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

$A_{\infty}$ -category      403
$cpl(\sum)$       41—45 135—137 139 161
$cpl(\sum)$ , $\sum$ -convex      43
$cpl(\sum)$ , definition in $A^{+}(\Xi)$       43
$cpl(\sum)$ , definition in $A^{+}_{n-1}(X_{\sum})\otimes\mathbb{R}$       39
$I_{\nu}$       see “Givental I-function
$J_{\nu}$       see “Givental J-function
      333 350 Gromov
for a critical bundle      352
in multiple cover case      352 355
      336 375
, fixed point loci      376
, fixed point loci, normal bundle of      377
, map ft to       375
      334
, fixed point loci      334 339
, fixed point loci, description of      334
, fixed point loci, graph $\Gamma$       339
, fixed point loci, normal bundle of      341
, map $\varphi$ to       336 337 339—341 344 375
, torus action on      334
, Gromov — Witten invariant for $\mathbb{P}^2$       198
, Gromov — Witten invariant for the quintic threefold      201
, projective space      334 375
, projective space, equivariant cohomology of      335
, projective space, equivariant hyperplane class of      335
, projective space, fixed points of      335 340
, projective space, torus action on      335
, projective space, weights of      335
$\har{Q}$       27 28 340 343 344 347 348 350 351 353
$\har{Q}$ for a concavex bundle      349
$\har{Q}$ in multiple cover case      352 355
$\har{Q}$ is Euler data      339
$\har{Q}$ , definition of      337
$\hat{P}$       27 28 101 164 343 346 353
$\hat{P}$ for a concavex bundle      350
$\hat{P}$ in multiple cover case      355
$\hat{P}$ is Euler data      339
$\hat{P}$ , definition of      337
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up      49 89
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in      427
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in A-model correlation function of      269 386 388
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in Givental function $I_{\nu}$       384
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in Givental function $J_{\nu}$       386 388
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in Gromov — Witten potential of      387 388
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in Hodge numbers of      146
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in instanton numbers of      269 270
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in Kahler cone of      146 269 383
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in Kahler moduli of      146 147
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in mirror map of      158 159 164—166 270 385 388
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in Mirror Theorem for      383; 386 388
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in moduli coordinates of      269
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in SCFT moduli of      147
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in toric Kahler cone of      148
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Calabi — Yau hypersurface in toric Kahler moduli of      148 269
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, GKZ decomposition of      50 124 146
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, Mirror Conjecture for      269
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in      88 95 103 123 145 164
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in A-system of      96 385
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in discriminant locus of      147
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in Frobenius method      165
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in Gauss — Manin connection of      269 385 386
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in Hodge numbers of      146
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in homogeneous coordinate ring of      89
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in moduli coordinates of      89 95 148 158 387
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in monodromy of      270 385 387
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in periods of      164 165 385
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in Picard — Fuchs equations of      90 96 103 158 165 385 386
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in satisfies integrality conjecture      270 385
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in simplified moduli of      147 148
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, mirror of Calabi — Yau hypersurface in Yukawa couplings of      103 104 125 385 388
$\mathbb{P}(1,1,2,2,2)$ and its toric blow-up, polytope of      88
$\mathcal{L}_i$       see “Cotangent line $\mathcal{L}_i$ ”
$\nu'_{d,k,i}$       358 373
$\nu'_{\beta,k,i}$       364 (see also $$\nu'_{d</a></span> <span class=subjpages><a href=$
$\nu_d$ for $\mathcal{O}_\mathbb{P}^n}(l)$       337 340 344
$\nu_d$ for a concavex bundle      349 351 358 375
$\nu_d$ for quintic threefold      181 184 201 288 299 302 332
$\nu_d$ in multiple cover case      352
$\nu_{d,k}$       357 358 360
$\nu_{\beta,k}$       364 (see also $$\nu_{d</a></span> <span class=subjpages><a href=$
$\Omega$ -Euler data      see “Euler data”
$\Sigma$ -convex      see “ $cpl(\Sigma)$
A-discriminant      122 125
A-hypergeometric equations      see “A-system”
A-model      7 422 423
A-model is a (l + 1)-dimensional TQFT      434
A-model, connection      82 111 258 259 265—267 271
