Moduli space, Homotopical algebra, Field theory

模空间、同伦代数、场论

基本信息

批准号：
13852001
负责人：
FUKAYA Kenji
金额：
$ 63.23万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (S)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13852001/
关键词：
symplectic topology Lagrangian sublmanifolds Floer homology Mirror symmetry A infinity algebra quasi-regular curve Topological field theory Hamiltonian dynamical system ループ空間シンプレクティック微分同相群曲率と基本群 open string モジュライ空間 Dブレイン A無限大

项目摘要

1. Lagrangian Floer theoryWe establish basic story of Floer homology of Lagrangian submanifolds in the book‘Lagrangian intersection Floer theory - anomaly and obstruction', written by Fukaya, Oh, Ohta, Ono.In this book, we aaaociate in a functorial way the structgure of filtered A infinity algebra to the cohomology group of relatively spin Lagrangian submanifolds.We gave several applications to symplectic topology (see 5) and to Mirror symmetry (see 3).2 Relation to string theoryWe generalized the theory to the case when there are interior marked points.3 Homological mirror symmetry.The result mentioned in 1 makes it possible to state the conjecture precisely. There are two approaches towerd its proof one is by asymptotic analysis and the other by rigi analytic geometry.The first one is proposed by Fukaya.4. Relation to contact homologyUsing the formulation of Lagrangian Floer theory by loop space we can state correct relation of Floer theory to contact homology, as a conjecture.5. Application to symplectic topologyFukaya proved that the prime oriented 3 manifold L can be embedded as a Lagrangian submanifold in C^3 if and only there L has a direct S^1 factor.He also proved Audin's conjecture together with its generalization to K pi one space using the same method. In FOOO we clafired the relation of Hofer distance to torsion of Floer homology.We proved Arnold Givental Conjecture for monotoneLagrangian submanifold in FOOO.Ono proved Flux conjecture which state that the group of exact symplectic diffeomorphisms are C^1 close in the group of symplectic diffeomorphisms.

1. 拉格朗日弗洛尔理论我们在 Fukaya、Oh、Ohta、Ono 所著的《拉格朗日交集弗洛尔理论 - 异常与阻碍》一书中建立了拉格朗日子流形的弗洛尔同调的基本故事。在这本书中，我们以函子的方式关联了结构过滤 A 无穷代数到相对自旋拉格朗日子流形的上同调群。我们给出辛拓扑（参见 5）和镜像对称（参见 3）的几种应用。2 与弦理论的关系我们将该理论推广到存在内部标记点的情况。3 同调镜像对称性。1 中提到的结果使得可以精确地陈述该猜想有两种方法，一种是渐近分析，另一种是rigi解析几何。第一种是Fukaya提出的。4．通过环空间对拉格朗日弗洛尔理论的表述，我们可以将弗洛尔理论与接触同调的正确关系作为猜想来表述。 5．在辛拓扑中的应用Fukaya证明了面向素数的3流形L可以作为拉格朗日子流形嵌入到C^3中。当且仅当L有一个直接的S^1因子。他还用同样的方法证明了Audin猜想及其对K pi 一个空间的推广。 FOOO我们阐明了Hofer距离与Floer同调挠率的关系。我们在FOOO中证明了单调拉格朗日子流形的Arnold Givetal猜想。Ono证明了Flux猜想，该猜想指出精确辛微分同胚群在辛微分同胚群中是C^1紧密的。