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Nash C. — Differential Topology and Quantum Field Theory

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Название: Differential Topology and Quantum Field Theory

Автор: Nash C.

Аннотация:

The remarkable developments in diferential topology and how these recent advances have been applied as a primary research tool on quantum field theory are presented in a style reflecting the genuinely two-sided interaction between mathematical physics and applied mathematics. The author, following on from his previous work (Nash/Sen: Differential Topology for Physicists, Academic Press, 1983), covers ellipitc differential and pseudo-differential operators, Atiyah-singer Index theory, Morse theory, instanntons and monopoles, topological quantum field theory, string theory and knot theory. The explanatory approach serves to illuminate and clarify these theories for graduate students and research workers entering the field for the first time.

Язык:

Рубрика: Математика/Математическая Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 386

Добавлена в каталог: 24.04.2005

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Предметный указатель

Gauge fixing      218 349
Gauge orbits      218 271 323
Gauge system      318
Gauge transformations      175 177 216—217 234 257—258 269 281 283 297
Gauge transformations, and ellipticity      230
Gauge transformations, generated by the Higgs field      252
Gauss — Bonnet theorem      104 134 152 163
Gauss’s law      297
Germs      58 (see also sheaves)
Ghosts      218 347
Global anomaly      270 286—290 294
Gradient flow      212 327—329
Grassmannian      71—73
Gravitational anomaly      280 (see also anomalies)
Green’s function      28—30 33 34 344
Gribov ambiguity      219
Group extensions      178—185
Group extensions, central      179—183
Group extensions, non-central      183—185
H-space      8—10 12
Hamiltonian and anomalies      291—300
Handle      1 139 143 156
Harmonic forms      148
harmonic oscillator      211
Harmonic spinor      115
Heat equation      33—35
Heat equation, and the index theorem      132
Height function      196—204
hessian      193 197 199 212
Hessian, and the Yang — Mills action      258
Hessian, unbounded and Floer theory      325—326
Higgs field      248 252—253 257
Highest weight representation      186—191
Highest weight representation, and conformal field theory      309
Hodge numbers      108 148
Hodge theory      93 110 115 147—148 164 332 350
Holomorphic sections      58 139 142 146
Holomorphic sections, and conformal field theory      317
Holomorphic sections, and sheaves      58—59
Holomorphic sections, and the Borel-Weil theorem      189—191
Holomorphic sections, and the Weierstrass gap theorem      166—168
Holonomy group      218 232
Homeomorphisms      2 19 340
Homfly polynomial      341 360
Homology sphere      337—338 (see also homology 3-sphere)
Homology, 3—sphere      323 325 329—330 336—338
Homotopy classes      5—10 16 21 72 88 90
Homotopy equivalence      4 12 221—222 293
Homotopy groups      3 7 9—14
Homotopy groups, and Fredholm operators      294
Homotopy groups, and gauge transformations      219—222
Homotopy groups, and K-theory      72—75
Homotopy groups, and obstruction theory      21—22
Homotopy lifting property      17—18
Homotopy quotient      202—203
Homotopy type      4—5 12 219
Homotopy type, and Bott periodicity      76
Homotopy type, and fibrations      18
Homotopy type, and Fredholm operators      293—294
Homotopy type, and Morse theory      196
Homotopy type, and spectral flow      296
Homotopy type, of $S^{4}$       24
Hopf algebra      222
Hurewicz isomorphism      14
Huyghen’s principle      30
Hyperbolic equations      31
Hyperbolic monopoles      see Monopoles
Hyperbolic plane      151 156
Hyperbolic space      247
Hyperbolic space, and hyperbolic monopoles      247—249
