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

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

Предметный указатель

Dubrovin connection, flatness of      240 244
Dubrovin connection, formal      242
Dubrovin connection, homogeneity of      241
Dubrovin connection, potential function of      239
Dubrovin connection, torsion of      240
Dubrovin connection, WDVV equation for      240 241
Dubrovin formalism      234 239—242 244 304
Dubrovin formalism, binary product of      239
Dubrovin formalism, connection of      see “Dubrovin connection”
Electromagnetic potential      417
Energy-momentum conservation      424
Energy-momentum tensor      424
Enumerative geometry      16 215 enumerative
Equivariant characteristic class      338
Equivariant Chern class      276 280 366
Equivariant cohomology      27 101
Equivariant cohomology of $\mathbb{P}^r$       277 278 335
Equivariant cohomology, definition of      276
Equivariant cohomology, explanation of parameter h      319
Equivariant cohomology, localization of      277 336 for
Equivariant cohomology, restriction map      343 344
Equivariant hyperplane class      28 101 335 346 366 368
Equivariant hyperplane class of       see “ $N_d$

projective equivariant
Equivariant hyperplane class, definition of      277
Equivariant integral      280 335 367 368 373 376 377
Equivariant integral, definition of      279
Equivariant vector bundle      276 298
Euler $\Gamma$ -function      162 163 165
Euler characteristic      310
Euler class      173 289 351 358 360
Euler class of zero bundle      358 365 372
Euler class, equivariant      276 292 336 338 341 342 344 377
Euler class, equivariant of $N_{\Gamma}$       285 287—289 291 293 370
Euler class, stack versus coarse moduli space      181 184 302 332 337
Euler data      27 28 345 348—350 380
Euler data, definition of      339
Euler data, examples of      see “P and Q”
Euler data, hypergeometric function of      346 (see also “HG of
Euler data, linked      343 345 353 355
Euler data, linked, definition of      342
Euler data, linked, mirror transformation of      348 352 353 381 mirror
Euler data, linked, multiple cover interpretation      343
Euler data, linked, uniqueness theorem for      344 348 353 355 380
Euler integral      160 (see also “GKZ system”)
Euler — Lagrange equation      412—414 418 421 422 424
Euler — Lagrange equation for electricity and magnetism      417
Euler’s constant      162
Evaluation map      172 179 189 205 210 222 298 301 332 357 360 365 375
Excess dimension      308
Excess dimension for moduli of stable maps      182
Excess dimension in intersection theory      182
Excess normal bundle      173 174
Expected dimension in Kuranishi theory      174
Expected dimension of moduli of stable maps      see “Stable maps moduli
Expected dimension of zero locus      173
Extremal transition      143 144 146 405 406
Extremal transition, compatibility with mirror symmetry      144 148
Extremal transition, dual      147 148 158 159 405
Extremal transition, example of      146—148 158
Extremal transition, mirror      see “Extremal transition dual”
Extremal transition, relation to SCFT moduli      144
Extremal transition, toric      145
Extremal transition, Type I      143 145
Extremal transition, Type II      143
Extremal transition, Type III      143 145
F-theory      404
Face of a cone      31
Fan      31 (see also “Projective subdivision”)
Fan, complete      32
Fan, flip of      see “Flip”
Fan, Grobner      see “Grobner fan”
Fan, secondary      see “Secondary fan”
Fan, simplicial      32
Fan, smooth      32
Fan, support of      31
Fan, supported on a linear circuit      see “Linear circuit supported
Fan, symmetries of      48
Fan, toric variety of      32 51
Fano variety      46 218 219 223 228 405
Fermion      2 419—421 425
Fermion, relation to supercommuting variables      230 419
Fermionic field      419 420 fermionic
Fermionic variables      419 427 428
Feynman diagram      196
Feynman integral      2 7
Field strength      417 418
Field, bosonic      see “Bosonic field”
Field, classical      411 414 416
Field, fermionic      see “Fermionic field”
Field, primary      424
Field, quantum      414 416
Field, theory      411 414
Flag manifold      219 228 254
