Cox D., Katz S. — Mirror symmetry and algebraic geometry :: Электронная библиотека попечительского совета мехмата МГУ

Главная Ex Libris Книги Журналы Статьи Серии Каталог Wanted Загрузка ХудЛит Справка Поиск по индексам Поиск Форум

Авторизация

Поиск по указателям

Cox D., Katz S. — Mirror symmetry and algebraic geometry

Обсудите книгу на научном форуме

Нашли опечатку?
Выделите ее мышкой и нажмите Ctrl+Enter

Название: Mirror symmetry and algebraic geometry

Авторы: Cox D., Katz S.

Аннотация:

Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kähler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.

Язык:

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 469

Добавлена в каталог: 16.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Gromov — Witten invariant, computing, other examples      201 214 405
Gromov — Witten invariant, computing, recursion for invariants of $\mathbb{P}^2$       198 200 211 220 232 237
Gromov — Witten invariant, computing, recursion for invariants of $\mathbb{P}^3$       213
Gromov — Witten invariant, enumerative significance of      195 196 198 202 211 212 214 262 400 enumerative
Gromov — Witten invariant, equivalence of algebraic and symplectic definitions      195
Gromov — Witten invariant, equivariant      196 298—300 319
Gromov — Witten invariant, equivariant of quintic threefold      299
Gromov — Witten invariant, equivariant of zero section of equivariant bundle      299
Gromov — Witten invariant, equivariant, definition of      298
Gromov — Witten invariant, equivariant, nonequivariant limit of      298
Gromov — Witten invariant, symplectic definition      184 188—191 205 208 209
Gromov — Witten invariant, symplectic definition, equivalence of definitions      191 195
Gromov — Witten invariant, symplectic definition, intuitive idea of      186
Gromov — Witten invariant, symplectic definition, Li — Tian approach      190
Gromov — Witten invariant, symplectic definition, relation between mixed and symplectic invariants      190
Gromov — Witten invariant, symplectic definition, Ruan — Tian mixed invariant      188 189 210 212
Gromov — Witten invariant, symplectic definition, Ruan — Tian symplectic invariant      189 210 212
Gromov — Witten invariant, tree-level      196 230
Gromov — Witten invariant, used to define A-model correlation function      221
Gromov — Witten invariant, used to define Gromov — Witten potential      230
Gromov — Witten invariant, used to define quantum cohomology      218
Gromov — Witten potential      28 109 230—234 244 247 303 304 389
Gromov — Witten potential of a Calabi — Yau threefold      238 239 256 261 318 326
Gromov — Witten potential of a Calabi — Yau threefold, expressible in terms of periods of mirror      333 389
Gromov — Witten potential of a Calabi — Yau threefold, toric part of      389
Gromov — Witten potential of an elliptic curve      231 235
Gromov — Witten potential of the quintic threefold      28 237 238 333
Gromov — Witten potential, classical part      235
Gromov — Witten potential, definition of      230
Gromov — Witten potential, equivariant      319 320
Gromov — Witten potential, genus 0 free energy      231
Gromov — Witten potential, genus g      231
Gromov — Witten potential, homogeneity of      235
Gromov — Witten potential, of $\mathbb{P}^2$       236
Gromov — Witten potential, quantum part      235
Gromov — Witten potential, structure of      234—236
Gromov — Witten potential, third partials of      231 234
Gromov — Witten potential, WDVV equation for      232—234 236 237 239
Gysin map      171
Gysin spectral sequence      59
Hamiltonian      412—415 430
Hamiltonian action      40 429
Hamiltonian action, moment map of      see “Moment map”
Hamiltonian for electricity and magnetism      418
Hamiltonian vector field      40
Hamilton’s equations      413 414
Hard Lefschetz theorem      250 409
Heisenberg picture      413
HG of Euler data      346
HG of P      28 101 164 347 352—354 357
HG of Q      28 347 350 352—354 357
HG, definition of      346
HG, mirror transformation of      348 349 353 354 linked mirror
Hilbert scheme      208 282
Hirzebruch — Riemann — Roch theorem      175
Hochschild cohomology      403
Hodge diamond      4 5
Hodge numbers      4 6 66 227
