Enumerative invariants in algebraic geometry and string theory: lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2008
|
| Series: | Lecture notes in mathematics
1947 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 201 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783540798132 3540798137 9783540798149 |
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MARC
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| 020 | |a 3540798137 |c Pb. : EUR 42.75 (freier Pr.), sfr 66.50 (freier Pr.) |9 3-540-79813-7 | ||
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| 245 | 1 | 0 | |a Enumerative invariants in algebraic geometry and string theory |b lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 |c Dan Abramovich ... ; Ed.: Kai Behrend ... |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2008 | |
| 300 | |a X, 201 S. |b graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1947 | |
| 650 | 4 | |a Géométrie algébrique - Congrès | |
| 650 | 4 | |a Modèles des cordes vibrantes (Physique nucléaire) - Congrès | |
| 650 | 4 | |a Geometry, Algebraic |v Congresses | |
| 650 | 4 | |a String models |v Congresses | |
| 650 | 0 | 7 | |a Algebraische Geometrie |0 (DE-588)4001161-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Stringtheorie |0 (DE-588)4224278-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Invariante |0 (DE-588)4128781-2 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Stringtheorie |0 (DE-588)4224278-2 |D s |
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| 700 | 1 | |a Abramovich, Dan |e Sonstige |0 (DE-588)136162762 |4 oth | |
| 700 | 1 | |a Behrend, Kai |d 1961- |e Sonstige |0 (DE-588)1024401200 |4 oth | |
| 830 | 0 | |a Lecture notes in mathematics |v 1947 |w (DE-604)BV000676446 |9 1947 | |
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Record in the Search Index
| _version_ | 1804137956222435328 |
|---|---|
| adam_text | Contents
Preface
........................................................
V
Lectures
on Gromov-
Witten
Invariants of Orbifolds
D. Abramovich
.................................................. 1
1
Introduction
.................................................. 1
1.1
What This Is
............................................. 1
1.2
Introspection
............................................. 1
1.3
Where Does All This Come
Erom?
.......................... 2
1.4
Acknowledgements
........................................ 2
2
Gromov-Witten Theory
........................................ 2
2.1
Kontsevich s Formula
...................................... 2
2.2
Set-Up for a Streamlined Proof
............................. 3
2.3
The Space of Stable Maps
.................................. 7
2.4
Natural Maps
............................................. 8
2.5
Boundary of Moduli
....................................... 9
2.6
Gromov-Witten Classes
.................................... 10
2.7
The WDW Equations
..................................... 11
2.8
Proof of WDW
........................................... 12
2.9
About the General Case
.................................... 15
3
Orbifolds/Stacks
.............................................. 16
3.1
Geometric Orbifolds
....................................... 16
3.2
Moduli Stacks
............................................ 17
3.3
Where Do Stacks Come Up?
................................ 19
3.4
Attributes of Orbifolds
..................................... 19
3.5
Étale Gerbes .............................................
20
4
Twisted Stable Maps
.......................................... 21
4.1
Stable Maps to a Stack
.................................... 21
4.2
Twisted Curves
........................................... 22
4.3
Twisted Stable Maps
...................................... 23
4.4
Transparency
25:
The Stack of Twisted Stable Maps
........... 24
4.5
Twisted Curves and Roots
................................. 25
4.6
Valuative Criterion for Properness
........................... 27
VIII Contents
5
Gromov-Witten Glasses
........................................ 29
5.1
Contractions
.............................................. 29
5.2
Gluing and Rigidified Inertia
................................ 29
5.3
Evaluation Maps
.......................................... 31
5.4
The Boundary of Moduli
................................... 32
5.5
Orbifold Gromov-Witten Classes
............................ 32
5.6
Fundamental Classes
...................................... 34
6
WDW, Grading and Computations
............................. 35
6.1
The Formula
.............................................. 35
6.2
Quantum Cohomology and Its Grading
...................... 36
6.3
Grading the Rings
......................................... 38
6.4
Examples
................................................ 38
6.5
Other Work
.............................................. 41
6.6
Mirror Symmetry and the
Crêpant
Resolution Conjecture
...... 42
A The Legend of String Cohomology: Two Letters of Maxim
Kontsevich to Lev
Borisov
...................................... 43
A.I The Legend of String Cohomology
........................... 43
A.
2
The Archaeological Letters
................................. 44
References
...................................................... 46
Lectures on the
Topologica!
