Enumerative Invariants in Algebraic Geometry and String Theory

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1 Dan Abramovich -. Marcos Marino Michael Thaddeus Ravi Vakil 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 Editors: Kai Behrend Marco Manetti

2 Preface Lectures on Gromov Witten Invariants of Orbifolds D. Abramovich 1 1 Introduction What This Is Introspection Where Does All This Come Prom? Acknowledgements 2 2 Gromov-Witten Theory Kontsevich's Formula Set-Up for a Streamlined Proof 3 ' 2.3 The Space of Stable Maps Natural Maps Boundary of Moduli :...:.: :'.! Gromov-Witten Classes The WDVV Equations.V. ; Proof of WDW... :./.....:....' About the General Case....: Orbifolds/Stacks..'..'...'.I : Geometric Orbifolds...:...! Moduli Stacks....' Where Do Stacks Come Up? Attributes of Orbifolds Etale Gerbes 20 4, Twisted Stable Maps Stable Maps to a Stack Twisted Curves Twisted Stable Maps 23 J 4.4- Transparency 25: The Stack of Twisted Stable Maps Twisted Curves and Roots.. ':'.:' Valuative Criterion for Properness. 27 V

3 VIII 5 Gromov-Witten Classes..._ Contractions Gluing and Rigidified Inertia Evaluation Maps The Boundary of Moduli : Orbifold Gromov-Witten Classes Fundamental Classes 34 6 WDW, Grading and Computations The Formula Quantum Cohomology and Its Grading Grading the Rings Examples Other Work "; Mirror Symmetry and the Crepant 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 Topological Vertex M. Marino 49 1 Introduction and Overview 49 2 Chern-Simons Theory Basic Ingredients Perturbative Approach Non-Perturbative Solution Framing Dependence...: The 1/N Expansion in Chern-Simons Theory 70 3 Topological Strings Topological Strings and Gromov-Witten Invariants Integrality Properties and Gopakumar-Vafa Invariants Open Topological Strings 77 4 Toric Geometry and Calabi-Yau Threefolds ; Non-Compact Calabi-Yau Geometries: An Introduction Constructing Toric Calabi-Yau Manifolds Examples of Closed String Amplitudes 87 5 The Topological Vertex The Gopakumar-Vafa Duality Framing of Topological Open String Amplitudes Definition of the Topological Vertex Gluing Rules Explicit Expression for the Topological Vertex Applications 96 A Symmetric Polynomials 99 References 100

4 IX Floer Cohomology with Gerbes M. Thaddeus " Floer Cohomology Newton's Second Law The Hamiltonian Formalism The Arnold Conjecture Floer's Proof Morse Theory Bott-Morse Theory Morse Theory on the Loop Space Re-Interpretation #1: Sections of the Symplectic Mapping Torus Re-Interpretation #2: Two Lagrangian Submanifolds Product Structures The Finite-Order Case Givental's Philosophy Gerbes Definition of Stacks Examples of Stacks Morphisms and 2-Morphisms Definition of Gerbes The Gerbe of Liftings The Lien of a Gerbe Classification of Gerbes Allowing the Base Space to Be a Stack Definition of Orbifolds Twisted Vector Bundles Strominger-Yau-Zaslow Orbifold Cohomology and Its Relatives Cohomology of Sheaves on Stacks The Inertia Stack Orbifold Cohomology Twisted Orbifold Cohomology The Case of Discrete Torsion The Fantechi-Gottsche Ring Twisting the Fantechi-Gottsche Ring with Discrete Torsion Twisting It with an Arbitrary Flat Unitary Gerbe The Loop Space of an Orbifold Addition of the Gerbe The Non-Orbifold Case The Equivariant Case A Concluding Puzzle Notes on the Literature Notes to Lecture Notes to Lecture Notes to Lecture 3 140

5 X The Moduli Space of Curves and Gromov-Witten Theory R. Vakil " Introduction The Moduli Space of Curves Tautological Cohomology Classes on Moduli Spaces of Curves, and Their Structure A Blunt Tool: Theorem * and Consequences Stable Relative Maps to P 1 and Relative Virtual Localization Applications of Relative Virtual Localization : ' Towards Faber's Intersection Number Conjecture 3.23 via Relative : Virtual Localization Conclusion 194 References : 194 List of Participants 199

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