| Project/Area Number |
16540084
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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 | Waseda University |
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Principal Investigator |
KORI Toshiaki Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (50063730)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | Chern-Simons quantization / Moduli space of flat connections / Symplectic structure / Abelian extension of 3-dim. Mapping group / Wess-Zumino theory / Chern-Simonsゲージ理論 / 幾何的量子化 / シンプレクティク幾何 / 電磁ヘリシティ / ディラック方程式 |
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| Research Abstract |
The reporter of this note studied from the year 2004 to 2006 the quantization of the Chern-Simons gauge theory on four-manifolds, he gave the geometric quantization of the moduli space of connections on a four-manifolds generally with boundary. He gave a pre- symplectic structure on the moduli-space of connections. He constructed an hermitian line bundle with connection on the moduli space whose curvature is given by the pre-symplectic form. The transition function of the line bundle is described by the 5- dimensional Chern-Simons functional. On the space of connections there is a Hamiltonian action of the group of gauge transformations that are identity on the boundary, whose symplectic reduction becomes the space of flat connections, this is the geometric quantization of the space of flat connections. The mapping group from the boundary three-manifold to the structure Lie group acts infinitesimally symplectic way on this moduli space of flat connections, and the author showed that the abelian extension of the mapping group lifts the action to the quantium line bundle. The last extension was constructed by Mickelsson and extended by the author's previous research on four-dimensional Wess-Zumino-Witten theory. The results were submitted to a journal of differential geometry. After this research the reporter investigated also the quasi-symplectic structure on the space of flat connections on three- manifolds and the condition for a connection to be extended to the four-manifold that cobord the first one and decided the class of such connections. Other than these research the author investigated the vortex representation of the Hamilton-Yang-Mills equation and the relation of it to the helicity of Hamiltonian flows.
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