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

By: Michael Thaddeus, Kai Behrend (Editor), Marcos Marino

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Paperback | 22 August 2008

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Starting in the middle of the 80s, there has been a growing and fruitful interaction between algebraic geometry and certain areas of theoretical high-energy physics, especially the various versions of string theory. Physical heuristics have provided inspiration for new mathematical definitions (such as that of Gromov-Witten invariants) leading in turn to the solution of problems in enumerative geometry. Conversely, the availability of mathematically rigorous definitions and theorems has benefited the physics research by providing the required evidence in fields where experimental testing seems problematic. The aim of this volume, a result of the CIME Summer School held in Cetraro, Italy, in 2005, is to cover part of the most recent and interesting findings in this subject.

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Non-Fiction Mathematics Geometry Algebraic Geometry Algebra Differential & Riemannian Geometry Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Analytic Geometry

Preface	p. v
Lectures on Gromov-Witten Invariants of Orbifolds	p. 1
Introduction	p. 1
What This Is	p. 1
Introspection	p. 1
Where Does All This Come From?	p. 2
Acknowledgements	p. 2
Gromov-Witten Theory	p. 2
Kontsevich's Formula	p. 2
Set-Up for a Streamlined Proof	p. 3
The Space of Stable Maps	p. 7
Natural Maps	p. 8
Boundary of Moduli	p. 9
Gromov-Witten Classes	p. 10
The WDVV Equations	p. 11
Proof of WDVV	p. 12
About the General Case	p. 15
Orbifolds/Stacks	p. 16
Geometric Orbifolds	p. 16
Moduli Stacks	p. 17
Where Do Stacks Come Up?	p. 19
Attributes of Orbifolds	p. 19
Etale Gerbes	p. 20
Twisted Stable Maps	p. 21
Stable Maps to a Stack	p. 21
Twisted Curves	p. 22
Twisted Stable Maps	p. 23
Transparency 25: The Stack of Twisted Stable Maps	p. 24
Twisted Curves and Roots	p. 25
Valuative Criterion for Properness	p. 27
Gromov-Witten Classes	p. 29
Contractions	p. 29
Gluing and Rigidified Inertia	p. 29
Evaluation Maps	p. 31
The Boundary of Moduli	p. 32
Orbifold Gromov-Witten Classes	p. 32
Fundamental Classes	p. 34
WDVV, Grading and Computations	p. 35
The Formula	p. 35
Quantum Cohomology and Its Grading	p. 36
Grading the Rings	p. 38
Examples	p. 38
Other Work	p. 41
Mirror Symmetry and the Crepant Resolution Conjecture	p. 42
The Legend of String Cohomology: Two Letters of Maxim Kontsevich to Lev Borisov	p. 43
The Legend of String Cohomology	p. 43
The Archaeological Letters	p. 44
References	p. 46
Lectures on the Topological Vertex	p. 49
Introduction and Overview	p. 49
Chern-Simons Theory	p. 51
Basic Ingredients	p. 51
Perturbative Approach	p. 55
Non-Perturbative Solution	p. 61
Framing Dependence	p. 68
The 1/N Expansion in Chern-Simons Theory	p. 70
Topological Strings	p. 73
Topological Strings and Gromov-Witten Invariants	p. 74
Integrality Properties and Gopakumar-Vafa Invariants	p. 76
Open Topological Strings	p. 77
Toric Geometry and Calabi-Yau Threefolds	p. 79
Non-Compact Calabi-Yau Geometries: An Introduction	p. 79
Constructing Toric Calabi-Yau Manifolds	p. 81
Examples of Closed String Amplitudes	p. 87
The Topological Vertex	p. 89
The Gopakumar-Vafa Duality	p. 89
Framing of Topological Open String Amplitudes	p. 89
Definition of the Topological Vertex	p. 91
Gluing Rules	p. 93
Explicit Expression for the Topological Vertex	p. 95
Applications	p. 96
Symmetric Polynomials	p. 99
References	p. 100
Floer Cohomology with Gerbes	p. 105
Floer Cohomology	p. 106
Newton's Second Law	p. 106
The Hamiltonian Formalism	p. 107
The Arnold Conjecture	p. 108
Floer's Proof	p. 108
Morse Theory	p. 109
Bott-Morse Theory	p. 110
Morse Theory on the Loop Space	p. 110
Re-Interpretation #1: Sections of the Symplectic Mapping Torus	p. 112
Re-Interpretation #2: Two Lagrangian Submanifolds	p. 113
Product Structures	p. 114
The Finite-Order Case	p. 115
Givental's Philosophy	p. 115
Gerbes	p. 117
Definition of Stacks	p. 117
Examples of Stacks	p. 118
Morphisms and 2-Morphisms	p. 118
Definition of Gerbes	p. 120
The Gerbe of Liftings	p. 121
The Lien of a Gerbe	p. 122
Classification of Gerbes	p. 123
Allowing the Base Space to Be a Stack	p. 123
Definition of Orbifolds	p. 124
Twisted Vector Bundles	p. 124
Strominger-Yau-Zaslow	p. 125
Orbifold Cohomology and Its Relatives	p. 126
Cohomology of Sheaves on Stacks	p. 126
The Inertia Stack	p. 127
Orbifold Cohomology	p. 128
Twisted Orbifold Cohomology	p. 129
The Case of Discrete Torsion	p. 129
The Fantechi-Gottsche Ring	p. 130
Twisting the Fantechi-Gottsche Ring with Discrete Torsion	p. 131
Twisting It with an Arbitrary Flat Unitary Gerbe	p. 131
The Loop Space of an Orbifold	p. 132
Addition of the Gerbe	p. 134
The Non-Orbifold Case	p. 135
The Equivariant Case	p. 135
A Concluding Puzzle	p. 136
Notes on the Literature	p. 137
Notes to Lecture 1	p. 137
Notes to Lecture 2	p. 139
Notes to Lecture 3	p. 140
The Moduli Space of Curves and Gromov-Witten Theory	p. 143
Introduction	p. 143
The Moduli Space of Curves	p. 145
Tautological Cohomology Classes on Moduli Spaces of Curves, and Their Structure	p. 154
A Blunt Tool: Theorem * and Consequences	p. 173
Stable Relative Maps to P[superscript 1] and Relative Virtual Localization	p. 177
Applications of Relative Virtual Localization	p. 186
Towards Faber's Intersection Number Conjecture 3.23 via Relative Virtual Localization	p. 190
Conclusion	p. 194
References	p. 194
List of Participants	p. 199
Table of Contents provided by Ingram. All Rights Reserved.

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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