge field theory 105
6.1 Gauge potential and Higgs field 106
6.1.1 Scaling argument 108
6.1.2 Principal bundles 109
6.2 Dirac monopole and flux quantization 110
6.2.1 Hopf fibration 112
6.3 Non-abelian monopoles 114
6.3.1 Topology of monopoles 115
6.3.2 Bogomolny–Prasad–Sommerfeld (BPS) limit 116
Contents ix
6.4 Yang–Mills equations and instantons 119
6.4.1 Chern and Chern–Simons forms 120
6.4.2 Minimal action solutions and the anti-self-duality
condition 122
6.4.3 Ansatz for ASD fields 123
6.4.4 Gradient flow and classical mechanics 124
7 Integrability of ASDYM and twistor theory 129
7.1 Lax pair 129
7.1.1 Geometric interpretation 132
7.2 Twistor correspondence 133
7.2.1 History and motivation 133
7.2.2 Spinor notation 137
7.2.3 Twistor space 139
7.2.4 Penrose–Ward correspondence 141
8 Symmetry reductions and the integrable chiral model 149
8.1 Reductions to integrable equations 149
8.2 Integrable chiral model 154
8.2.1 Soliton solutions 157
8.2.2 Lagrangian formulation 165
8.2.3 Energy quantization of time-dependent unitons 168
8.2.4 Moduli space dynamics 173
8.2.5 Mini-twistors 181
9 Gravitational instantons 191
9.1 Examples of gravitational instantons 191
9.2 Anti-self-duality in Riemannian geometry 195
9.2.1 Two-component spinors in Riemannian signature 198
9.3 Hyper-Kähler metrics 202
9.4 Multi-centred gravitational instantons 206
9.4.1 Belinskii–Gibbons–Page–Pope class 210
9.5 Other gravitational instantons 212
9.5.1 Compact gravitational instantons and K3 215
9.6 Einstein–Maxwell gravitational instantons 216
9.7 Kaluza–Klein monopoles 221
9.7.1 Kaluza–Klein solitons from Einstein–Maxwell
instantons 222
9.7.2 Solitons in higher dimensions 226