Dunajski M. Solitons, Instantons, and Twistors
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356 Index
complexified Minkowski space, 129,
133, 136, 139, 145, 150, 182–184,
186–188, 230
complex manifold, 129, 135, 136, 139,
182, 203, 232, 233, 238, 266–267,
269, 272, 281–284, 300–303,
308–309
complex structure, 178, 202–203, 207,
211, 232–235, 249–250, 263–264,
268–269, 273–274
conformal Killing vector, 149, 246, 248,
253, 255, 258, 283
conformal structure, 198, 201, 229–286
conformal transformation, 136, 149,
246–247, 272
connection, 49, 54, 67, 87, 109, 112,
115, 119–122, 125–127, 129, 131,
141–142, 152, 155, 166, 182,
195–196, 199, 201, 203–204, 214,
230–232, 246–247, 251, 253–254,
268, 283, 310, 313, 323–328,
330–331
correspondence space, 140, 183, 266,
269–270, 275, 279, 282
curvature, 21, 43–60, 109–110, 113,
121, 126–127, 130–132, 151,
153, 168, 194, 196, 198, 201,
203, 205, 225, 227, 229, 231–247,
263, 265, 270–272, 277, 310,
323–331
Darboux theorem, 15, 47, 89
deformation theory, 274, 308–309
degree of a map, 294–299
Derrick argument, 99
Diff(S
1
) equation, 250–253, 283, 285
Dirac monopole, 109–115
dispersionless integrable systems,
256–262, 283
dispersionless Kadomtsev–Petviashvili
(dKP) equation, 250, 257–258,
261–262, 283
distribution, 110, 140, 183, 202, 231,
235, 247, 253–254, 257, 266,
268–269, 273, 282, 284, 318,
318n4, 319
double fibration, 140, 266–267, 269
dressing method, 50, 52–54, 147,
237
Einstein equations, 191, 193–194, 205,
215, 221, 237
Einstein–Maxwell equations, 216–226,
228
Einstein–Weyl (EW) space, 246
exterior differential system, 207, 311,
314–320
Frobenius theorem, 130, 230–231, 233,
239, 247, 253, 312, 314–320, 323,
332, 335–336
fundamental solution, 50, 52, 124, 131,
157
gauge
field, 100, 105–127, 141, 152
group, 109, 114–115, 141, 143, 146,
150, 152, 155, 231, 251
potential, 105–111, 124
transformation, 50, 54, 105, 107, 109,
111–112, 115, 117, 119–121, 124,
144, 150–151, 155–156, 177, 184,
222, 281
Gelfand–Levitan–Marchenko (GLM)
equation, 34, 35, 40
generalised monopole equation, 248–249
Gibbons, Gary, 12n3, 194
Gibbons–Hawking ansatz, 206, 208
gradient flow, 124–127
Grassmannian, 338, 341
gravitational instantons
asymptotically locally euclidean (ALE),
193–194, 209–210, 213, 218–219
asymptotically locally flat (ALF), 194,
210, 214, 218, 219
compact, 215–216
Eguchi–Hanson metric, 192–193, 195,
209–210, 212
Euclidean Schwarzchild metric, 191,
194
Taub–NUT, 193–194, 198, 208–209,
221, 226
Hamiltonian formalism, 1–4, 14, 16,
43–60, 85, 88, 287
Hamiltonian system, 4, 10, 18, 56, 240
harmonic function, 147, 206, 208,
211–212, 218–219, 221, 224, 227,
316
heavenly equations
first, 238, 242–244, 281
hierarchy, 242–244
recursion operator, 240–242
second, 239–242, 244, 258, 275–277,
279–280, 282, 325
Higgs field, 100, 106–110, 115, 118,
150, 152–153, 155, 184, 185, 252
higher dimensions, 98–99, 226–227
Index 357
Hitchin, Nigel, 214, 283–284
Hodge operator, 117, 119, 126, 132,
155, 197, 206, 227, 316
homogeneous function, 305–306
homotopy, 99–100, 102, 112, 119, 122,
164n2, 168–170, 172–173,
175–176, 287, 294–299
homotopy group, 100, 102, 122, 172,
287, 297–298
Hopf fibration, 112–114
horizontal distribution, 110
hyper–complex structure, 232, 233
hyper–Hermitian structure, 232–234, 270
hyper–Kähler metric, 202–206, 210–211,
