## Symplectic Geometry and Mirror Symmetry: Proceedings of the 4th KIAS Annual International Conference, Korea Institute for Advanced Study, Seoul, South Korea, 14-18 August 2000In 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 CalabiOCoYau manifolds. Kontsevich''s proposal uses Fukaya''s construction of the A Oex -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 StromingerOCoYauOCoZaslow 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 GromovOCoWitten invariants. Kontsevich-Soibelman describe their solution to the mirror conjecture on the abelian variety based on the deformation theory of A Oex -categories, and Ohta describes recent work on the Lagrangian intersection Floer theory by FukayaOCoOhOCoOhtaOCoOno which takes an important step towards a rigorous construction of the A Oex -category. There follow a number of contributions on the homological mirror symmetry, D-branes and the GromovOCoWitten 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. Contents: Estimated Transversality in Symplectic Geometry and Projective Maps (D Auroux); Local Mirror Summetry and Five-Dimensional Gauge Theory (T Eguchi); The Toda Conjecture (E Getzler); Examples of Special Lagrangian Fibrations (M Gross); Linear Models of Supersymmetric D-Branes (K Hori); The Connectedness of the Moduli Space of Maps to Homogeneous Spaces (B Kim & R Pandharipande); Homological Mirror Symmetry and Torus Fibrations (M Kontsevich & Y Soibelman); Genus 1-Virasoro Conjecture on the Small Phase Space (X Liu); Obstruction to and Deformation of Lagrangian Intersection Floer Cohomology (H Ohta); Topological Open p-Branes (J-S Park); Lagrangian Torus Fibration and Mirror Symmetry of CalabiOCoYau Manifolds (W-D Ruan); More About Vanishing Cycles and Mutation (P Seidel); Moment Maps, Monodromy and Mirror Manifolds (R Thomas). Readership: Researchers and graduate students in differential geometry, topology and theoretical physics." |

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### Contents

Politics and Philosophy | 25 |

Politics and Philosophy | 112 |

Further Illustrations of Shakespeares | 192 |

NOTES | 269 |

BIBLIOGRAPHY | 393 |

Local mirror symmetry and fivedimensional gauge theory 31 | 31 |

Examples of special Lagrangian fibrations 81 | 81 |

The connectedness of the moduli space of maps to homogeneous spaces 187 | 187 |

Homological mirror symmetry and torus fibrations 203 | 203 |

Genus1 Virasoro conjecture on the small phase space 265 | 265 |

Topological open pbranes 311 | 311 |

More about vanishing cycles and mutation | 429 |

Moment maps monodromy and mirror manifolds | 467 |

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action functional algebra asymptotically holomorphic Banquo boundary condition boundary interaction Calabi-Yau manifolds chiral chiral superfield codimension complex structure connection consider construction coordinates Cordelia corresponding curves D-branes defined Definition deformation denote derived differential Duke Edmund equivalent example F-term fibration field theory Floer cohomology formula Fukaya category gauge theory ghost number given Goneril graph Hamiltonian holomorphic homotopy human implies integral intersection invariant isomorphic Kahler King Lear Kontsevich lago Lagrangian submanifold Lear's Lemma Macbeth Macduff master equation metric mirror symmetry moduli space morphisms Morse natural observes one's Othello parameter philosophical Phys Plato's play political problem proof quantization quantum quantum cohomology quasi-isomorphism rational Republic respect satisfies sequence Shakespeare sigma model singular locus smooth Sokrates soul speak special Lagrangian fibrations string theory superfield superpotential supersymmetry symplectic Theorem things thou tion topological toric transversality trivial vanishing vector field Virasoro conjecture witches zero