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

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

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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

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Предметный указатель

Mirror theorems, Givental, how to find change of variables      359
Mirror theorems, Givental, quintic      26 361—363
Mirror theorems, Givental, recursion lemma      369 371 372 374 37 378 380
Mirror theorems, Givental, regularity lemma      369 373 375 378
Mirror theorems, Givental, toric      381 382 387 389 391 405 small of
Mirror theorems, Givental, uniqueness lemma      378 380
Mirror theorems, Lian — Liu — Yau      28 263 332 334 349 354 380 398
Mirror theorems, Lian — Liu — Yau, application to multiple covers      355
Mirror theorems, Lian — Liu — Yau, toric      398
Mirror theorems, quintic      24—29 263 332 334 352 354 362
Mirror transformation      see “Euler data linked mirror mirror
Missing mirrors      72
Moduli      1 5
Moduli, complex      3—5 9 56 81 82 113—128 131 144 148 149 264 423
Moduli, complex of a Calabi — Yau threefold      78 79 82 102 105 107 109 121 257 258 260
Moduli, complex of a Calabi — Yau toric hypersurface      92 93 116—127
Moduli, complex, 1-dimensional      78 79 82 84 87 102 105 107 109 257
Moduli, complex, compactification of      114 115 117 119—121
Moduli, complex, coordinates      114 123 124 155 159 267
Moduli, complex, dimension of      118 119
Moduli, complex, example of      123—127 139 147
Moduli, complex, Kahler      3—5 11 12 56 82 116 127—149 152 218 245 264 423
Moduli, complex, Kahler of a Calabi — Yau threefold      132—134 142 257 258
Moduli, complex, Kahler, 1-dimensionai      149
Moduli, complex, Kahler, boundary      143
Moduli, complex, Kahler, compactification of      132 135 136 142—144 152
Moduli, complex, Kahler, coordinates      128 129 267
Moduli, complex, Kahler, enlarged      132—134 136 142
Moduli, complex, Kahler, example of      139—142 146
Moduli, complex, Kahler, global      142
Moduli, complex, Kahler, large radius limit point      see “Large radius limit point”
Moduli, complex, Kahler, of a Calabi — Yau toric hypersurface      134—143
Moduli, complex, Kahler, partial compactification of      128—132 134 138 141 142 150
Moduli, complex, Kahler, toric      60 120 134 135 138 141 155 156 267 271
Moduli, complex, maximally unipotent boundary point      see “Maximally unipotent boundary
Moduli, complex, polynomial      61 116—119 124—126 138 147 383 427
Moduli, complex, SCFT      3—5 128 144 145 264 401
Moduli, complex, SCFT, example of      147
Moduli, complex, simplified      95 117—127 135 138 142 155 156 267 271
Moduli, complex, stable maps      see “Stable maps moduli
Moment map      41 44 45 49 143 429
Moment map, example of      50
Monodromy theorem      75
Monodromy transformation      75 77 81 114 316
Monodromy transformation of A-model connection      see “A-model connection monodromy
Monodromy transformation, logarithm of      75 81 111 114 142 246 247 249 316
Monodromy transformation, quasi-unipotent      75
Monodromy transformation, unipotent      75 77 78 114 246 247
Monodromy transformation, unipotent, maximaliy      see “Maximally unipotent”
Monodromy weight filtration      76 80 81 104 105 115 149 150 154 161 250 268
Monodromy weight filtration of the A-model connection      see “A-model connection monodromy
Monomial ordering      225
Monomial-divisor mirror map      57 60 158 268 271 272
Mori cone      39 98 228
Mori cone of a toric variety      see “Toric variety Mori
Movable cone      133 142
Movable cone, reflected      134 142
Multiple covers      202 207 209 288 343
Multiple covers of non-isolated curves      208
Multiple covers, contribution to Gromov — Witten, invariant      202 204 288—294 352
Multiple covers, contribution to Gromov — Witten, invariant, Lian — Liu — Yau’s proof      355
Multiple covers, contribution to Gromov — Witten, invariant, Pandharipande’s proof      291—293
Multiple covers, contribution to Gromov — Witten, invariant, proof for d = 2      289—291
Multiple covers, contribution to virtual fundamental class      202 205 294
Multiple covers, elliptic      209 214 300