Hodge numbers, Calabi — Yau toric complete intersection      63
Hodge numbers, Calabi — Yau toric hypersurface      56—60 119 145 146
Hodge numbers, string theoretic      60 63 273 402
Hodge structure of an orbifold      409
Hodge structure of weight k      73
Hodge structure, degeneration of      75
Hodge structure, Hodge filtration of      73 105 153
Hodge structure, Hodge filtration of limiting      76 111
Hodge structure, integer lattice of      73 76 260 268
Hodge structure, mixed      76 253
Hodge structure, mixed, Hodge — Tate      80 105 255 264 266 268
Hodge structure, mixed, Hodge — Tate, limiting      75 105 255 266
Hodge structure, polarized      74 107 253 258 267
Hodge structure, pure      59
Hodge structure, variation of      25 74 107 109 152 158 253 254 258 267
Hodge structure, variation of $\mathbb{C}$ -variation      268 271 272
Hodge structure, variation of formally degenerating      261
Hodge structure, variation of real      255 264
Hodge structure, variation of, comes from geometry      262
Hodge — Riemann bilinear relation      74 254 268
Hodge — Tate      see “Hodge structure mixed Hodge
Holomorphically symplectic      402
Homogeneous change of coordinates in the Mirror Theorem      see “Weights of variables”
Hopf fibration      49
Hyper-Kahler      402
Hypergeometric equations      90—101 161 333
Hypergeometric functions      95 164
Hypergeometric functions, generalized      20
Hypergeometric functions, HG      see “HG”
Hypergeometric functions, p-adic      20
Indicial equation      77 79 80 163
Instanton number      8 203—210 226 308
Instanton number for quintic threefold      see “Quintic threefold instanton
Instanton number, definition of      203
Instanton number, divisibility properties of      24
Instanton number, elliptic analog of      213 214
Instanton number, enumerative significance of      205—208
Instanton number, example of      205—208 (see also “Quintic threefold instanton
Instanton number, symplectic definition of      208—210 400
Instanton, corrections      9 423
Instanton, holomorphic      8 430
Instanton, naive      430
Instanton, sum      422 423
Instanton, worldsheet      422
Integrality conjecture      105—107 109 110 112 151 155 157 250 258 259 262 266 268 270 385 386 399
Integrality conjecture, statement of      82 150
Intersection homology      409
Intrinsic normal cone      see “Normal cone intrinsic”
Intrinsic normal sheaf      178 179 intrinsic”)
J-holomorphic curve      209
J-holomorphic curve, definition of      208
J-holomorphic map      185
J-holomorphic map, moduli of, dimension of      186 187
J-holomorphic map, moduli of, fundamental class of      188
J-holomorphic map, moduli of, Li — Tian approach      188 190
J-holomorphic map, moduli of, Ruan approach      186
J-holomorphic map, moduli of, Ruan — Tian approach      187
J-holomorphic map, multiple cover      186
J-holomorphic map, perturbed      187 190 205
J-holomorphic map, simple      186 208 210
J-holomorphic map, stable      188
Jacobian ideal      84 86 90
Jacobian ring      427 428
Jones polynomial      435
K3 surface      359
K3 surface with involution      65—67
K3 surface, mirror symmetry for      401
K3 surface, rational curves on      400
K3 surface, vector bundles on      404
Kahler, cone      12 120 127 128 130 150 225 227 365
Kahler, cone of a Calabi — Yau threefold      132 144 146
Kahler, cone, behavior when complex structure varies      134
Kahler, cone, closure of      128 132 135 142 153 243 245 267 316 317 381
Kahler, cone, face of      132—134 144 146 158 227
Kahler, cone, locally polyhedral      132
Kahler, cone, polyhedral      39 129
Kahler, cone, simplicial      97
Kahler, cone, toric part of      135 136 138 139 399
Kahler, form on an orbifold      409
Kahler, moduli      see “Moduli Kahler”
Kahler, space      see “Complexified Kahler
Klein — Gordon equation      412 414 415 418
Kodaira — Spencer class      133
Kodaira — Spencer map      153
Kuranishi map      175 177
Lagrange lift      349 (see also “Euler data linked mirror
Lagrangian      1 412 420
Lagrangian category      403
Lagrangian density      411 412 414—416
Lagrangian density for electricity and magnetism      417 418
Lagrangian submanifold      403
Lagrangian submanifold, special      11 402 403

