2000 Fiscal Year Final Research Report Summary
DEFORMATION QUANTIZATION AND ITS APPLICATION
| Project/Area Number |
11640095
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | SCIENCE UNIVERSITY OF TOKYO |
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Principal Investigator |
YOSHIOKA Akira SCIENCE UNIVERSITY OF TOKYO MATHEMATICS ASISTANT PROFESSOR, 理学部, 助教授 (40200935)
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| Co-Investigator(Kenkyū-buntansha) |
MIYAZAKI Naoya KEIO UNIVERSITY, MATHEMATICS, ASSISTANT PROFESSOR, 経済学部, 助教授 (50315826)
MAEDA Yoshiaki KEIO UNIVERSITY, MATHEMATICS, PROFESSOR, 理工学部, 教授 (40101076)
OMORI Hideki SCIENCE UNIVERSITY OF TOKYO MATHEMATICS PROFESSOR, 理工学部, 教授 (20087018)
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| Project Period (FY) |
1999 – 2000
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| Keywords | DEFORMATION QUANTIZATION / STAR PRODUCT / NONCOMMOTATIVE GEOMETRY / QUANTIZATION / ASYHPTOTIC ANALYSIS / SYMPLECTIC GEOMETRY / HAMILTONIAN MECHANICS |
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| Research Abstract |
This researchment is two fold ; (i) geometric aspect of Deformation quantization via Weyl manifolds, (ii) investigation of convergent deformation quantization with repect to the deformation parameter h. We obtain the following. (i) The moduli space of Weyl manifolds are the formal power serires with coefficients in the 2nd cohomology classes of the base manifol. Using the cohomolgy corresponding to the Weyl manifold, we construct a contact Weyl manifold which contains Weyl manifold as a subbundle. On contact Weyl manifold, we also constuct a connection whose curvature form determines the cohomology class of the Weyl manifold. We show this connection is an extension of Fedosov connection and proved that the cohomolgy class given by the curvature coincides with the cohomology class of the Weyl manifold, hence we show the Poincare-Cartan class of Weyl manifold and the cohomology class of the curvature of Fedosov connection are the same thing. (ii) Using the Moyal product formula, we set certain Frechet space of certain holomophic functions on the multidimensional complex plane where the Moyal products are absolutely convergent. Singular exponent of holomorphic functions are introduced with respect which the star products breaks the associativity of product. We also investigate a star exponential functions of quadratic functions. Althoug the Frechet space does not contain the exponentials of the quadratic functions, the star product is well defined between the quadratic exponentials and holomorphic function having the exponent less than the singular one. Certain properties are investigated for the group generated by the quadratic exponential functions. Especially, the group is considered an extension of the special linear groups.
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