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McDuff D., Salamon D. — J-Holomorphic Curves and Quantum Cohomology
McDuff D., Salamon D. — J-Holomorphic Curves and Quantum Cohomology



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Название: J-Holomorphic Curves and Quantum Cohomology

Авторы: McDuff D., Salamon D.

Аннотация:

$J$-holomorphic curves revolutionized the study of symplectic geometry when Gromov first introduced them in 1985. Through quantum cohomology, these curves are now linked to many of the most exciting new ideas in mathematical physics. This book presents the first coherent and full account of the theory of $J$-holomorphic curves, the details of which are presently scattered in various research papers. The first half of the book is an expository account of the field, explaining the main technical aspects. McDuff and Salamon give complete proofs of Gromov's compactness theorem for spheres and of the existence of the Gromov-Witten invariants. The second half of the book focuses on the definition of quantum cohomology. The authors establish that this multiplication exists, and give a new proof of the Ruan-Tian result that is associative on appropriate manifolds. They then describe the Givental-Kim calculation of the quantum cohomology of flag manifolds, leading to quantum Chern classes and Witten's calculation for Grassmannians, which relates to the Verlinde algebra. The Dubrovin connection, Gromov-Witten potential on quantum cohomology, and curve counting formulas are also discussed. The book closes with an outline of connections to Floer theory.


Язык: en

Рубрика: Математика/Алгебра/Квантовые группы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 207