Landau — Ginzburg, model      426
Landau — Ginzburg, orbifold      6 143 427 429
Landau — Ginzburg, potential      428
Landau — Ginzburg, theory      426—429
Large radius limit point      115 128—131 136 138 149 150 152 153 155 156 245 251 257 264 267 269
Large radius limit point, equivalence of      131 152
Large radius limit point, example of      141
Large radius limit point, tangent space at      158
Lattice of linear relations      39 41 44 136 138 140 155 159 224 390
Lattice of linear relations, example of      50 93 95 124 126 160 164
Lattice of linear relations, relation to ,4-system      91 97 99
Laurent monomial      35 118 126
Laurent polynomial      35 116 117 122 155
Lefschetz decomposition      259 264 361
Lefschetz Theorem      361
Left-moving, fermions      2 421
Left-moving, sector      425
Levi — Civita connection      420
Limiting weight filtration      see “Monodromy weight filtration”
Linear circuit      136 138
Linear circuit, example of      136 140 141
Linear circuit, shrink the GKZ decomposition      139
Linear circuit, supported by a fan      137 139
Linear sigma model      see “Gauged linear sigma model”
Local operator      421 423
localization      279 280 366 382
Localization in $\bar{M}_{0,n}(\mathbb{P}^r,d)$ , fixed point loci      282 340 348 367 368 376 377
Localization in $\bar{M}_{0,n}(\mathbb{P}^r,d)$ , fixed point loci, description of      283
Localization in $\bar{M}_{0,n}(\mathbb{P}^r,d)$ , fixed point loci, example of      284 285 289
Localization in $\bar{M}_{0,n}(\mathbb{P}^r,d)$ , fixed point loci, graph of      283 289 290 367 368 370
Localization in $\bar{M}_{0,n}(\mathbb{P}^r,d)$ , fixed point loci, normal bundle of      285—287 289 291 293 370 equivariant of
Localization in $\bar{M}_{0,n}(\mathbb{P}^r,d)$ , fixed point loci, Type A graph      368 369 374
Localization in $\bar{M}_{0,n}(\mathbb{P}^r,d)$ , fixed point loci, Type B graph      368 374
Localization in $\bar{M}_{0,n}(\mathbb{P}^r,d)$ , torus action on      282
Localization of stacks      289
Localization theorem for equivariant cohomology      277 319 338 339 347—349
Localization theorem for polynomials in Chern classes      281 291—293
Localization theorem for smooth stacks      280 281 289 368 373 375 377
Localization theorem for the equivariant integral      279 335 339—342 345 348 376
Localization theorem, applications to enumerative geometry      282
Localization theorem, applied to $\mathbb{P}^r$       278 279
Localization theorem, computing instanton number       288
Localization theorem, computing lines on quintic threefold      282
Localization theorem, proving multiple cover formula      289
Lorentz transformation      415
Low energy limit      10
m(X)      see “Mori cone”
Macaulay      297
Mathematical mirror pair      4 11 25 56 242 387 397
Mathematical mirror pair of Calabi — Yau threefolds      257—262
Mathematical mirror pair of Calabi — Yau threefolds in terms of correlation functions      259 262
Mathematical mirror pair of Calabi — Yau threefolds in terms of potential functions      260 262
Mathematical mirror pair of Calabi — Yau threefolds, definition of      258
Mathematical mirror pair, global      258
Mathematical mirror pair, higher dimensional      264—267
Mathematical mirror pair, higher dimensional for fourfolds      265 266
Mathematical mirror pair, higher dimensional, definition of      264
Mathematical mirror pair, higher dimensional, limitations of      265
Mathematical mirror pair, Hodge-theoretic      258 264
Mathematical mirror pair, quintic mirror and quintic threefold      262—263 333
Matrix integral      310
Matrix model      310
Maximal degeneracy      121 (see also “Maximally unipotent boundary
Maximal projective crepant partial desingularization      55
Maximal projective subdivision      see “Projective subdivision maximal”
Maximally unipotent, boundary point      82 93 102 104 105 109 110 113—115 119 121 128 131 138 149 151—156 159—161 251 257 266—269
Maximally unipotent, boundary point, definition of      81
Maximally unipotent, boundary point, equivalence of      115 120
Maximally unipotent, boundary point, example of      89 90 125 127 142
Maximally unipotent, boundary point, existence of      116 120 399
Maximally unipotent, boundary point, tangent space at      158
Maximally unipotent, monodromy      18 19 78—82 109 113 127 153 163 264
Maximally unipotent, monodromy, 1-dimensional moduli      78—81