213, 229, 238, 249, 251, 263–264,
272
instanton, 109, 119–127, 129, 137,
143, 147, 175–176, 191–227,
229, 272
integrable chiral model, 149–189
integrable system, 1, 4–6, 43, 48, 65n2,
85, 106, 130, 137, 149, 229, 231,
232, 248, 252, 256–262, 283, 309
integral element, 336–342
integral flag, 338
integral manifold, 311–319, 332–340,
342
inverse scattering, 20–40, 57, 93, 129
involutivity, 320–324
Jacobi identity, 2, 3, 14–15, 17, 47, 57,
66, 69, 88, 131, 290
jet space, 74–75, 320, 324, 332, 334, 342
JNR ansatz, 124
John transform, 133–134
Jones–Tod construction, 249–250
K3, 216, 229, 263
Kähler
form, 172, 176, 203–207, 248, 265,
270–271, 299, 316
manifold, 203
Kaluza–Klein theory, 221
Kepler problem, 11
Killing vector, 149, 155, 206–207, 211,
214, 216, 219, 220, 222, 225, 246,
248–250, 253, 255–256, 258, 278,
283, 313, 315–316, 320, 326–328
kink, 24, 25, 72, 93–100, 104, 109
Korteweg–de Vries (KdV)
equation, 18, 20, 22–23, 27, 32,
43–44, 46, 48, 56, 58–59, 61, 71,
73–74, 82–83, 151–152
hierarchy, 56–57, 59, 62, 243
Lax pair, 41, 55–56
modified, 41, 82–83, 246
Lagrangian, 85–103, 105, 114–115, 126,
155, 157, 165–168, 174, 222, 228,
244–245, 339
Large Hadron Collider (LHC), 26n2,
64n1, 106
Lax pair
ASD conformal structure, 231, 255
ASD Null Kähler, 234–237, 239–240,
250, 271
ASDYM, 129–133
heavenly, 238–245
integrable chiral model, 154, 156–157,
161, 164, 181–182, 184, 185
KdV, 27, 32, 56
NLS, 49, 252–253
Sine-Gordon, 49
Legendre transform, 87–88, 258,
277–278
Levi-Civita connection, 87, 195, 199,
203, 230, 247, 268, 313, 325,
329–330
Lie algebra, 6n1, 64–67, 70–71, 77–78,
105, 107, 115, 118, 123, 155, 180,
251, 252, 290–292, 326
Lie group, 290–293, 298
Lie point symmetry, 72–73, 76–77, 82
Lie, Sophus, 65
linear system, 35, 49–50, 52, 63, 74,
130–132, 158, 236, 242–243, 256,
267, 325, 331
line bundle, 114, 146, 182, 266, 270,
273, 278, 303–306
Liouville theorem, 5, 8n2, 53, 58–59,
142, 273, 276, 289, 300, 306
lumps, 93, 100–103, 159, 178–180
magnetic field, 110, 116–117, 173,
176–181, 217, 292
Manakov–Santini
system, 259
manifold 287–289
complex, 232–233, 238, 266–267,
269, 272, 281–284, 300–303,
308
Riemannian, 193–197, 203–204,
325
smooth, 88, 289, 314, 333, 338n8
Manton, Nicholas, 92, 173
Mason, Lionel, 152
Maximum Modulus theorem, 300–301,
303
358 Index
metric
flat, 90, 119, 129, 191–194, 205–209,
211, 215–216, 221, 227, 240–241,
243, 251, 256, 258, 263–265,
272–282
Fubini–Study, 215, 299
Hermitian, 203, 203n3, 207, 232–233
hyper–Hermitian, 229, 232–234, 238,
250–253, 270
Kähler, 202–206, 210–211, 213, 215,
229, 235–236, 238, 248–251,
263–265, 270, 272, 315, 343
moduli space, 176–180
Ooguri-Vafa, 264
Sparling–Tod, 240
mini–twistors
compactified, 185–188
Ward correspondence, 141–147
Miura transformation, 41
moduli space, 92, 173–181, 183, 214,
267, 277, 309
monopole, 93, 107, 109–118, 155, 193,
206, 214, 221–227, 248–251, 257,
264, 278
M–theory, 226
Nahm equations, 153
Newlander–Nirenberg theorem, 202
Noether charges, 91, 167–168
Noether theorem, 3, 45, 92
non–Abelian monopole, 107, 109,
114–118, 155
Nonlinear Schrödinger equation (NLS),
49, 189, 252, 253
normal bundle, 183, 266–267, 270,
272–273, 277, 282–284, 308–309
null plane, 132, 181–182, 187, 188
null surface, 283
null tetrad, 198–199, 219–220, 228, 230,
233–234, 238, 241, 281
