2000 Fiscal Year Final Research Report Summary
Moduli space and infinite dimensional geometry
| Project/Area Number |
09304008
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| Research Category |
Grant-in-Aid for Scientific Research (A).
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
FUKAYA Kenji Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (30165261)
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| Co-Investigator(Kenkyū-buntansha) |
FURUTA Mikio Univ. of Tokyo Dept. of Math. Sci. Professor, 大学院・数理科学研究科, 教授 (50181459)
ONO Kaoru Hokkaido Univ. Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20204232)
OHTA Hiroshi Nagoya Univ. Depart. of PolyMath. Associate Professor, 大学院・多元数理科学研究科, 助教授 (50223839)
NAKAJIMA Hiraku Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00201666)
KONO Akira Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00093237)
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| Project Period (FY) |
1997 – 2000
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| Keywords | Symplectic Geometry / Floer homology / Homotopical Algebra / moduli space / Mirror symmetry / pseudoholomorphic curve / Kuranishi map / Lagrangian submanifold |
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| Research Abstract |
Fukaya, Ono, Ohta together with Oh constructed an obstruction theory for Lagrangian intersection Floer homology to be well defined. Based on it, an (algebraic) deformation theory of Lagrangian submanifold and quantum deformation of Lagrangian submanifold is constructed. We finished a preliminary version of the book (of 350 pages) describing them in Dec. 2000. After than we made a progress on homotopical algebra part and clarify the relation to classical homotopy type etc. So we are now adding more material (approximately 100 pages).
Our Lagrangian intersection Floer theory is an open string version of the theory of Gromov-Witten invariant which are completed by several mathematicians including Fukaya-Ono.
In our case of open string version, we need more careful treatment on homological algebra part for example so that we need to develop homotopical algebra itself for this purpose.
The orientation of the moduli space is also a delicate question since the moduli space of pseudoholomorphic d
… More
isks do not carry an almost complex structure. Moreover we need additional argument to work out analytic detail mainly because we need to work in the chain level.
While working out the detail of the Lagrangian intersection Floer theory, we have a better understanding of the relation between quantum field theory and various notions developed in the late half of the 20 th century. For example we observed a close relation between Feynman diagram, homotopical algebra and of local deformation theory.
Nakajima pursuit his study to construct interesting algebraic structures based on moduli spaces. In his study, various example which are supposed to play the central role in the theory is studied in detail explicitly and various interesting new algebraic structures are constructed.
Furuta and his coauthors further studied a relation between moduli space in 4 dimensional gauge theory and stable homotopy theory. This point of view is already appeared in Furuta's proof of 10/8 theorem of intersection form of 4 manifolds. By recent development, several interesting applications such as construction of new invariant of homology 3 spheres and study of embedding of surface in the connected sum of two K3 surfaces. are obtained.f embedding of surface in the Less
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