Multiple covers, of 6-nodal quintics      293 294
Multiple mirrors      45 60 121
Nef complete intersection      see “Toric complete intersection nef”
Nef-partition      62 63 97 100
Nef-partition, dual      63
Nets of quadrics      366
Nilpotent orbit      254
Nilpotent orbit theorem      76
Node deformations      341
Noether’s theorem      419
Non-linear sigma model      see “Sigma model”
Non-perturbative state      402
Non-renormalization theorem      7 8 423
Nonequivariant limit      28 346 350—354 357 366 372 379 380
Nonequivariant limit, definition of      276
Normal cone      172 174 177 178 181 360
Normal cone, intrinsic      178 179
Normal cone, intrinsic, example of      179
Normal cone, refines Euler class      173
Normal cone, virtual      177
Normalization exact sequence      290 292
Normalized 3-form      21 104—106 110 112 151 258 259 268 328 385 normalized”)
Normalized d-form      154
Novikov ring      229
Observable      1 2 36 414
Obstruction bundle      208 289 290 292 obstruction”)
Obstruction space      175
Obstruction theory      178 300
Obstruction theory, perfect      174 180 299
Obstruction theory, perfect for virtual fundamental class      180
Obstruction theory, perfect, definition of      179
Obstruction theory, perfect, equivariant      299
Obstruction theory, perfect, example of      179
octahedron      51
Open conjectures      399 400
Open conjectures, Calabi — Yau threefold, B-model potential function in terms of $I_{\nu}$       389 399
Open conjectures, Calabi — Yau threefold, Yukawa coupling in terms of $I_{\nu}$       389 399
Open conjectures, complex geometry, integrality conjecture      82 399
Open conjectures, complex geometry, maximally unipotent boundary points in simplified moduli space      120 399
Open conjectures, convergence of quantum cohomology      218 400
Open conjectures, enumerative geometry, Clemens conjecture      202 400
Open conjectures, enumerative geometry, enumerative significance of a Gromov — Witten invariant      196 400
Open conjectures, enumerative geometry, symplectic definition of instanton numbers      208 400
Open conjectures, Kahler geometry, cone conjecture      130 399
Open conjectures, Kahler geometry, toric part of Kahler cone of a Calabi — Yau toric hypersurface      138 399
Open conjectures, Picard — Fuchs equations, $I_{\nu}$ is a solution      101 399
Open conjectures, Picard — Fuchs equations, missing equations in A-system      94 399
Open conjectures, relation between and       386 400
Open conjectures, virtual fundamental class for a complete intersection      386 400
Open conjectures, virtual fundamental class, equivalence of definitions      176 399
Operator product expansion      424 425
Orbifold      12 32 38 54 171 407
Orbifold, Calabi — Yau      see “Calabi — Yau orbifold”
Orbifold, definition of      408
Orbifold, diffeomorphism      42 45 410
Orbifold, Goreilstein      407 408 410
Orbifold, Hodge theory on      409
Orbifold, infinitesimal deformations of      410
Orbifold, integration over      340
Orbifold, Kahler form on      409
Orbifold, symplectic      42 410
Orbifold, tangent sheaf of      410
Pair of pants      433 434
Partition function      310
path integral      416—418 422
Period      19 20 27 75 92 94 95 159 333 solutions
Period domain      254
Perturbative state      402
Phase transition      429
Phases      46 143
Physicist’s Euler number      60 402 string
Picard — Fuchs equation      19 75 77 80 81 84 102 103
Picard — Fuchs equation of a Calabi — Yau threefold      79 105 327
Picard — Fuchs equation of a Calabi — Yau toric hypersurface      see “Calabi — Yau toric
Picard — Fuchs equation of mirror of a Calabi — Yau toric complete intersection      100 101
Picard — Fuchs equation of the quintic mirror      see “Quintic mirror Picard
Picard — Fuchs equation, computing via A-system      see “A-system”
Picard — Fuchs equation, computing via Griffiths — Dwork method      see “Griffiths — Dwork method”
Picard — Fuchs equation, examples of      87—90
Picard — Fuchs equation, missing equations in A-system      94 95 101 383 385 399
Picard — Fuchs equation, normalized      27 105—107
Picard — Fuchs equation, solutions of      19 96 101 149 156 160 161 263 268