Flag manifold, complete intersection in      364 366
Flag manifold, complete intersection in, Calabi — Yau      405
Flag of graph $\Gamma$       285 (see also “Localization in $$\bar{M}_{0</a></span> <span class=subjpages><a href=$ d)$"/> fixed
Flip      137 139
Flip, example of      141
Flip, geometric significance of      137 (see also “Flop generalized”)
Flip, trivial      137 138 141 142 156
FLOP      43 133 147 225 226
Flop, example of      141
Flop, generalized      137
Flop, trivial      137
Fourier transform      414 418
Frobenius algebra      221 234 240 430 435
Frobenius algebra, definition of      432
Frobenius algebra, gives (l + 1)-dimensional TQFT      432 434
Frobenius algebra, n-point function of      432 434
Frobenius algebra, three-point function of      432
Frobenius algebra, trace map of      432
Frobenius manifold      241
Frobenius method      26 77 161—163 333
Frobenius method in cohomology ring of $\mathbb{P}^4$       164
Frobenius method, example of      162 165
Fundamental class      172
Gale transform      44
Gauge choice      418 419 424
Gauge fixing      417
Gauge group      417 419 420 424 429
Gauge group for electricity and magnetism      418
Gauge theory      417 428
Gauged linear sigma model      12 42 45 334 428—430
Gauged linear sigma model, action of      428
Gauged linear sigma model, action of, bosonic part of      428
Gauged linear sigma model, correlation function of      364
Gauged linear sigma model, example of      429
Gauged linear sigma model, gauge group of      428
Gauged linear sigma model, mirror symmetry for      430
Gauged linear sigma model, phases of      143 (see also “Phases”)
Gauged linear sigma model, phases of Calabi — Yau      429
Gauged linear sigma model, phases of Landau — Ginzburg      430
Gauged linear sigma model, relation to toric geometry      429
Gauss — Manin connection      8 25 75 82 151 250 259 265—267
Gauss — Manin connection of a Calabi — Yau threefold      106 108 109 251 257 258
Gauss — Manin connection, connection matrix      see “Connection matrix”
Gauss — Manin connection, definition of      74
Gauss — Manin connection, fiat integral section      82 106 111 150 151 387
Gauss — Manin connection, relation to normalized Yukawa coupling      105
Generalized Berglund-Hiibsch transposition rule      70
Generalized fiop      see “Flop generalized”
Generalized Fredholm orbifold bundle      191
Generalized Fredholm orbifold bundle, oriented Euler class of      191
generated by global sections      33
Genus g coupling      303 312
Genus g gravitational Gromov — Witten potential      303
Genus g gravitational Gromov — Witten potential, definition of      304
Geometric quotient      37 117
Geometrically engineered theory      406
Ghosts      424

GIT quotient      117 120 122
Giventai connection      316
Giventai connection, definition of      311
Giventai connection, dual      321 329
Giventai connection, flat sections of      311—313 319
Giventai connection, flat sections of, restriction of      314 322
Giventai connection, flat sections of, symbolic notation for      312
Giventai connection, relation to Dubrovin connection      245
Giventai connection, restriction to $H^0(X,\mathbb{C}) \otimes H^2(X, \mathbb{C})$       314 320
Giventai connection, restriction to H^2(V,\mathbb{C})      315
Giventai I-function for $\mathcal{F}_2$       393 394
Giventai I-function for $\mathcal{P}^1$       357
Giventai I-function for $\mathcal{P}^n$       357 365 371
Giventai I-function for a toric variety      389
Giventai I-function when is nef      393
Giventai I-function, $I_{\nu}$       26 27 100 358 379 380
Giventai I-function, $I_{\nu}$ for a convex variety      381
Giventai I-function, $I_{\nu}$ for a toric variety      381 383
Giventai I-function, $I_{\nu}$ for hypersurface in P(1,1,2,2,2)      384
Giventai I-function, $I_{\nu}$ for quintic threefold      26 98 101 164 361 362
Giventai I-function, $I_{\nu}$ , definition of      356
Giventai I-function, $I_{\nu}$ , example of      364
Giventai I-function, $I_{\nu}$ , satisfies Picard — Fuchs equations of mirror      101 121 128 383 385 399
Giventai I-function, $\bar{I}$       98—101 128 164 359
Giventai I-function, $\bar{I}$ , satisfies A-system      98 391