Добавлена в каталог: 26.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(J, J')$-holomorphic      2
$D_u$, definition      24
$D_u$, ellipticity of      28
$D_u$, formula for      28
$D_u$, its adjoint $D_u^*$      29
$D_u$, right inverse      29 173—176
$D_u$, surjectivity of      29 38
a priori estimate for energy density      44
Action of short loop      45
Action on loop space      153
Action, Morse — Novikov theory for      154
Adjunction formula      39 101
Almost complex structure, $\omega$-compatible      25
Almost complex structure, $\omega$-tame      2 42
Almost complex structure, condition to be regular      38
Almost complex structure, generic      5
Almost complex structure, good      98
Almost complex structure, integrable      3
Almost complex structure, K-positive      59
Almost complex structure, K-semi-positive      59
Almost complex structure, positive      59
Almost complex structure, regular      24
Almost complex structure, semi-positive      59 143
Almost complex structure, smooth homotopy      25
Almost complex structure, tame versus compatible      26 60
Arnold conjecture      158 165
Aronszajn      15 197
Aronszajn's theorem      14 35
Aspinwall      ix 107 151 197
Aspinwall — Morrison formula      151
Astashkevich      119 197
Atiyah      197
Atiyah — Floer conjecture      163 165
Audin      ix 1 197
Bertram      123 197
Bubbling      41 46—49 155
Calabi — Yau manifold      1 12 137 141 145 148—151
Calabi — Yau manifold, examples      148 149
Calderon — Zygmund inequality      186—189
Callaghan      163 197
Candelas      141 148 197
Cauchy — Riemann equation      2 5 13—15 190
Cauchy — Riemann equation, perturbed      118 144 193
Cauchy — Riemann operator      24
Cauchy — Riemann operator, fundamental solution      190
Chern character      128
Chern class      2
Chern class, quantum      121
Chern number      38
Chern number, minimal      10 61
Chern number, negative      86 165
Chern number, zero      148
Ciocan — Fontanine      119 197
Cohomology, quantum      108
Compactness      1 5—6
Compactness Theorem      51 63 65 68
Compactness theorem, proof      52 81—87
Composition rule      137
Condition on dimension      see dimension condition
Condition, $(JA_3)$      67
Condition, $(JA_4)$      68 86 142
Condition, $(JA_p)$      67—69 85—87 101
Condition, (H1)      93 101
Condition, (H2)      93 96
Condition, (H3)      96
Condition, (H4)      101 103 114
Condition, (H5)      101 103 114
Condition, finiteness      see finiteness condtion
Conformal map      13 41
Conformal rescaling      41
Conley      155
Convergence problem      137 139 141 149 151
Critical point      15
Critical point, finite number of      15
Critical point, stable and unstable manifolds      161
Cross-ratio      105
Cup product, deformed      110 147
Cup product, given by triple intersection      107
Cup product, pair-of-pants      160
Cup product, weakly monotone case      141
Curve      3—4
Curve of negative Chern number      86 165
Curve, approximate J-holomorphic      30
Curve, as symplectic submanifold      3
Curve, counting discrete curves      101 138 151
Curve, critical point      15—18
Curve, cusp      see cusp-curve
Curve, determined by $\infty$-jet      14
Curve, energy      6
Curve, graph of      142
Curve, implicit function theorem for      30
Curve, injective point      4 18
Curve, intersections of      17
Curve, isolated      89
Curve, multiply-covered      4 18 119 142—144 165
Curve, parametrized      3
Curve, positivity of intersections      21 39 100 101
Curve, reducible      42
Curve, regular      24 38
Curve, simple      4 18—21
Curve, singular point of sequence      53
Curve, somewhere injective      4 18
Curve, somewhere injective, is simple      18
Curve, weak convergence      50
Cusp-curve      42 50 62
Cusp-curve with two components      80
Cusp-curve, framing D      63 76
Cusp-curve, label T      65 81
Cusp-curve, moduli space of      75
Cusp-curve, type D      76
Darboux's theorem      2
Daskalopoulos      123 197
Deformed cup product      see cup product
Determinant bundle      33
Dimension condition for $\Phi$      93
Dimension condition for $\Psi$      102
Dimension condition for a*b      110
Dimension condition for a*b*c      114
Donaldson      33 162 167 168 197
Donaldson's quantum category      162—165
Donaldson's quantum category for Lagrangians      164
Donaldson's quantum category for mapping tori      163
Dostoglou      30 155 162 163 197
Dubrovin      107 113 131 139 198
Dubrovin connection      113 132
Dubrovin connection and quantum products      133
Dubrovin connection, explicit formula      134
Dubrovin connection, flatness      136
Dubrovin connection, potential function      135
Elliptic bootstrapping      41 43 192—196
Elliptic regularity      25—27 29 35 191 192
Energy      6 42 44
Energy level      145
Energy, bounds derivative      44
Energy, conformal invariance      41
Energy, identity      41 43
Evaluation map      6—8 49
Evaluation map and orientations      98
Evaluation map as pseudo-cycle      94—98
Evaluation map for cusp-curves      79—81
Evaluation map for marked curves      66—69
Evaluation map is submersion      71—74
Evaluation map p-fold      8 65 66
Evaluation map p-fold, compactification      8 65
Evaluation map, domain of      64
Evaluation map, image of      62—65
Finiteness condition      144 147 156
Finiteness result      155
First Chern class      2
Flag manifold      see quantum cohomology
Floer      ix 12 15 41 153 155—158 162 164 198
Floer (co)homology      1 11 12 153—159