A-model, connection has maximally unipotent monodromy      250
A-model, connection of a Calabi — Yau threefold      248 255—257 318 328 329
A-model, connection, asymptotics of      245
A-model, connection, canonical extension of      249 251 254 316
A-model, connection, convergence assumption      245 249 261 316
A-model, connection, definition of      244
A-model, connection, flat sections of      248 249 315—318
A-model, connection, flatness of      244
A-model, connection, monodromy of      246 248 249 316
A-model, connection, monodromy weight filtration of      249 250 253
A-model, connection, Picard — Fuchs operator for      see “Picard — Fuchs operator”
A-model, connection, role of basis       251
A-model, connection, satisfies integrality conjecture      250 258
A-model, correlation function      see “Correlation function A-mode”l
A-system      90 (see also “GKZ system and hypergeometric equations”)
A-system of a Calabi-Yau toric hypersurface      see “Calabi — Yau toric
A-system of a smooth Fano toric variety      390 391
A-system, definition of      91
A-system, example of      93—96 125 162
A-system, holomorphic solution at maximally unipotent boundary point      160
A-system, solutions of      98 100 164
A-variation of Hodge structure      25 107 222 253 254 connection”)
A-variation of Hodge structure, definition of      251 255
A-variation of Hodge structure, Griffiths transversality for      252 254
A-variation of Hodge structure, Hodge filtration of      251
A-variation of Hodge structure, integer lattice of      249 251 254 260
A-variation of Hodge structure, middle      258
A-variation of Hodge structure, middle has maximally unipotent monodromy      255
A-variation of Hodge structure, middle, definition of      255
A-variation of Hodge structure, middle, mixed Hodge structure is Hodge — Tate      255 264
A-variation of Hodge structure, middle, real structure on      255 264
A-variation of Hodge structure, middle, toric      271
A-variation of Hodge structure, mixed Hodge structure of      253 255 connection monodromy
A-variation of Hodge structure, polarization of      252 254 258
Abel — Jacobi mapping      67 427
action      412 416
Action of sigma model      420 421
Action of sigma model, bosonic part of      420
Action of sigma model, fermionic part of      420
Algebraic space      170
Algebraic stack      169
Algebraic stack, bundle on      see “Euler class stack
Algebraic stack, Deligne — Mumford      178
Algebraic stack, intersection theory on      178 279
Algebraic stack, quotient of      178
Algebraic stack, smooth      170 279
Almost complex structure      205 208
Almost complex structure, compatible      185 188
Almost complex structure, tamed      185
ample      33 35 40
Annihilation operator      415 418
Anticanonical divisor      46 48 49 51 54
Anticanonical linear system      55 56
Anticommutator      422
Arnold strange duality      401
B-model      8 422 423
B-model, correlation function      see “Correlation function B-model”
B-model, potential function      108 109 261 263
B-model, potential function in terms of periods      389 399
B-model, potential function of the quintic mirror      see “Quintic mirror B-model
B-model, potential function, toric      269 388
Batyrev mirror      see “Mirror construction Batyrev”

Batyrev quantum ring      223 225 389
Batyrev quantum ring for nef      393
Batyrev quantum ring for $\mathbb{F}_2$       393—395
Batyrev quantum ring for $\mathbb{P}^r$       224
Batyrev quantum ring for a Fano toric variety      392
Batyrev quantum ring, definition of      224
Bipyramid construction      70 72
Birational cone conjecture      see “Cone conjecture birational”
Bivariant Chow group      180 182
Black holes      145 404
Black holes, massless      145
Boson      2 419 420 425
Bosonic components      427 429
Bosonic field      419
Bosonic variables      427 428
BRST cohomology      302 422 423
BRST operator      422 423
Caiabi-Yau cone      139
Caiabi-Yau cone, example of      140
Calabi — Yau, automorphism group of      12 127 129—131 138
Calabi — Yau, constructed from K3 surfaces      66
Calabi — Yau, generalized      65
Calabi — Yau, manifold      1 10 185 218 222 243 246 264—267 315 317 359 381 421 427
Calabi — Yau, manifold with K3 or elliptic fibration      404
Calabi — Yau, minimal      267
Calabi — Yau, moduli of      see “Moduli”
Calabi — Yau, orbifold      12 17 55 408 410
Calabi — Yau, rigid      65 261 267
Calabi — Yau, threefold      3 4 6 7 9 10 53 74 79 102—112 130 132—134 143 151 201—210 225—227 238 247 248 251 255—262 267 282 288—293 308 318 325—329 352 366 388 389 405 423 425 427 A-model;
Calabi — Yau, toric complete intersection      see “Toric complete intersection Calabi
Calabi — Yau, toric hypersurface      53 56 60 116—127 134—143 155—166 267 270 383 430
Calabi — Yau, toric hypersurface, A-system of      91—93
Calabi — Yau, toric hypersurface, complex moduli of      see “Moduli complex”
Calabi — Yau, toric hypersurface, Kahler moduli of      see “Moduli Kahler”
Calabi — Yau, toric hypersurface, nondegenerate      122
Calabi — Yau, toric hypersurface, Picard — Fuchs equations of      91 92 383
Calabi — Yau, variety      55 60 63 407
Calabi — Yau, variety, definition of      11
Calabi — Yau, variety, minimal      11 55 63 116 407 409 410
Calabi — Yau, weighted projective hypersurface      70 207 405
Calabi — Yau, weighted projective hypersurface, list of      55 60 72 145