Vertex
M. Marino
...................................................... 49
1
Introduction and Overview
..................................... 49
2
Chern-Simons Theory
......................................... 51
2.1
Basic Ingredients
.......................................... 51
2.2
Perturbative Approach
..................................... 55
2.3
Non-Perturbative Solution
.................................. 61
2.4
Framing Dependence
...................................... 68
2.5
The 1/JV Expansion in Chern-Simons Theory
................. 70
3
Topological Strings
............................................ 73
3.1
Topological Strings and Gromov-Witten Invariants
............ 74
3.2
Integrality Properties and Gopakumar-Vafa Invariants
......... 76
3.3
Open Topological Strings
................................... 77
4
Toric Geometry and Calabi-Yau Threefolds
....................... 79
4.1
Non-Compact Calabi-Yau Geometries: An Introduction
........ 79
4.2
Constructing Toric Calabi-Yau Manifolds
.................... 81
4.3
Examples of Closed String Amplitudes
....................... 87
5
The Topological Vertex
........................................ 89
5.1
The Gopakumar-Vafa Duality
.............................. 89
5.2
Framing of Topological Open String Amplitudes
............... 89
5.3
Definition of the Topological Vertex
......................... 91
5.4
Gluing Rules
............................................. 93
5.5
Explicit Expression for the Topological Vertex
................ 95
5.6
Applications
.............................................. 96
A Symmetric Polynomials
........................................ 99
References
...................................................... 100
Contents
IX
Floer Cohornology
with
Gerbes
M.
Thaddens....................................................105
1
Floer Cohomology
.............................................106
1.1
Newton s
Second
Law
......................................106
1.2
The Hamiltonian Formalism
................................107
1.3
The Arnold Conjecture
....................................108
1.4
Floer s Proof
.............................................108
1.5
Morse Theory
.............................................109
1.6
Bott-Morse
Theory
........................................110
1.7
Morse Theory on the Loop Space
............................110
1.8
Re-Interpretation
#1:
Sections of the Symplectic Mapping Torus
112
1.9
Re-Interpretation
#2:
Two Lagrangian Submanifolds
..........113
1.10
Product Structures
........................................114
1.11
The Finite-Order Case
.....................................115
1.12
Giventaľs
Philosophy
......................................115
2
Gerbes
.......................................................117
2.1
Definition of Stacks
........................................117
2.2
Examples of Stacks
........................................118
2.3
Morphisms and 2-Morphisnis
...............................118
2.4
Definition of
Gerbes
.......................................120
2.5
The
Gerbe
of Liftings
......................................121
2.6
The Lien of
a Gerbe
.......................................122
2.7
Classification of
Gerbes
....................................123
2.8
Allowing the Base Space to Be a Stack
.......................123
2.9
Definition of Orbifolds
.....................................124
2.10
Twisted Vector Bundles
....................................124
2.11
Strominger-Yau-Zaslow
....................................125
3
Orbifold Cohomology and Its Relatives
...........................126
3.1
Cohomology of Sheaves on Stacks
...........................126
3.2
The Inertia Stack
.........................................127
3.3
Orbifold Cohomology
......................................128
3.4
Twisted Orbifold Cohomology
..............................129
3.5
The Case of Discrete Torsion
...............................129
3.6
The
Fantechi-Göttsche
Ring
................................130
3.7
Twisting the
Fantechi-Göttsche
Ring with Discrete Torsion
.....131
3.8
Twisting It with an Arbitrary Flat Unitary
Gerbe.............131
3.9
The Loop Space of an Orbifold
..............................132
3.10
Addition of the
Gerbe .....................................134
3.11
The Non-Orbifold Case
....................................135
3.12
The Equivariant Case
......................................135
3.13
A Concluding Puzzle
......................................136
4
Notes on the Literature
........................................137
4.1
Notes to Lecture
1.........................................137
4.2
Notes to Lecture
2.........................................139
4.3
Notes to Lecture
3.........................................140
X
Contents
The Moduli Space of Curves and Gromov-Witten Theory
R. Vakil
........................................................143
1
Introduction
..................................................143
2
The Moduli Space of Curves
....................................145
3
Tautological Cohomology Classes on Moduli Spaces
of Curves, and Their Structure
..................................154
4
A Blunt Tool: Theorem
*
and Consequences
......................173
5
Stable Relative Maps to P1 and Relative Virtual Localization
.......177
6
Applications of Relative Virtual Localization
......................186
7
Towards Faber s Intersection Number Conjecture
3.23
via Relative
Virtual Localization
...........................................190
8
Conclusion
...................................................194
References
......................................................194
List of Participants
............................................199
|
| adam_txt |
Contents
Preface
.
V
Lectures
on Gromov-
Witten
Invariants of Orbifolds
D. Abramovich
. 1
1
Introduction
. 1
1.1
What This Is
. 1
1.2
Introspection
. 1
1.3
Where Does All This Come
Erom?
