Integrability, self-duality, and twistor theory

Cover......Page 1
Title Page......Page 3
Copyright Page......Page 4
Preface......Page 5
Contents......Page 7
1.1 Examples of integrability......Page 11
1.2 Outline of the book......Page 17
Notes on Chapter 1......Page 19
Part I Reductions of the ASDYM......Page 21
2.1 Space-time......Page 23
2.2 Differential forms......Page 24
2.3 Conformal transformations and compactified space-time......Page 26
2.4 Bundles, connections, and curvature......Page 29
2.5 The Yang-Mills equations......Page 33
2.6 Gauge theories......Page 39
Notes on Chapter 2......Page 40
3.1 ASD electromagnetic fields......Page 42
3.2 Lax pairs......Page 43
3.3 Yang\'s equation and the K-matrix......Page 44
3.4 Lagrangians for the ASDYM equation......Page 46
3.5 The Hamiltonian formalism......Page 48
Notes on Chapter 3......Page 52
4.1 Classification of reductions......Page 53
4.2 Reductions of the linear ASD equation......Page 55
4.3 Conformal reduction in the non-Abelian case......Page 56
4.4 Invariant connections and Higgs fields......Page 57
4.5 The space of orbits......Page 59
4.6 Backlund transformations......Page 65
Notes on Chapter 4......Page 66
5.1 The Bogomolny equation......Page 69
5.2 Hyperbolic monopoles and other generalizations......Page 70
5.3 Reduction by a null translation......Page 73
Notes on Chapter 5......Page 76
6.1 Two-dimensional groups of conformal motions......Page 77
6.2 Reductions by H++......Page 78
6.3 Reduction by H+o......Page 83
6.4 Reduction by HSD......Page 92
6.6 The Ernst equation......Page 94
6.7 Reduction of Yang\'s equation......Page 99
6.8 Liouville\'s equation......Page 101
Notes on Chapter 6......Page 102
7.1 Abelian reduction to one-dimension......Page 105
7.2 Nahm\'s equations and tops......Page 108
7.3 The motion of an n-dimensional rigid body......Page 111
7.4 The Painleve equations......Page 112
7.5 Non-Abelian reductions......Page 118
Notes on Chapter 7......Page 119
8.1 The KdV flows......Page 121
8.2 The recursion operator for the ASDYM equation......Page 124
8.3 Hamiltonian formalism......Page 125
8.4 ASDYM and Bogomolny hierarchies......Page 128
8.5 Reductions of the ASDYM flows......Page 133
8.6 The generalized ASDYM equation......Page 137
Notes on Chapter 8......Page 142
Part II Twistor methods......Page 145
9.1 Projective spaces and flag manifolds......Page 147
9.2 Twistor space......Page 148
9.3 Birkhoff\'s factorization theorem......Page 155
9.4 Holomorphic vector bundles: the Cech description......Page 159
9.5 e-operators......Page 163
9.6 Cohomology......Page 165
9.7 The Grassmannian......Page 167
9.8 Scattering on the real line......Page 168
9.9 Spinors......Page 170
Notes on Chapter 9......Page 178
10.1 The concrete form of the Penrose-Ward transform......Page 181
10.2 The abstract form of the transform......Page 186
10.3 The Painleve property......Page 189
10.4 Global solutions in Euclidean signature......Page 190
10.5 Global solutions in ultrahyperbolic signature......Page 197
10.6 The GASDYM equation......Page 204
10.7 The truncated GASDYM hierarchy......Page 205
10.8 The linear Penrose transform......Page 206
 Notes on Chapter 10......Page 211
11 Reductions of the Penrose-Ward transform......Page 214
11.1 Symmetries of the twistor correspondence......Page 215
11.2 Symmetries of the twistor bundle......Page 216
11.3 Reduced twistor spaces......Page 221
11.4 The KdV and NLS equations......Page 228
11.5 The initial value problem and inverse scattering......Page 230
11.6 Isomonodromy and the Painlev6 equations......Page 241
11.7 The Schlesinger equation......Page 249
 Notes on Chapter 11......Page 251
12 Twistor construction of hierarchies......Page 254
12.1 Transformations of the patching matrix......Page 255
12.2 DS operators and the GASDYM hierarchy......Page 260
12.3 The twistor construction of the DS flows......Page 264
12.4 Explicit construction of solutions from twistor data......Page 274
12.5 Hamiltonian formalism......Page 279
12.6 The KP equation and the KP hierarchy......Page 284
 Notes on Chapter 12......Page 292
13.1 Self-duality in curved space-time......Page 294
13.2 The Levi-Civita connection......Page 296
13.3 Spinors and the correspondence space......Page 299
13.4 ASD conformal structures......Page 304
13.5 Curved twistor spaces......Page 310
13.6 Reductions......Page 315
13.7 ASDYM fields and the switch map......Page 317
 Notes on Chapter 13......Page 326
A Active and passive gauge transformations......Page 327
B The Drinfeld-Sokolov construction......Page 329
Notes on Appendix B......Page 336
C Poisson and symplectic structures......Page 338
Notes on Appendix C......Page 348
D Reductions of the ASDYM equation......Page 349
References......Page 353
A note on notation......Page 366
Index of notation......Page 367
Index......Page 369