Maximally unipotent, monodromy, r-dimensionai moduli      81—82
Maxwell’s equations      417 418
McKay correspondence      402
Minimal model      6
Minkowski sum      33 63
Mirror conjecture      328
Mirror conjecture for Calabi — Yau threefolds      258 261
Mirror conjecture for quintic threefold      262 333
Mirror conjecture, higher dimensional      267
Mirror conjecture, relation to Givental Mirror Theorem      383
Mirror conjecture, toric      112 267—273 383 Giventai toric”)
Mirror conjecture, toric for threefolds      267
Mirror conjecture, toric, example of      269 270
Mirror conjecture, toric, higher dimensional      273
Mirror conjecture, toric, Hodge-theoretic      56 112 268 272 273 387
Mirror conjecture, toric, mathematical      271
Mirror conjecture, toric, statement of      268 272
Mirror construction      261
Mirror construction for Calabi — Yau complete intersections in flag manifolds      405
Mirror construction, Batyrev      53 56 60 70 72 120 123—125 135 139 143 144 146 148 155 261 267 268 271 272 381 383
Mirror construction, Batyrev — Borisov      62 63 100 147 405
Mirror construction, Batyrev, quintic threefold      61
Mirror construction, Greene — Plesser      53 56 89
Mirror construction, Voisin — Borcea      65 68—69 72 401 402
Mirror construction, Voisin — Borcea, complex and Kahler moduli      68 69
Mirror construction, Voisin — Borcea, relation to Batyrev construction      70
Mirror construction, Voisin — Borcea, Yukawa coupling      69
Mirror coordinates      see “Mirror map”
Mirror map      5 9 56 60 82 109 128 143 147—166 257 385
Mirror map for quintic threefold      see “Quintic threefold mirror
Mirror map, definition of      150 157
Mirror map, definition of, choices to make      150 152 154
Mirror map, definition of, data needed to define      150
Mirror map, definition of, indeterminacy in definition      150
Mirror map, definition of, justification from physics      158
Mirror map, definition of, role of integrality conjecture      150 151
Mirror map, derivative of      152—154 158 266 267 definition choices
Mirror map, mirror coordinates      110 112 152 259 265
Mirror map, mirror coordinates, definition of      149
Mirror map, toric      148 151 155—159 267 268 270 272
Mirror map, toric for quintic threefold      156 157
Mirror map, toric, definition of      155 156
Mirror map, toric, not the identity      148
Mirror map, toric, resolving the lack of uniqueness      156
Mirror map, via hypergeometric functions      159—166
Mirror pair      5 6 9 53 56 402 421
Mirror pair, Hodge-theoretic      see “Mathematical mirror pair Hodge-theoretic”
Mirror pair, mathematical      see “Mathematical mirror pair”
Mirror pair, physics definition of      4
Mirror Symmetry      1 4—6 9 10 16 35 46 53 82 107 111 115 116 128 129 131 142 144 147 152 250 256 261 266 328 423 426
Mirror symmetry for a K3 surface      401
Mirror symmetry for gauged linear sigma models      430
Mirror symmetry for quintic threefold      see “Quintic threefold mirror
Mirror symmetry for rigid Calabi — Yau      65
Mirror symmetry, categorical      see “Mirror symmetry homological”
Mirror symmetry, classical      9 13 24 27 333 363 385
Mirror symmetry, earliest evidence for      60
Mirror symmetry, enumerative aspects of      214
Mirror symmetry, Hodge-theoretic      25 154 362 Hodge-theoretic”)
Mirror symmetry, homological      11 403
Mirror symmetry, mathematical      11 112 217 242 257 274 397 402
Mirror symmetry, relation to Giventai mirror theorem      383
Mirror symmetry, toric      56
Mirror theorems      6 273 274
Mirror theorems, broader conception of      355 398 403
Mirror theorems, Givental      356 398
Mirror theorems, Givental for $\mathbb{F}_2$       394
Mirror theorems, Givental for $\mathbb{P}^1$       324 357
Mirror theorems, Givental for $\mathbb{P}^n$       325 357 366 371 372
Mirror theorems, Givental for hypersurface in $\mathbb{P}(1,1,2,2,2)$       383—386 388
Mirror theorems, Givental for nef complete intersection in $\mathbb{P}^n$       356 358 359 373 379 380
Mirror theorems, Givental, class $\mathcal{P}$       378—380
Mirror theorems, Givental, computes genus 0 gravitational, correlators      361
Mirror theorems, Givental, coordinate change lemma      379—381
Mirror theorems, Givental, double construction lemma      375 378 380
Mirror theorems, Givental, example of      359 364
Mirror theorems, Givental, genus 1 version      381

1 2 3 4 5

О проекте