2005 Fiscal Year Final Research Report Summary
Poisson Geometry, Contact Geometry and Quantization Problems
| Project/Area Number |
15204005
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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 | KEIO UNIVERSITY |
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Principal Investigator |
MAEDA Yoshiaki Keio University, Faculty of Science and Technology, Prof., 理工学部, 教授 (40101076)
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| Co-Investigator(Kenkyū-buntansha) |
MORIYOSHI Hitoshi Keio University, Faculty of Science and Technology, Associ. Prof., 理工学部, 助教授 (00239708)
NAKADA Hitoshi Keio University, Faculty of Science and Technology, Prof., 理工学部, 教授 (40118980)
GUEST Martin Tokyo Metropolitan University, Faculty of City Environments, Prof., 都市教養学部, 教授 (10295470)
ONO Kaoru Hokkaido University, Faculty of Science, Prof., 大学院・理学研究科, 教授 (20204232)
WATAMURA Satoshi Tohoku University, Faculty of Science, Associ. Prof., 大学院・理学研究科, 助教授 (00201252)
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| Project Period (FY) |
2003 – 2005
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| Keywords | deformation quantization / Noncommutative geometry / Gerbe / Symmplectic geometry / Contact geometry / quantization problems / cyclic cohomology / Loop spaces |
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| Research Abstract |
This research project aims several problems in Poisson geometry and contact geometry, and quantization problems. Specially, our project focus on the treatments for the quantization problems geometrical point of view. The noncommutative differential geometry is one of our target of our research project. In this project, we had several results on convergent deformation quantization problem. Grebes appears naturally from the construction of the star exponential functions of quadratic forms. Namely, we consider the set of quadratic forms in the complex Weil algebra which forms a Lie algebra isomorphic to sp(n, C). When we consider the esponential functions for these objects, we might expect the complex version of the metaplectic Lie group. Since this is simply connected, we could not handle it. We invested this object by using the explicit computations and have it is in the category of grebes with multiplications. However, it can be described more geometry in terms of the connections. The second problem is to study the invariant deformation quantization problems and construct a convergent star product for ax+b group case. We obtain the universal star product formula. The third result is to study the closed star product. We describe a general settings for obtain how to get the Hochschild cocycle via Stokes formula. This results is still working on and we expect it should be related to the deformation quantization problems for infinite dimensional case. To obtain these results, we have lot of workshops by inviting overseas and domestic researchers together with the research partners. As conclusions, we have fruitful research results which have high evaluation internationally, and also establish the international research network for this area by this grant. This project is still working and will continue for the next project.
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[Book] 量子的な微分・積分2005
Author(s)
大森英樹, 前田吉昭
Total Pages
335
Publisher
シュプリンガー・フェアラーク
Description
「研究成果報告書概要(和文)」より
