## 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 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. |

### Contents

Politics and Philosophy | 25 |

Politics and Philosophy | 112 |

Further Illustrations of Shakespeares | 192 |

NOTES | 269 |

Speakers V | 1 |

Local mirror symmetry and fivedimensional gauge theory 31 | 31 |

The Toda conjecture 51 | 51 |

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 |

Obstruction to and deformation of Lagrangian intersection Floer 281 | 281 |

Topological open pbranes 311 | 311 |

Lagrangian torus fibration and mirror symmetry of CalabiYau 385 | 385 |

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 complex structure connection consider construction coordinates Cordelia corresponding CP¹ critical curves D-branes defined deformation denote differential Duke equivalent fibration fibre field theory Floer cohomology formula Fukaya category gauge theory ghost number given Goneril Hamiltonian holomorphic homotopy human Iago integral intersection invariant isomorphic Kähler King Lear Kontsevich Lagrangian submanifold Lear's Lemma Macbeth Macduff master equation Math metric mirror symmetry moduli space moment map morphisms Morse natural observes one's Othello parameter philosophical Phys Plato's play political proof quantization quantum quantum cohomology rational Republic respect satisfies sections Shakespeare sigma model singular locus 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