Giventai I-function, $\bar{I}_{\nu}$       365 366
Giventai I-function, equivariant version of $I_{P^n}$       371
Giventai I-function, equivariant version of $I_{\nu}$       379 382
Giventai J-function      25 320—325 358 359
Giventai J-function for $\mathbb{F}_2$       393 394
Giventai J-function for $\mathcal{P}^1$       324 357
Giventai J-function for $\mathcal{P}^n$       324 325 346 357 365
Giventai J-function for a Calabi — Yau threefold      325—329
Giventai J-function for a Calabi — Yau threefold, formula for      325 326
Giventai J-function for a Calabi — Yau threefold, in terms of Gromov — Witten potential      326
Giventai J-function for a Calabi — Yau threefold, second partials of      326 363
Giventai J-function for a toric variety      389
Giventai J-function for a toric variety, when is nef      393
Giventai J-function for hypersurface in P(1,1,2,2,2)      386 388
Giventai J-function for quintic threefold      25 362
Giventai J-function, $J_{\nu}$       26 358 359 379 380
Giventai J-function, $J_{\nu}$ for a convex variety      365
Giventai J-function, $J_{\nu}$ for a toric variety      381—383
Giventai J-function, $J_{\nu}$ for hypersurface in P(l,1,2,2,2)      384 388
Giventai J-function, $J_{\nu}$ for quintic threefold      26 362
Giventai J-function, $J_{\nu}$ , definition of      358
Giventai J-function, $J_{\nu}$ , example of      364
Giventai J-function, $J_{\nu}$ , gives relations in modified quantum cohomology      363
Giventai J-function, $J_{\nu}$ , gives relations in quantum cohomology      364
Giventai J-function, $J_{\nu}$ , symbolic notation for      358
Giventai J-function, definition of      320
Giventai J-function, equivariant version of $J_{P^n}$       366
Giventai J-function, equivariant version of $J_{P^n}$ , definition of       367
Giventai J-function, equivariant version of $J_{P^n}$ , definition of S      366
Giventai J-function, equivariant version of $J_{\nu}$       372 382
Giventai J-function, equivariant version of $J_{\nu}$ , definition of $S_{\nu}$       372 373
Giventai J-function, equivariant version of $J_{\nu}$ , definition of $Z_{i,\nu}      373
Giventai J-function, equivariant version of $J_{\nu}$ , definition of $\tilde{S}_{\nu}$       372 373
Giventai J-function, formulas for      322 356
Giventai J-function, gives relations in quantum cohomology      321
Giventai J-function, primitive part of      361
Giventai J-function, relation between and $J_{\nu}$       359 360 382 386 400
Giventai J-function, symbolic notation for      322 356 360
Giventai J-function, toric part of      382
GKZ decomposition      43 44 60 120 121 139 143 155 161 225 430
GKZ decomposition, enlarged      43 430
GKZ decomposition, examples of      50—52 124 127 140
GKZ system      95 96 100 101 122—123 125 158—166 333 383
global sections of      34?
GLSM      see “Gauged linear sigma model”
Gluing lemma      338 342 380
Grassmannian      167 173 219 227 228 281 293 405
Grassmannian, Caiabi — Yau complete intersection in      405
Grassmannian, Caiabi — Yau complete intersection in threefold      366
Gravitational class      306 309
Gravitational class, definition of      305
Gravitational correlators      25 196 211 301—311 320 382 383
Gravitational correlators, axioms for      see “Gravitational class”
Gravitational correlators, axioms for Degree Axiom      304 310 311 317 318 325 360
Gravitational correlators, axioms for Dilaton Axiom      306—308 310 318
Gravitational correlators, axioms for Divisor Axiom      305—308 310 314 318 357
Gravitational correlators, axioms for Fundamental Class Axiom      305—308 318 324 325
Gravitational correlators, axioms for Splitting Axiom      305—307 313
Gravitational correlators, coefficients      303
Gravitational correlators, definition of      302
Gravitational correlators, genus 0 correlators in terms of Gromov — Witten invariants      306
Gravitational correlators, genus 0 correlators of $\mathbb{P}^1$       306 357
Gravitational correlators, genus 0 correlators of $\mathbb{P}^1$ , recursion for      307 308
Gravitational correlators, genus 0 correlators of Calabi — Yau threefolds      308
Gravitational correlators, genus 1 correlator $<\tau_1>_{1,0}$       308 309