Picard — Fuchs ideal      74 77 94
Picard — Fuchs ideal, associated $\mathcal{D}$ -module      74 75 101
Picard — Fuchs operator      328 329

Picard — Fuchs operator, gives relation in quantum cohomology      329
Picard — Fuchs operator, h-homogenization of      328 329
Picard — Fuchs operator, principal part of      328
Picard — Lefschetz transformation      see “Monodromy transformation”
Planck’s constant h      413
Poincare duality      82 381
Poincare duality for orbifolds      171 172 183 305 408
Poisson bracket      413 414
Polytope      33
Polytope, facet of      33
Polytope, integral      33
Polytope, Newton      35 60
Polytope, normal fan of      34 51 53 135
Polytope, normalized volume      59
Polytope, polar      34 46 51 53 120
Polytope, polar, duality      35
Polytope, reflexive      60 (see also “Reflexive polytope”)
Polytope, ring      34 36 37
Polytope, simplicial      409
Polytope, toric variety of      34
Potential function      see “B-model potential
Prestable maps      179
Primitive cohomology      83 86 253 264 361
Primitive cohomology, definition of      74
Primitive collection      38 50 223—225
Primitive collection, ampleness criterion      40
Primitive contraction      132 143 148
Primitive contraction, Type I      133 134 143 144 147 226 227
Primitive contraction, Type II      133 134 143
Primitive contraction, Type III      133 134 143 146 227
Principal A-determinant      122
Projective subdivision      54 55 137
Projective subdivision, definition of      53
Projective subdivision, maximal      54—56 60 62 63 116 135 267 382
Projective subdivision, maximal, definition of      53
Projective subdivision, simplified      136 139—141 143 146
Projective subdivision, simplified, definition of      135
Pseudo-manifold      189
Quantization      10 413 415 418 419 424 425 430
Quantization, canonical      413 414
Quantization, radial      424
Quantum $\mathcal{D}$ -module      405
Quantum anomaly      423—425
Quantum cohomology      7 25 82 101 200 422 430 435
Quantum cohomology of $\mathbb{F}_2$       393—395
Quantum cohomology of $\mathbb{P}^r$       222 223 237 325
Quantum cohomology of a Calabi — Yau manifold      218 222 252
Quantum cohomology of a Calabi — Yau threefold      225—227 239 327 329
Quantum cohomology of a Grassmannian      227 405
Quantum cohomology of a smooth toric variety      223 390 392 393
Quantum cohomology of the quintic threefold      225 226 238
Quantum cohomology, big      229 230 304 312 314
Quantum cohomology, big, associativity of      232 234 237 WDW
Quantum cohomology, big, convergence of      230
Quantum cohomology, big, definition of      231
Quantum cohomology, big, degree condition for      235
Quantum cohomology, big, Kunneth formula for      234
Quantum cohomology, big, of $\mathbb{P}^2$       237
Quantum cohomology, big, of a Calabi — Yau threefold      239
Quantum cohomology, big, of the quintic threefold      238
Quantum cohomology, big, relation to small quantum product      242 243
Quantum cohomology, big, restriction of      242—244 314 320 327
Quantum cohomology, big, ring structure of      233 (see also “Frobenius algebra”)
Quantum cohomology, big, used to define Givental connection      311
Quantum cohomology, coefficients      228—230
Quantum cohomology, definition of      218
Quantum cohomology, degree condition for      221 222
Quantum cohomology, effect of a flop      226 227
Quantum cohomology, gravitational      see “Gravitational quantum product”
Quantum cohomology, modified      363
Quantum cohomology, modified, relations in      363
Quantum cohomology, of (l + 1)-dimensional TQFT      434
Quantum cohomology, other examples      227 228
Quantum cohomology, properties of      221
Quantum cohomology, relation to big quantum product      242 243
Quantum cohomology, relations in      321 322 325 364 387 390 392
Quantum cohomology, ring structure of      219 (see also “Frobenius algebra”)
Quantum cohomology, small      217—228 230 314 315 320 326 359 363 364
Quantum cohomology, small, associativity of      219 228
Quantum cohomology, small, convergence of      218 227 228 244 245 400
Quantum cohomology, used to define A-model connection      243 244
Quantum corrections      12 18 148