. 2
1.4
Acknowledgements
. 2
2
Gromov-Witten Theory
. 2
2.1
Kontsevich's Formula
. 2
2.2
Set-Up for a Streamlined Proof
. 3
2.3
The Space of Stable Maps
. 7
2.4
Natural Maps
. 8
2.5
Boundary of Moduli
. 9
2.6
Gromov-Witten Classes
. 10
2.7
The WDW Equations
. 11
2.8
Proof of WDW
. 12
2.9
About the General Case
. 15
3
Orbifolds/Stacks
. 16
3.1
Geometric Orbifolds
. 16
3.2
Moduli Stacks
. 17
3.3
Where Do Stacks Come Up?
. 19
3.4
Attributes of Orbifolds
. 19
3.5
Étale Gerbes .
20
4
Twisted Stable Maps
. 21
4.1
Stable Maps to a Stack
. 21
4.2
Twisted Curves
. 22
4.3
Twisted Stable Maps
. 23
4.4
Transparency
25:
The Stack of Twisted Stable Maps
. 24
4.5
Twisted Curves and Roots
. 25
4.6
Valuative Criterion for Properness
. 27
VIII Contents
5
Gromov-Witten Glasses
. 29
5.1
Contractions
. 29
5.2
Gluing and Rigidified Inertia
. 29
5.3
Evaluation Maps
. 31
5.4
The Boundary of Moduli
. 32
5.5
Orbifold Gromov-Witten Classes
. 32
5.6
Fundamental Classes
. 34
6
WDW, Grading and Computations
. 35
6.1
The Formula
. 35
6.2
Quantum Cohomology and Its Grading
. 36
6.3
Grading the Rings
. 38
6.4
Examples
. 38
6.5
Other Work
. 41
6.6
Mirror Symmetry and the
Crêpant
Resolution Conjecture
. 42
A The Legend of String Cohomology: Two Letters of Maxim
Kontsevich to Lev
Borisov
. 43
A.I The Legend of String Cohomology
. 43
A.
2
The Archaeological Letters
. 44
References
. 46
Lectures on the
Topologica!
Vertex
M. Marino
. 49
1
Introduction and Overview
. 49
2
Chern-Simons Theory
. 51
2.1
Basic Ingredients
. 51
2.2
Perturbative Approach
. 55
2.3
Non-Perturbative Solution
. 61
2.4
Framing Dependence
. 68
2.5
The 1/JV Expansion in Chern-Simons Theory
. 70
3
Topological Strings
. 73
3.1
Topological Strings and Gromov-Witten Invariants
. 74
3.2
Integrality Properties and Gopakumar-Vafa Invariants
. 76
3.3
Open Topological Strings
. 77
4
Toric Geometry and Calabi-Yau Threefolds
. 79
4.1
Non-Compact Calabi-Yau Geometries: An Introduction
. 79
4.2
Constructing Toric Calabi-Yau Manifolds
. 81
4.3
Examples of Closed String Amplitudes
. 87
5
The Topological Vertex
. 89
5.1
The Gopakumar-Vafa Duality
. 89
5.2
Framing of Topological Open String Amplitudes
. 89
5.3
Definition of the Topological Vertex
. 91
5.4
Gluing Rules
. 93
5.5
Explicit Expression for the Topological Vertex
. 95
5.6
Applications
. 96
A Symmetric Polynomials
. 99
References
. 100
Contents
IX
Floer Cohornology
with
Gerbes
M.
Thaddens.105
1
Floer Cohomology
.106
1.1
Newton's
Second
Law
.106
1.2
The Hamiltonian Formalism
.107
1.3
The Arnold Conjecture
.108
1.4
Floer's Proof
.108
1.5
Morse Theory
.109
1.6
Bott-Morse
Theory
.110
1.7
Morse Theory on the Loop Space
.110
1.8
Re-Interpretation
#1:
Sections of the Symplectic Mapping Torus
112
1.9
Re-Interpretation
#2:
Two Lagrangian Submanifolds
.113
1.10
Product Structures
.114
1.11
The Finite-Order Case
.115
1.12
Giventaľs
Philosophy
.115
2
Gerbes
.117
2.1
Definition of Stacks
.117
2.2
Examples of Stacks
.118
2.3
Morphisms and 2-Morphisnis
.118
2.4
Definition of
Gerbes
.120
2.5
The
Gerbe
of Liftings
.121
2.6
The Lien of
a Gerbe
.122
2.7
Classification of
Gerbes
.123
2.8
Allowing the Base Space to Be a Stack
.123
2.9
Definition of Orbifolds
.124
2.10
Twisted Vector Bundles
.124
2.11
Strominger-Yau-Zaslow
.125
3
Orbifold Cohomology and Its Relatives
.126
3.1
Cohomology of Sheaves on Stacks
.126
3.2
The Inertia Stack
.127
3.3
Orbifold Cohomology
.128
3.4
Twisted Orbifold Cohomology
.129
3.5
The Case of Discrete Torsion
.129
3.6
The
Fantechi-Göttsche
Ring
.130
3.7
Twisting the
Fantechi-Göttsche
Ring with Discrete Torsion
.131
3.8
Twisting It with an Arbitrary Flat Unitary
Gerbe.131
3.9
The Loop Space of an Orbifold
.132
3.10
Addition of the
Gerbe .134
3.11
The Non-Orbifold Case
.135
3.12
The Equivariant Case
.135
3.13
A Concluding Puzzle
.136
4
Notes on the Literature
.137
4.1
Notes to Lecture
1.137
4.2
Notes to Lecture
2.139
4.3
Notes to Lecture
3.140
X
Contents
The Moduli Space of Curves and Gromov-Witten Theory
R. Vakil
.143
1
Introduction
.143
2
The Moduli Space of Curves
.145
3
Tautological Cohomology Classes on Moduli Spaces
of Curves, and Their Structure
.154
4
A Blunt Tool: Theorem
*
and Consequences
.173
5
Stable Relative Maps to P1 and Relative Virtual Localization
.177
6
Applications of Relative Virtual Localization
.186
7
Towards Faber's Intersection Number Conjecture
3.23
via Relative
