| Project/Area Number |
15K13437
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Kyoto University |
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Principal Investigator |
Shimizu Tatsuro 京都大学, 数理解析研究所, 特定助教 (00738859)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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| Keywords | Chern-Simons摂動論 / 量子不変量 / 配置空間積分 / 有限型不変量 / 3次元多様体の不変量 / Morseホモトピー / Casson不変量 / 特異点論 / propagator / 不足符号数 / 非自明な平坦接続 / グラフ複体 / SU(2) / Morse homotopy |
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| Outline of Final Research Achievements |
I gave several new invariant related to the Chern-Simons perturbation theory and I investigated the relationship between these invariants.
I found a gap on the construction of the Chern-Simons perturbation theoretical invarinat given by Bott and Cattaneo. Then I remove the gap and I refined their construction to more flexible one. I also gave a Morse homotopy theoretical description of the invariant. I gave a new description of the signature defect which plays an important role of the Chern-Simons perturbation theoretical description of the Casson invariant.
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| Academic Significance and Societal Importance of the Research Achievements |
Chern-Simons摂動論は物理学のChern-Simons量子場の理論を，数学的に近似し記述する手法であり，3次元多様体論(数学の一分野)に多くの知見をもたらす．Chern-Simons量子場の理論は数学的にはまだ正当化されていない．一方，厳格な言語である数学によって記述されたChern-Simons摂動論は人類にとって確固たる数学的知見を与える．本研究ではその記述をより広げたとともに，物理から切り離された純粋な数学的視点からその一部をとらえなおすことに成功した．これらの成果は3次元多様体論のみならず，数学全体においても重要な指導原理・知見を与えうるものである．
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