Quantum differential equation      321 324 327 329 356 386 390 405
Quantum differential operator      321 322 329 392 395
Quantum differential operator, derived from relations in quantum cohomology      322 324
Quantum differential operator, example of      324 325 327
Quantum differential operator, gives relations in quantum cohomology      321 (see also “Quantum cohomology small relations
Quantum field theory      413—419 422 430
Quantum field theory, Hamiltonian formulation      416
Quantum field theory, Lagrangian formulation      416
Quantum field theory, two-dimensional      1
Quantum hyperplane section principle      26 364—366 405
Quantum hyperplane section principle for $\mathbb{P}^n$       365
Quantum hyperplane section principle for a convex variety      365
Quantum hyperplane section principle, relation to Givental mirror theorem      366
Quasi-smooth      60 86 116 408
Quintic mirror      17 122 333
Quintic mirror, A-system of      93 94
Quintic mirror, A-system of, solutions of      101
Quintic mirror, B-model correlation function of      17 21—23 27 29 B-model”)
Quintic mirror, B-model potential function of      263 333
Quintic mirror, complex moduli of      18 262
Quintic mirror, construction of      17 61
Quintic mirror, Frobenius method      162
Quintic mirror, Gauss — Manin connection of      25
Quintic mirror, homogeneous coordinate ring of      62
Quintic mirror, monodromy of      111 112 262
Quintic mirror, Picard — Fuchs equation of      20 21 26 81 87 88 95 163 333
Quintic mirror, Picard — Fuchs equation of, solutions of      23 101 110 157 160 162 164 333 353 362
Quintic mirror, reflexive polytope of      61
Quintic mirror, satisfies integrality conjecture      82 110 150 157
Quintic mirror, Yukawa coupling of      21 157
Quintic mirror, Yukawa coupling of, normalized      21—23 2j| 110—111 262 363 B-model
Quintic threefold      15—29 352 429
Quintic threefold, 2875 lines on      23 24 157 173 281 282
Quintic threefold, 6-nodal quintics on      16 206 293 295 297
Quintic threefold, 6-nodal quintics on, normal bundle of      294
Quintic threefold, A-model correlation function of      15 16 22 23 27 28 203 226 262 333 363 A-model”)
Quintic threefold, big quantum cohomology of      238
Quintic threefold, Fermat      143 174
Quintic threefold, Gromov — Witten invariants of      173 174 201—203 293 332
Quintic threefold, Gromov — Witten invariants of equivariant      299
Quintic threefold, Gromov — Witten invariants of, formula for      184 201 281 332 350
Quintic threefold, Gromov — Witten invariants of, via localization      288
Quintic threefold, Gromov — Witten potential of      28 237 238 333 350 362 388
Quintic threefold, instanton numbers of      16 23—25 205 207 262 293 333 363
Quintic threefold, instanton numbers of, computation of $n_{4}$       288
Quintic threefold, instanton numbers of, subtleties of $n_{10}$       16 206 297
Quintic threefold, Kahler moduli of      15 18 22 157
Quintic threefold, mirror map of      18—20 22 23 26 28 110 156 157 162 263 333 353 362
Quintic threefold, mirror of      see “Quintic mirror”
Quintic threefold, mirror symmetry for      15 18 20 22 23 262 263 288 333 quintic”)
Quintic threefold, Mirror theorem      see “Mirror theorems quintic”
Quintic threefold, rational curves on      1 15 16 24 202 206 288 293
Quintic threefold, small quantum cohomology of      225 226
Quintic threefold, virtual fundamental class of      173 180 181 184
Raising indices      218 239 417
Rational polyhedral cone      31
Rational polyhedral cone, strongly convex      31
Reciprocity lemma      344
Reconstruction theorem      196 197 200
Recursion lemma      see “Mirror theorems Givental recursion
Recursion relation, differential equations      95 162 163
Recursion relation, Gromov — Witten invariants      see “Gromov — Witten invariant computing recursion”
Reflexive Gorenstein cone      62 64
Reflexive Gorenstein cone, duality of      64
Reflexive Gorenstein cone, index of      64
Reflexive Gorenstein cone, relation to nef-partitions      64
Reflexive Gorenstein cone, relation to reflexive polytopes      64
Reflexive polytope      6 35 47 53—55 63 96 116 118 134 135 144 145 155 267 382 383 405

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