Virtual Localization
.190
8
Conclusion
.194
References
.194
List of Participants
.199 |
| any_adam_object | 1 |
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| author_GND | (DE-588)136162762 (DE-588)1024401200 |
| building | Verbundindex |
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| callnumber-search | QA564 |
| callnumber-sort | QA 3564 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SI 850 |
| classification_tum | PHY 014f MAT 145f MAT 517f |
| ctrlnum | (OCoLC)227033882 (DE-599)DNB988160765 |
| dewey-full | 516.3/5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 |
| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| format | Book |
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| genre | (DE-588)1071861417 Konferenzschrift 2005 Cetraro gnd-content |
| genre_facet | Konferenzschrift 2005 Cetraro |
| id | DE-604.BV035027387 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:48:30Z |
| indexdate | 2024-07-09T21:20:32Z |
| institution | BVB |
| isbn | 9783540798132 3540798137 9783540798149 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016696423 |
| oclc_num | 227033882 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-824 DE-83 DE-19 DE-BY-UBM DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-824 DE-83 DE-19 DE-BY-UBM DE-11 DE-188 |
| physical | X, 201 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 Dan Abramovich ... ; Ed.: Kai Behrend ... Berlin [u.a.] Springer 2008 X, 201 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1947 Géométrie algébrique - Congrès Modèles des cordes vibrantes (Physique nucléaire) - Congrès Geometry, Algebraic Congresses String models Congresses Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Stringtheorie (DE-588)4224278-2 gnd rswk-swf Invariante (DE-588)4128781-2 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2005 Cetraro gnd-content Algebraische Geometrie (DE-588)4001161-6 s Stringtheorie (DE-588)4224278-2 s Invariante (DE-588)4128781-2 s DE-604 Abramovich, Dan Sonstige (DE-588)136162762 oth Behrend, Kai 1961- Sonstige (DE-588)1024401200 oth Lecture notes in mathematics 1947 (DE-604)BV000676446 1947 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016696423&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 Lecture notes in mathematics Géométrie algébrique - Congrès Modèles des cordes vibrantes (Physique nucléaire) - Congrès Geometry, Algebraic Congresses String models Congresses Algebraische Geometrie (DE-588)4001161-6 gnd Stringtheorie (DE-588)4224278-2 gnd Invariante (DE-588)4128781-2 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4224278-2 (DE-588)4128781-2 (DE-588)1071861417 |
| title | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 |
| title_auth | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 |
| title_exact_search | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 |
| title_exact_search_txtP | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 |
| title_full | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 Dan Abramovich ... ; Ed.: Kai Behrend ... |
| title_fullStr | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 Dan Abramovich ... ; Ed.: Kai Behrend ... |
| title_full_unstemmed | Enumerative invariants in algebraic geometry and string theory lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 Dan Abramovich ... ; Ed.: Kai Behrend ... |
| title_short | Enumerative invariants in algebraic geometry and string theory |
| title_sort | enumerative invariants in algebraic geometry and string theory lectures given at the c i m e summer school held in cetraro italy june 6 11 2005 |
| title_sub | lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 6 - 11, 2005 |
| topic | Géométrie algébrique - Congrès Modèles des cordes vibrantes (Physique nucléaire) - Congrès Geometry, Algebraic Congresses String models Congresses Algebraische Geometrie (DE-588)4001161-6 gnd Stringtheorie (DE-588)4224278-2 gnd Invariante (DE-588)4128781-2 gnd |
| topic_facet | Géométrie algébrique - Congrès Modèles des cordes vibrantes (Physique nucléaire) - Congrès Geometry, Algebraic Congresses String models Congresses Algebraische Geometrie Stringtheorie Invariante Konferenzschrift 2005 Cetraro |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016696423&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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