2005 Fiscal Year Final Research Report Summary
Moduli space, Homotopical algebra, Field theory
Grant-in-Aid for Scientific Research (S)
|Allocation Type||Single-year Grants |
|Research Institution||Kyoto University |
FUKAYA Kenji Kyoto University, Graduate School of Science, Professor -> 京都大学, 大学院・理学研究科, 教授 (30165261)
NAKAJIMA Hiraku Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00201666)
MOCHIZUKI Takuro Kyoto University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (10315971)
YAMAGUCHI Takao University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (00182444)
ONO Kaoru Hokkaido Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20204232)
OHTA Hiroshi Nagoya Univ., Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (50223839)
|Project Period (FY)
2001 – 2005
|Keywords||symplectic topology / Lagrangian sublmanifolds / Floer homology / Mirror symmetry / A infinity algebra / quasi-regular curve / Topological field theory / Hamiltonian dynamical system|
1. Lagrangian Floer theory
We 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 theory
We 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 homology
Using 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 topology
Fukaya 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 monotone
Lagrangian 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.
Research Products (12 results)