Floer (co)homology and quantum cohomology      160
Floer (co)homology of (H, J)      156
Floer (co)homology of symplectomorphism      162
Floer (co)homology, cochain complex      155
Floer (co)homology, Euler characteristic      164
Floer (co)homology, pair-of-pants product      160
Floer (co)homology, ring structure      159—160
Floer — Donaldson theory      163
Framing, D for cusp-curve      63 76
Fredholm operator      5
Fredholm operator, determinant bundle      33
Fredholm operator, index      5 24
Fredholm operator, regular value      5 33
Fredholm theory      1 4 5
Frobenius algebra      113 131
Frobenius condition      133
Gilbarg      186 188 198
Givental      ix 11 107 119 121 198
Green      141 148 197
Griffiths      33 38 100 198
Gromov      ix 7 41 50 89 107 143 198
Gromov invariant $\Phi$      66 89 93—94
Gromov invariant $\Phi$ for 6-manifolds      100
Gromov invariant $\Phi$ for conies in $\mathbb{C}P^2$      99
Gromov invariant $\Phi$ for discrete curves      101
Gromov invariant $\Phi$ for lines in $\mathbb{C}P^n$      98
Gromov invariant $\Phi$ on blow up      100
Gromov invariant $\Phi$, dimension condition      93
Gromov — Witten invariant $\Psi$      66 69 101—105
Gromov — Witten invariant $\Psi$, dimension condition      102
Gromov — Witten invariants      1 8—9
Gromov — Witten invariants, compared on $\mathbb{C}P^2$      104
Gromov — Witten invariants, composition rule      117
Gromov — Witten invariants, composition rule, on $\mathbb{C}P^2$      119
Gromov — Witten invariants, deformation invariance      96
Gromov — Witten invariants, mixed      90 138
Gromov — Witten invariants, signs      98
Gromov — Witten invariants, weakly monotone case      141
Gromov — Witten invariants, well defined for Kahler manifolds      96
Gromov — Witten potential      131—139
Gromov — Witten potential for $\mathbb{C}P^2$      138
Gromov — Witten potential for $\mathbb{C}P^n$      138
Gromov — Witten potential, non-monotone case      139
Gromov's compactness theorem      50—58
Grothendieck      38 198
Guillemin      92 198
Hamiltonian differential equation      153
Harmonic function      185
Harmonic function, mean value property      185
Harris      33 38 100 198
Hartman      15 198
Heegard splitting      165
Hoelder inequality      169
Hoelder norm      183
Hofer      12 15 40 41 51 54 146 153 155 157 158 162 198
Homology class of pure degree      108 131
Homology class, framed      63
Homology class, indecomposable      49
Homology class, J-effective      67
Homology class, spherical      6 47
Hurewicz homomorphism      6
Hurewicz homomorphism, implicit function theorem      27—33
Hurewicz homomorphism, integrable systems      113 121
Hurewicz homomorphism, isoperimetric inequality      41 45
J-holomorphic curve      see curve
J-holomorphic map      13
john      185 186 199
Kaehler manifold      61
Kaehler manifold and symplectic deformations      100
Kaehler manifold, Fano variety      60
Kaehler manifold, Gromov — Witten invariants      96
Kim      11 107 119 121 198
Kobayashi      199
Kobayaski      28
Kodaira vanishing theorem      38
Kontsevich      90 113 131 137—139 199
Kronhcimer      167 168 197
Labelling, T for cusp-curve      65 81
Lafontaine      ix 1 197
Lalonde, ix      7 199
Landau — Ginzburg potential      126 127
Laplace's equation, fundamental solution      185
Laurent polynomial ring      10
Lawson      44 199
Liu      118 199
Lizan      40
Loop space of manifold      146 154
Loop space, universal cover      154
Loop, action of      45
Lorek      40 98 199
Manin      90 113 131 137—139 199
Mapping tori and Atiyah — FIoer conjecture      163
Marked sphere      105
Maslov index      155
Mass of singular point      53
McDuff      ix 1 2 7 17 21 28 33 39 40 59 62 74 77 89 98—101 156 199 200
MEnor      92 129 200
Micallef      21 200
Minimal Chern number      10 61
Mirror symmetry conjecture      122 148 149
Moduli space      4—5
Moduli space of cusp-curves      75—78
Moduli space of flat connections      163
Moduli space of N-tuples of curves      74—75
Moduli space of two-component cusp-curves      80
Moduli space of unparametrized curves      52
Moduli space, cobordism of      25
Moduli space, compactifications      59—69
Moduli space, complex structure on      33
Moduli space, integration over      97 103
Moduli space, main theorems      23—25
Moduli space, orientation      32 33 98
Moduli space, regular point of      24
Moduli space, top strata in the boundary      82
Moduli space, universal      33 77
Moduli space, Yang — Mills      33
Monotone      see symplectic manifold
Morrison      ix 107 151 197
Morse — Novikov theory      146 154
Morse — Witten coboundary      158
Morse — Witten complex      161
Moser      121 122 200
Nijenhuis      74 200
Nijenhuis tensor      28
Nijenhuis tensor, anti-commutes with J      32
Nomizu      28 199
Non-squeezing theorem      7
Novikov      146 154 200
Novikov ring      11 12 141 144—148
Novikov ring as Laurent series ring      146
Number of curves in $\mathbb{C}P^2$      105 138
Number of curves in Calabi — Yau manifold      101 151
Oh      200
Ono      153 158 200
Ossa      141 148 197
Pansu      41 44 51 200
Parker      41 51 200
Parkes      141 148 197
Piunikhin      157 161 200
Poisson's identity      185
Pollack      92 198
Potential function      135
PRODUCT      see cup product
Pseudo-cycle      8 63 66 68 90—93
Pseudo-cycle of dimension k      90
Pseudo-cycle, bordant      90
Pseudo-cycle, transverse      91
Pseudo-cycle, weak representative      93
1 2
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