Canonical extension      75 154 247 249 260
Canonical extension of A-model connection      see “A-model connection canonical
Categorical quotient      37
Cauchy — Riemann equation      185 187 191
Cauchy — Riemann equation, inhomogeneous      187
Cauchy — Riemann equation, perturbed      187
Cayley trick      64
Central charge      310 425
Chiral ring      36 217 427
Chiral superfield      426—428
Chow quotient      117 119 120 148
Class $\mathcal{P}$       see “Mirror theorems Givental class
Clemens conjecture      15 16 202 206 214 400
Clemens conjecture, statement of      202
Cohomological field theory      241 (see also “Dubrovin formalism”)
Cohomological field theory for higher genus      241
Cohomological field theory, correlation functions of      241
Cohomological field theory, Kunneth formula for      241
Combinatorial duality      42 46
Complete intersection      364
Complete intersection in $\mathbb{P}^n$       215 364
Complete intersection in a convex variety      366
Complete intersection in a convex variety nef      365
Complete intersection in a flag manifold      see “Complete intersection in a convex varietyflag manifold”
Complete intersection in a Grassmannian      see “Grassmannian”
Complete intersection toric      see “Toric complete intersection”
Complete intersection virtual fundamental class of      see “Virtual fundamental class of
Complex reflection      408
Complexified, Kahler class      3 6—9 11 12 15 18 22 204 217 218 225 228 230 303 304 434
Complexified, Kahler moduli      see “Moduli Kahler”
Complexified, Kahler space      12 18 127 128 152 243 245 251
Concave bundle      349
Concave bundle in multiple cover case      352
Concavex bundle      337 349 350 352
Concavex bundle, critical      350 352
Concavex bundle, critical, definition of      351
Cone conjecture      130 134 150 152 245 399
Cone conjecture, birational      133
Cone conjecture, consequences of      130
Cone conjecture, when known      130
Cone, Calabi — Yau      see “Calabi — Yau cone”
Cone, dual      see “Dual cone”
Cone, face of      see “Face of a cone”
Cone, Gorenstein      64
Cone, Gorenstein, reflexive      see “Reflexive Gorenstein cone”
Cone, homogeneous self-adjoint      131
Cone, rational polyhedral      see “Rational polyhedral cone”
Conformai algebra      2 423 424
Conformal field theory      423—426
Conformal field theory of a sigma model      425
Conformal field theory, gauge choice in      424
Conformal field theory, gauge group of      424
Conformal group      424 425
Conformal invariance      420 424
Conifold      122 143
Conifold, transition      143 404—406 type
Conifold, transition, example of      144
Conjugate momentum      412 414
Connection $\nabla^c$       247 (see also “A-model connection”)
Connection $\nabla^c$ , flat sections of      247—249 254 316 318
Connection matrix      25 27 76 79 106 108 246
Conserved current      419
Convex function      39 (see also “ $cpl(\sum)$ ”)
Convex function, strictly      39 42 43 45
Convex, bundle      349 (see also “Concavex bundle”)
Convex, line bundle      364
Convex, variety      298 358 365 366 381 382
Convex, variety, definition of      180
Correlation function      1 6 56 152 158 241 260 303 364 416 421
Correlation function of a Frobenius algebra      432
Correlation function, A-model      7—9 11 107 203 259 266 268 385 387 389 422 423 A-model
Correlation function, A-model of a Calabi — Yau threefold      204 226 255 256 327
Correlation function, A-model, definition of      221
Correlation function, A-model, effect of a flop      227
Correlation function, A-model, in terms of instanton numbers      226
Correlation function, B-model      8 9 110 256 259 265 268 385 387 423 potential
Correlation function, B-model, toric      268
Correlation function, genus g      434
Correlation function, n-point      416 434
Correlation function, quintic      see “Quintic mirror and quintic threefold”
Correlation function, three-point      6—10 221 226 227 255 256 422
Correlation function, Voisin — Borcea mirror construction      69
Cotangent complex      178
Cotangent line $\mathcal{L}_r$       302 369
Creation operator      415 418
Crepant      54
Critical bundle      see “Concavex bundle critical”
Curvature tensor      420
D-brane      402 404
Deformations, example of      125 147 158 159
Deformations, infinitesimal      57 153 175
Deformations, infinitesimal of a minimal Calabi — Yau orbifold      410
Deformations, infinitesimal of an orbifold      410
Deformations, non-polynomial      58 147
Deformations, polynomial      57 61
Deformations, principal component      122 125
Discriminant locus      104 122 toric nondegenerate”)
Dominance conjecture      117 155
Double covers of 6-nodai quintics      293—297
Double covers of 6-nodai quintics, contribution to Gromov — Witten invariant and virtual fundamental class      297
Double covers of 6-nodai quintics, covers factoring through normalization      295—297
Double covers of 6-nodai quintics, covers not factoring through normalization      295 296
Double covers of 6-nodai quintics, moduli of      295
Double covers of 6-nodai quintics, moduli of tangent space of      296
Dual cone      31
Dualizing sheaf      11 46 407 409
Dubrovin connection      239 311 312
Dubrovin connection, definition of      239
Dubrovin connection, Euler vector field of      241

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