Symplectic geometry and mirror symmetry : proceedings of the 4th KIAS Annual International Conference, Korea Institute for Advanced Study, Seoul, South Korea, 14-18 August 2000

Proceedings of the 4th KIAS Annual International Conference, held in August 14-18, 2000, Seoul, South Korea. Leading experts in the field explore the more recent developments in relation to homological mirror symmetry, Floer theory, D-branes and Gromov-Witten invariants. Speakers 6 Preface 8 Contents 10 Estimated transversahty in symplectic geometry and projective maps 12 1 Introduction 12 2 Ample bundles over almost-complex manifolds 13 3 Estimated transversality in jet bundles 18 4 Proof of the main result 25 5 Examples and applications 32 References 40 Local mirror symmetry and five-dimensional gauge theory 42 1 Introduction 42 2 SU(2) Gauge Theory without Matter 43 3 Small and Large Radius Limits 48 4 SU(2) theory with matter 49 5 Discussions 53 References 59 The Toda conjecture 62 1 Witten's conjecture 63 2 The Toda lattice 68 3 Hamiltonian operators and the Toda lattice 73 4 The Toda conjecture 77 5 The Toda conjecture and the Virasoro conjecture 81 6 Propagating the Toda conjecture 87 Acknowledgments 88 References 88 Examples of special Lagrangian fibrations 92 Introduction 92 1 The Basic Construction 93 2 Resolutions of Toric Singularities 96 3 Deformations of toric singularities 104 4 Local mirror symmetry and connections with the work of Ruan and Joyce 109 5 The Future of the SYZ Conjecture 115 Acknowledgments 119 References 119 Linear models of supersymmetric D-branes 122 1 Introduction 122 2 N - 2 Boundary Superspace 125 3 A Linear Model: An Example 132 4 A-Type D-Branes in Linear Sigma Model 142 5 B-type D-Branes and Tachyom Condensation 160 6 B-Type D-Branes in Landau-Ginzburg Model 179 Acknowledgement 191 References 192 The connectedness of the moduli space of maps to homogeneous spaces 198 0 Introduction 198 1 The torus action on G/P 200 2 The C*-fiow 203 3 Connectedness 205 4 Rationality 208 References 211 Homological mirror symmetry and torus fibrations 214 1 Introduction 214 2 Degenerations of unitary Conformal Field Theories 217 3 Calabi-Yau manifolds in the large complex structure limit 225 4 Aoo-algebras and Aoo-categories 233 5 Fukaya category and its degeneration 239 6 Morse-Smale complex and the category of Morse functions 246 7 Aoo-structure for the derived category of coherent sheaves 258 8 Homological mirror conjecture 261 9 Appendix: constructions in the case of complex numbers 267 References 271 Genus-1 Virasoro conjecture on the small phase space 276 1 Genus-1 Virasoro conjecture on the small phase space 277 2 Compatibility of functions ok 280 3 Some sufficient conditions for genus-1 Virasoro conjecture 284 References 289 Obstruction to and deformation of Lagrangian intersection Floer cohomology 292 0. Introduction 293 1. Preliminaries and problems to overcome 293 2. Orientations and obstruction classes 297 3. Construction of the obstruction classes 300 4. Aoo-deformation of Lagrangian submanifold 303 5. Bounding cochains and deformation 312 6. Some applications 318 References 320 Topological open p-branes 322 1 Introduction 322 2 Preliminary 334 3 Topological Open Membrane 353 4 Back to the strings 375 Acknowledgement 389 References 389 Lagrangian torus fibration and mirror symmetry of Calabi-Yau manifolds 396 1 Introduction 396 2 Quintic case 401 3 The mirror of quintic 408 4 Calabi-Yau hyper surfaces and complete Intersections in toric variety 413 5 Conifold transition 416 6 Calabi-Yau complete Intersection In flag manifold 427 References 436 More about vanishing cycles and mutation 440 1. INTRODUCTION 440 2. BRANCHED COVERS 442 3. EXAMPLES FROM ALGEBRAIC GEOMETRY 447 4. FLOER COHOMOLOGY FOR AUTOMORPHISMS 453 5. HOCHSCHILD COHOMOLOGY 457 6. HOCHSCHILD COHOMOLOGY AND GLOBAL MONODROMY 459 7. MORSE CATEGORIES 465 8. REAL STRUCTURES 468 9. MATCHING PAIRS AND MATCHING PATHS 470 REFERENCES 475 Moment maps monodromy and mirror manifolds 478 1 Introduction 478 2 Chern-Simons-type functionals and critical points 481 3 Gauge equivalence and moment maps 485 4 Relationship to Kontsevich's mirror conjecture 500 5 Stability 503 6 The 2-torus 504 References 506 In 1993, M. Kontsevich proposed a conceptual framework for explaining the phenomenon of mirror symmetry. Mirror symmetry had been discovered by physicists in string theory as a duality between families of three-dimensional Calabi-Yau manifolds. Kontsevich's proposal uses Fukaya's construction of the A∞-category of Lagrangian submanifolds on the symplectic side and the derived category of coherent sheaves on the complex side. The theory of mirror symmetry was further enhanced by physicists in the language of D-branes and also by Strominger-Yau-Zaslow in the geometric set-up of (special) Lagrangian torus fibrations. It rapidly expanded its scope across from geometry, topology, algebra to physics. In this volume, leading experts in the field explore recent developments in relation to homological mirror symmetry, Floer theory, D-branes and Gromov-Witten invariants. Kontsevich-Soibelman describe their solution to the mirror conjecture on the abelian variety based on the deformation theory of A∞-categories, and Ohta describes recent work on the Lagrangian intersection Floer theory by Fukaya-Oh-Ohta-Ono which takes an important step towards a rigorous construction of the A∞-category. There follow a number of contributions on the homological mirror symmetry, D-branes and the Gromov-Witten invariants, e.g. Getzler shows how the Toda conjecture follows from recent work of Givental, Okounkov and Pandharipande. This volume provides a timely presentation of the important developments of recent years in this rapidly growing field