Gravitational correlators, genus 1 correlator $<\tau_1>_{1,0}$ , relation to Euler characteristic      310
Gravitational correlators, recursions coming from TRRs      311 (see also “Topological recursion relation”)
Gravitational correlators, relation to Gromov — Witten invariants      303
Gravitational correlators, symbolic notation for      315 356
Gravitational descendants      302 311
Gravitational descendants, relation to gravitational correlators      303
Gravitational Gromov — Witten potential      see “Genus g gravitational Gromov — Witten potential”
Gravitational potential      310
Gravitational quantum product      234 304
Gravitational quantum product, coefficients      304
Griffiths transvereality      74 76 102 103 111 252 254
Griffiths transvereality for projective hypersurfaces      83
Griffiths transvereality for the A-variation of Hodge structure      see “A-variation of Hodge structure Griffiths
Griffiths — Dwork method      83—87 94 96
Griffiths — Dwork method for projective hypersurfaces      83—85
Griffiths — Dwork method for toric hypersurfaces      85-87
Griffiths — Dwork method for toric hypersurfaces, limitations of      86
Griffiths — Dwork method for weighted projective hypersurfaces      85
Griffiths — Dwork method, description of      84
Griffiths — Dwork method, examples of      87—90
Grobner basis techniques      84 86—88 90 225
Grobner fan      121 225
Gromov — Uhlenbeck compactification      187 209
Gromov — Witten class      200 212 220 232 305 309
Gromov — Witten class of $\mathbb{P}^2$       198 199
Gromov — Witten class, axioms for      191—195 304
Gromov — Witten class, axioms for Composition Law      194
Gromov — Witten class, axioms for Deformation Axiom      194 395
Gromov — Witten class, axioms for Degree Axiom      192 198 201 265
Gromov — Witten class, axioms for Divisor Axiom      193 198 200—203 212 235 236 239 246 265 307
Gromov — Witten class, axioms for Effectivity Axiom      192 228
Gromov — Witten class, axioms for Equivariance Axiom      192 199 201 219—221 233 241
Gromov — Witten class, axioms for Fundamental Class Axiom      193 198 199 201 212 219 234 235 265
Gromov — Witten class, axioms for Linearity Axiom      192
Gromov — Witten class, axioms for Motivic Axiom      194
Gromov — Witten class, axioms for Point Mapping Axiom      193 197 200 201 204 211 219 231 234 235 239 307 310 314
Gromov — Witten class, axioms for Reduction Axiom      194 212 310
Gromov — Witten class, axioms for Splitting Axiom      194 196 198 199 219 220 233 241
Gromov — Witten class, definition in simplest case      171
Gromov — Witten class, definition of      183 (see also “Gromov — Witten invariant algebraic
Gromov — Witten class, intuitive definition of      168
Gromov — Witten class, role of fundamental class      172
Gromov — Witten class, symplectic      190
Gromov — Witten class, tree-level      196
Gromov — Witten invariant      7 9 15 16 24 25 28 265 266 273 281 298—301 307
Gromov — Witten invariant algebraic definition in general case      183
Gromov — Witten invariant algebraic definition in simplest case      171
Gromov — Witten invariant, algebraic definition      168—184 (see also “Virtual fundamental class”)
Gromov — Witten invariant, algebraic definition, intuitive idea of      168
Gromov — Witten invariant, algebraic definition, role of fundamental class      172
Gromov — Witten invariant, arises in physics      422
Gromov — Witten invariant, axioms for      192 193 axioms
Gromov — Witten invariant, computing      196—215
Gromov — Witten invariant, computing elliptic invariants of $\mathbb{P}^2$       210—212 310
Gromov — Witten invariant, computing elliptic invariants of $\mathbb{P}^3$       213
Gromov — Witten invariant, computing elliptic invariants of the quintic threefold      213 214
Gromov — Witten invariant, computing genus 0 invariants of $\mathbb{P}^2$       197—201 236 237 310
Gromov — Witten invariant, computing genus 0 invariants of Calabi — Yau threefolds      201—210 308 325
Gromov — Witten invariant, computing genus 0 invariants of the quintic threefold      184 201—203 Gromov
Gromov — Witten invariant, computing genus 2 invariants of $\mathbb{P}^2$       311

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