Hyperbolic space, and instanton moduli      241
Hyperk $\ddot{a}$ hler manifold      255
Hypo-elliptic operators      33
Hypo-ellipticity      32—35
Hypo-ellipticity, and supports of pseudo-differential operators      47
Index bundle      293—294
Index of elliptic operators      41 89—136
Index of elliptic operators, acting on functions      100—101
Index of elliptic operators, analytical      see analytical index
Index of elliptic operators, and families of operators      118—122
Index of elliptic operators, and fixed point theory      122—127
Index of elliptic operators, and manifolds with boundary      127—136 288
Index of elliptic operators, and odd dimensional manifolds      98—99
Index of elliptic operators, and the Gauss — Bonnet theorem      104—106
Index of elliptic operators, and the Hirzebruch signature theorem      112
Index of elliptic operators, and the Lefschetz fixed point theorem      123—126
Index of elliptic operators, and the Riemann — Roch theorem      108
Index of elliptic operators, cohomological formula for      98 103 237
Index of elliptic operators, the operator      103 117
Index of elliptic operators, the $\bar{\partial}$ operator      106—108 117
Index of elliptic operators, the Dirac operator      112—116 118 270
Index of elliptic operators, the signature operator      108—112 118
Index of elliptic operators, topological      see Topological index
Index of elliptic operators, twisted operators      116—118
Index theorem of Atiyah and Singer      97
Inductive limit      71
Infinite dimensional groups      170—191
Infinite dimensional groups, the diffeomorphism group      174
Infinite dimensional groups, the group       171—174
Infinite dimensional groups, the group of gauge transformations      175—177
Infinite dimensional groups, the loop group      174—175
Infinite dimensional groups, the Virasoro group      174
Infinite dimensional manifolds      170—171 178 323
Instanton number      230
instantons      25—26 213—215 227—242 258
Instantons, and Donaldson invariants      332—339
Instantons, and Floer homology      327—330
Instantons, and Morse theory      213—215 258
Instantons, axially symmetric      246—250 253
Instantons, moduli space of      227—242
Instantons, time independent      245—246
Integrable system      173
Integral curve      159 174 212
Intersection form      23—25 108 240 338
Intersection number      143
Invariant polynomial      223 226
Inverse limit      19
Ising model      312—313
Isothermal coordinates      157
Isotopy      339—341
Jacobian variety      143
Jones polynomial      341—342
Jones polynomial, and Chern — Simons theory      343 350—359
K $\ddot{a}$ hler manifold      144—147 189 291 350
K $\ddot{a}$ hler metric      147 255—
K-theory      11 65—75
K-theory, and anomalies      275
K-theory, and Bott periodicity      75—78
K-theory, and Fredholm operators      88 293—294
K-theory, and index theory      89—91 95—96 101—102 119-126
K-theory, and the Chern character      81
Kac — Moody algebra      173 177—178 180—183 187—188 311 317
Kac — Moody algebra, and current algebra      177 178
Kac — Moody algebra, Casimir operator for      181—183
Kac — Moody algebra, representations of      187—188
Kernel      40—41 46 54 69 89 92 122 132—133 162—165 176 179 230 236 252 276 281 292 297—298
Kernel, and Fock space      297—298
Kernel, and gauge transformations      176
Kernel, for a real elliptic family      122
Kernel, Gaussian      46
Kernel, of a group extension      179
Kernel, of an elliptic operator      89 92 122 132—133
Kernel, of Laplacians      40—41 92
Kernel, of the map       69
Knot invariants      339—340
Knot invariants, Alexander polynomial      340—342
Knot invariants, Homfly polynomial      341 360
Knot invariants, Jones polynomial      341—343 350—359
Knot invariants, knot group      340
Knots      2 338—360

