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Gaussian Grassmann integrals 129
Euler“Lagrange equations 106
General linar groups 341
Evolution operator 114, 135, 154, 155
General relativity 285“289, 300, 308,
311, 312, 314, 336, 338, 339
Faddeev“Popov determinant 52
Generating functional 52“54, 77, 78, 92
Faithful representations 343
Generators of groups 136, 142, 143, 145,
Fermi“Bose doubling 247, 248
156, 159
Fermion determinant 176
Geodesic equation 287
Fermion doubling problem 169, 182,
Georgi“Glashow model 39, 43, 45, 47, 93
184, 216, 222, 234
Ghost ¬eld action 178
Fermion number 252, 254
Ghost ¬elds 176, 178
Fermions, euclidean 217
Ghost permutation operator 144
Field theories, bosonic 181
Ghost terms 177
Field-strength tensor 104
Ghost-number operator 137
First class constraints 112
Ghosts 130, 132, 133, 137, 143, 144, 146,
Flux tubes 237, 238
148, 149, 157, 159, 161
Fokker“Planck equations 222
Ginsparg“Wilson relation 169, 170, 184,
Formula, the Russian 162
216, 217, 221, 235
Fundamental group 21, 23, 66
Ginzburg“Landau model 9, 16
Fundamental representations 343
Ginzburg“Landau parameter 17
Glueball mass 75
Gauß law 42, 43, 205
Gluons 297“299, 332, 334“336
Gauge condition 69
Goldstone bosons 211
Gauge background 167“169, 175, 177,
Goldstone ¬elds 263
217, 218, 222, 234
Golfand“Likhtman superalgebra 257,
Gauge condition 12, 51, 52, 54, 67, 70,
259
92
Grand Uni¬ed Theory 328
Gauge copy 34, 51
Grassmann algebra 127, 128
Gauge couplings 291, 296, 297, 335, 337
Grassmann di¬erentiation 129
Gauge ¬xing 51, 76, 118
Gauge group 45 Grassmann integration 129
354 Index

Grassmann sources 175 Irreducibility 343
Grassmann variables 127“129, 143 Isometries 266
Graviton 302 Isotopic invariance 278
Green™s functions 154 Isotropy group 33, 44, 46, 50, 68, 71
Gribov horizon 53, 80
Jackiw“Rebbi phenomenon 254
Group representations 342
Jacobi identity 10, 40, 111, 113, 131,
Haar measure 78, 81, 83 133, 134, 136, 143, 145
Hamilton™s equations of motion 109 Jacobian, graded 155
Hamiltonian density 12, 42, 58 Jones“Witten invariant 64
Hamiltonian formalism 13 Julia“Zee dyon 49
Heat kernel expansion 313, 314, 321
Kaluza“Klein model 316
Hedgehog 36, 48, 55
Killing metric 143, 145
Heisenberg™s uncertainty relation 304,
Kink mass 243, 247, 250, 269“272
336
Kink, classical 239, 245“247
Helical ¬‚ow 63
Kinks 237, 238, 242, 244, 254, 258, 263,
Hierarchy problem 281
264, 270
Higgs boson 281, 295, 319, 339, 340
Kinks, critical 242
Higgs couplings 291, 296, 297, 301
Klein“Gordon action 295
Higgs ¬eld 9
Higgs mass 46
Lagrange multipliers 115, 116, 119, 124,
Higgs mechanism 314
150, 151
Higgs phase 13, 45
Landau orbit 84
Higgs potential 9, 10, 15, 19, 46, 49, 68,
Landau“Ginzburg models 268
92, 291, 295, 301, 327, 337
Langevin equations 222
Higher homotopy group 23
Hodge —-operator 144, 292 Large gauge transformation 55, 67
Legendre transformation 115, 116, 124,
Hodge duality 144
149
Holomorphic vectors 204
Lepto-quarks 336
Homogeneous space 33, 44
Levi“Civita connection 287, 312
Homotopic equivalence 20, 23
Lich´rowicz formula 321
e
Homotopic map 20
Lie algebra 108, 116, 134, 143, 342
Homotopy 20
Lie algebra, compact 122
Homotopy classes 20, 28, 203, 208, 210
Lie group 30, 341
Homotopy group 48, 51, 203, 208
Lie-algebra cohomology 143
Hopf algebra 338, 349
Light-cone co-ordinates 126
Hopf invariant 25, 92
Line defect 34
Hyperbolic defect 37
Link invariant 62
Hypercharge 297, 299, 322, 333, 335
Link variables 182
Index of the Dirac operator 197 Linking number 7, 28, 63
Instanton gas 60 Liquid crystal 27
Instantons 55, 58, 87, 88, 198, 199, 201, Little group 44, 291, 314
204“207, 210, 226, 234, 235, 273 London depth 14
Internal connection 318 Loop 21
Invariant subgroup 31 Lorentz gauge 53, 88
Inverse melting 75 Lorentz group 341
Inverse Noether theorem 110
Involution 304“306, 317, 323, 335, 345 Magnetic charge 7, 19, 48, 88
Index 355