null vector, 139, 200, 246–247, 250, 269,
273, 277, 283
NUT singularity, 193, 194, 215
Painlevé equations, 78–83
Painlevé property (PP), 79
Painlevé test, 82–83
Penrose, Roger, 106, 133–134, 139, 141,
145, 146, 148, 231, 267, 272,
277
Penrose transform, 134, 145
Penrose–Ward correspondence, 141–147
Pfaff theorem, 333–334
Picard theorem, 310–312
α-plane, 139–141, 230–231, 254
Pleba
´
nski equations, 237–239, 242,
265
Poisson bracket, 1, 3, 5, 14–15, 17,
41, 46–47, 88–89, 243, 256, 293
Poisson structure, 14–18, 46–48, 88,
243, 245, 282
polar space, 337, 340, 342
projective space, 136, 172, 176, 215,
298, 302
projective structure
metrisability, 330–331
prolongation, 74–78, 313, 321, 324–332
Przanowski equation, 237
quadric ansatz, 259–262
quantum mechanics, 25, 133
rational map, 103, 173–175, 181
reality conditions
Euclidean, 140, 147, 150, 151
Lorentzian, 251
neutral, 151, 248, 250
recursion operator, 47–48, 240–242,
244–245, 279–282
reflectionless potentials, 33–40
Ricci scalar, 196, 198, 201, 222, 236
Ricci tensor, 196, 201, 229, 247
Riemann–Hilbert problem, 50–51,
53–54, 143, 229
Riemann sphere, 113, 135, 168, 269,
302, 305
Riemann surface, 58–60, 101, 264–265
Riemann tensor, 192–193, 196–198, 201,
216, 229, 327
Russel, Scott, 20
Schrödinger equation, 26–29, 31–32, 35,
40, 42, 44, 49, 125
section
holomorphic, 139, 142, 183, 267,
304–305, 307
self–dual (SD), 108, 122–123, 130,
132–133, 140, 197–198, 200,
204–206, 211, 214–215, 219, 229,
231, 236–238, 258, 272–273, 275,
278, 281
sheaf cohomology, 134
sigma model, 99–103, 168
Skyrme model, 99
soliton, 4, 20–40, 49, 73–74, 93–94,
98–99,
102, 106–107, 109, 112,
Index 359
115, 137, 154–155, 157–165,
167–168, 173, 179–180, 221–227,
229, 256–257, 297
Sparling, George, 152, 240
spectral parameter, 48, 52, 131–132,
149, 161, 168, 232, 258
spin bundle, 182, 199, 230, 231, 233,
236, 266, 268, 275, 278, 280
spinor, 137–139, 143–144, 148, 182,
182n6, 185, 198–202, 204–205,
219–221, 230–231, 234, 236–237,
253–254, 263, 266, 268–271, 273,
275, 280, 306
splitting formula, 308
α-surface, 230–231
suspension, 162, 163, 169, 173, 295, 296
SU(∞)–Toda equation, 248–249, 261,
283
symmetry
hidden, 243, 281–282
null, 152, 255
point, 72–73, 76–77, 82
symplectic form, 88, 138, 182, 244–245,
265
t’Hooft ansatz, 124
Tod, Paul, 219, 240, 248–250, 259, 264
torsion, 166–167, 195–196, 202, 246,
254, 330
trivial scattering, 161–165, 169, 185
twistor, 106, 114, 129–147, 149,
181–189, 237, 257, 265–285, 300,
308
twistor function, 140, 145, 266, 279–282
twistor space, 129, 134, 136, 139–143,
145, 148, 182–183, 185–188, 230,
265–270, 272–275, 278–280,
282–284, 309
Tzitzeica equation, 84, 153
Uhlenbeck, Karen, 101n1, 119
unitons, 163–165, 164n2, 166, 168–170,
172, 173
vector field, 67–71, 70n4, 72–75, 77–78,
88–89, 110, 141, 166, 182, 195,
199, 202, 206, 214, 230–231,
233–235, 238–240, 242–243,
246–248, 251–254, 257, 259, 266,
268–269, 274, 278, 281, 283–284,
289–293, 302, 306, 311, 314,
317–319, 334
wave equation, 21–22, 134, 145, 147,
185, 240–242, 245, 264, 277, 279
wave geometry, 205
Wess–Zumino–Witten (WZW), 165
Weyl curvature, 198, 265, 277
Weyl spinor, 228, 231, 237, 280
Witten, Edward, 165
Woodhouse, Nick, 88
Yang–Mills equations, 119–127
Yang–Mills–Higgs system, 155, 185
zero curvature representation, 43–60,
130–131, 151
zero–rest–mass equations, 145, 147