Knots, and Chern-Simons theory      342—360
Knots, and skein relation      342 351 353—355
Knots, mirror image of      341
Lagrange multiplier      347
Laplacian      27—28 39—41 46 92—93 127—128 148 163—164 209 236 260 282 332 349-350
Laplacian, and Hodge theory      92—93
Laplacian, and Ray — Singer torsion      349—350
Laplacian, on a K $\ddot{a}$ hler manifold      148
Laurent polynomial      see $L$

-polynomial
Leading symbol      33—39 42 54 93 99—100 273
Lefschetz fixed point theorem      123—126
Level, curve      29 31
Level, of a representation      188 311 317 359
Level, set      29
Level, surface      195
Lexicographic ordering      291
Lie derivative      159—160 264
Lie group      9 11 73 75 125 170—171 174 178 180 185 188 190 192 201 207 221 226—227 241
Lie group, and instanton moduli      241
Lie group, and rational homotopy      221
Lie group, infinite dimensional      171—191
Light-cone      30
links      206 205 256 339 341—345 355—356
Liouville’s theorem      58
Local coefficients      21
Local cohomology      278 280
Locally convex      171
Loop group      173—174 181—183 191
Loop group, representations of      187—188
Lusternik — Schnirelmann category      198
M $\ddot{o}$ bius group      155 165 310—311 315
Magnetic charge      245 248—249 257
Magnetic monopole      see Monopole
Mapping class group      152 155—156
Minima      25 229—230 244 256 258 336
Minima, and Yang — Mills action      25 229—230 258 336
Minima, and Yang — Mills — Higgs energy      244 256
Modular form      161 312
Modular functor      321
Modular group      155 161 170 317
Modular invariance      313 321
Modular transformation      155 311 359
Moduli space      26 139 152—165
Moduli space, instanton      26 227—242 333—339
Moduli space, monopole      250—256
Moduli space, of flat connections      350 360
Moduli space, Riemann      139 152—165 262 268 315
Moduli space, universal Riemann      318—319
Monopoles      242—250
Monopoles, and Morse theory      256—257
Monopoles, and rational functions      253
Monopoles, hyperbolic      246—250
Monopoles, moduli space of      250—256
Monopoles, Prasad — Sommerfield      243 252—253
Monopoles, scattering of      250 253—256
Morse theory      192—215
Morse theory, and Floer theory      322 324—326 330
Morse theory, and gauge theories      256—258
Morse theory, and the Morse inequalities      194—195 200 203 211—212 215
Morse theory, and the Morse series      193 197 199—200 203—205
Morse theory, equivariant      201—205
Morse theory, for critical sub-manifolds      199—201
Morse theory, via quantum mechanics      207—215
Multi-index notation      32
Neumann problem      128
Normal bundle      66 94 199
Nullity      193
Obstruction theory      21—23 26
Operator product expansion      301—310 314—315
Operator products      181
Orbifold      152 156
Parabolic equation      31 33—34
Parallel transport      216—218
Parametrix      55
Partition function      218 260—268 271 275
Partition function, and anomalies      271—275 278
Partition function, and axiomatic conformal field theory      318—321
Partition function, and Donaldson invariants      337
Partition function, and the Ising model      311 313
Partition function, for strings      260—268
Partition function, for the Chern-Simons action      343—351 357
Partition of unity      48 61
Path connected      11 14 218
Path ordering      343—344
Perfect Morse function      195
Period matrix      143—144
Peter — Weyl theorem      190
Pfaffian      86
Phase space      350
Phase transition      301
Picard group      146
Piecewise-linear manifold      see PL manifold
PL manifold      2 20—22
Poincar $\acute{e}$ conjecture      24 192
Poincar $\acute{e}$ duality      23 144 238 337
Poincar $\acute{e}$ polynomial      199 323
Poincar $\acute{e}$ polynomial, upper-half-plane      151
Pointed space      5—6 9
Polyakov string formulation      259
Polygonal knot      340
Pontrjagin class      16 84—85 98 100—101 116 271
Positive line bundle      163—164 (see also Vanishing theorems)
Prasad-Sommerfield monopole      243 252—253
Primary field      302—317
Primary field, and knots      355—356
Principal bundle      15 33 175 219 281
Product metric      135
Projective connection      320
Projective limit      19
Projective representation      185—186
Pseudo-differential operators      41—44 89 91 94 131
Pseudo-differential operators, and Sobolev spaces      47—55
Pseudo-differential operators, on odd dimensional manifolds      99
Pseudo-locality      47
Quadratic differential      161—162
Quantised mass      227
Quaternionic projective space      323
Rank of a vector bundle      15 36 38 42—43 65 70—74 79—87 100 105 116 138—139 145—146 187 199 203 239 248 276
Rank of a vector bundle, and the stable range      72 74
Rank of a vector bundle, and virtual dimension      68—69
Rarita — Schwinger operator      289
Rational cohomology      81 98 121 123
Rational conformal field theory      313
Rational function      253
Real vector bundle      see under Vector bundle
Reduced join      6
Reduced product      6
Renormalisation group      301
Representations      185—191
Representations, and Wilson lines      343
Representations, of the fundamental group      330
Representations, of the Kac-Moody algebra      187—188
Representations, of the Virasoro algebra      185—187 309—310
Riemann surface      1 10 133—134 139—144 147 148—169 197 257—268 302 316—322 350 358
Riemann surface, and Chern-Simons theory      350 358
Riemann surface, and conformal field theory      302 316—322 350 358
Riemann surface, and divisors      139—144
Riemann surface, and K $\ddot{a}$ hler manifolds      147
Riemann surface, and moduli space      152—165
Riemann surface, and Morse theory      197
Riemann surface, and strings      259—268
Riemann surface, and Teichm $\ddot{u}$ ller space      148—151
Riemann surface, and the $\bar{\partial}$ operator      133—134
Riemann surface, and the Weierstrass gap theorem      166—169
Riemann surface, Yang-Mills theory on      257—258
Riemann uniformisation      150 152
Riemann — Roch theorem      98 108 168
Rigid rotation      183
Root space      190
Roots      190 290

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