Magnetic ¬‚ux 15, 48, 62, 73, 82 O(d) 180
Magnetic helicity 63 O(N) 341
Majorana representation 238, 263 Octonions 324
Majorana spinor 238, 240, 263, 280 On shell quantities 107
Master equation 152 Orbifold 34
Matrix groups 341 Orbit 33
Maxwell equations 10, 99, 102 Order parameter 14, 34
Maxwell Lagrangian 292 Ordered media 34
Maxwell“London equation 14 Ordering problem 169
Meissner E¬ect 13 Orientability 316
Meron 61 Orthogonal group 341
Metric tensor 288 Orthogonal transformation 33
Metric, ¬‚uctuating 333, 334 Overlap fermions 170, 184, 221
Minimal coupling 294“296, 312
Parallel transport 193
Minimal supersymmetric standard model
Parity symmetry 179
280
Parity transformation 293
Minkowskian geometry 285
Parity violation 293, 295, 301, 335
Modulus 263, 268, 270
Path ordered integral 41
Modulus, translational 263
Path-integral quantization 154
Momentum, canonical generalized 106
Pauli-matrix 29
Monopole 34, 36, 48, 51, 86, 88
Penetration length 14, 17, 18
Monopole charge 35
Periodic potentials 199
Monopole, magnetic 238, 270
Phase transition 46, 86
Morse theory 244, 245, 283
Photino 278, 280
Moyal plane 338
Photon mass 13, 17
Multiplet shortening 250, 253, 274
Planck time 336
Nematic liquid crystal 27, 35, 73 Planck™s law 76
Neutrino oscillations 338 Plaquette action 182
Neutrinos 298, 331, 338 Plasma phase 47, 64, 71, 75, 83, 90
Neutrons 308, 312, 346 Poincar´ duality 306, 316, 331
e
Nielsen“Olesen vortex 9, 19, 55, 73 Poincar´ group 116, 344
e
Noether charge 146, 147 Point defect 23, 34
Noether™s theorems 105 Poisson brackets 110, 111, 116, 118,
Non-abelian gauge theories 177 120“122, 125, 130, 135, 146, 151, 158,
Non-abelian Higgs model 29, 39, 43, 45, 166
73, 92 Polyakov action 124
Non-linear σ Model 205 Polyakov loop 69“71, 74, 75, 77“79, 81,
Non-linear σ model 172, 178, 179, 181, 82, 87, 89“91
205 Prasad“Sommer¬eld monopole 49
Non-renormalization theorem 260, 261 Projective space 27
Noncommutative geometry 286, 303, Punctured plane 35
304, 307, 309, 316, 327“329, 335, 336, Pure gauge 11, 15, 41, 55, 59, 60, 71, 72
338“340
QCD Lagrangian 39
Nonzero mode 245, 247“249
Quantum Chromodynamics (QCD) 39,
Normal subgroup 31
292
O(1,3) 341 Quantum Electrodynamics (QED) 292,
O(3) 264, 275“277 294
356 Index

Quantum Hall e¬ect 338 Self interactions 158, 177
Quantum phase transition 47, 65 Self-duality equations 209
Quasiclassical approximation 244, 245 Selfdual 58
Quasiclassical quantization 273 Semi-direct product 344
Quaternion 37, 315, 317, 323, 325, 326, Short representations 260
333, 335 Simply connected 22
Quotient Space 26
Singular gauge 60
SO(10) 329
Radiative corrections 271
SO(3) 31, 206, 207, 299, 308, 342
Receptacle group 317, 318
SO(4) 208, 309
Reconstruction theorem 306, 316
SO(5) 314
Redundant variable 11, 34, 46, 51, 54,
SO(6) 215
71, 76, 88
SO(N) 341
Regularity 306, 316
Solitons 226, 228, 235, 237
Regularization, dimensional 152, 168,
170, 178, 179, 188, 233, 251 Source terms 172
Regularization, infrared 245 Special orthogonal group 341
Regularization, lattice 167“169, 175, Spectral action 311
179“183, 233 Spectral ¬‚ow 60
Regularization, mode 197, 198 Spectral triple 306“309, 311, 315“318,
Regularization, momentum cut-o¬ 168, 322“324, 327“329, 331, 332, 334, 338,
170, 188, 233 339
Regularization, Pauli“Villars 171, 180
Sphere S n 20
Regularization, Point-splitting 192
Spin connection 312
Regularization, ultraviolet 250
Spin groups 308
Regulator ¬elds 168, 173, 174, 178, 186,
Spin system 21
191
Spin“statistics connection 168, 173
Renormalization 167, 168, 170, 233
Spontaneous orientation 36, 44
Reparametrization invariance 101, 115,
Spontaneous symmetry breakdown 32,
118, 125, 126, 156
44, 71, 73, 76
Residual gauge symmetry 43, 45, 64, 66,
SQED 280
68, 71
Stability group 44
Resolvent 224
Ricci tensor 264, 288, 302 Staggered fermions 184, 234
Riemann tensor 288 Standard model 329
Riemann sphere 202 Stefan“Boltzmann law 65, 85
Riemannian geometry 285“288, 305, Stereographic mapping 206
316, 323, 338“340 Stereographic projection 264
Riemannian spin geometry 305 Stokes theorem 207
Rydberg constant 289
String breaking 83
String tension 75, 83
Scalar ¬elds 170, 173, 174, 181
String theory 158
Schr¨dinger equation 114, 294
o
String, relativistic 124, 127, 135
Schwarzschild horizon 336
Strings 238, 282

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