2010 Fiscal Year Final Research Report
New Aspect of Probabilistic Approach to Chern-Simons Theory
| Project/Area Number |
20540120
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Saga University |
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Principal Investigator |
MITOMA Itaru Saga University, 大学院・工学系研究科, 教授 (40112289)
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| Co-Investigator(Kenkyū-buntansha) |
ICHIKAWA Takashi 佐賀大学, 大学院・工学系研究科, 教授 (20201923)
HANDA Kenji 佐賀大学, 大学院・工学系研究科, 准教授 (10238214)
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| Co-Investigator(Renkei-kenkyūsha) |
TANIGUCHI Setsuo 九州大学, 大学院・数理学研究院, 教授 (70155208)
AIDA Shigeki 東北大学, 大学院・理学研究科, 教授 (90222455)
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| Project Period (FY) |
2008 – 2010
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| Keywords | Chern-Simonns 理論 / 抽象ウィナー空間 / ホロノミー / 漸近展開 / スーパー・フィールド / 量子不変量 / 超対称的場の量子論 / 集合値確率積分 |
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| Research Abstract |
To give a mathematical meaning to the Chern-Simons functional integral of its total Lagrangian in an abstract Wiener space setting, we regularize the 3rd terms including the infinite determinant appeared in the Lagrangian modified by the method of super fields, along the suggestions of my research colleague, Prof. Albeverio atthe University of Bonn in Germany and Prof. Funakubo in my University, and give themathematical definition as a Wiener functional. We did the estimate of the remainder terms in the asymptotic expansion by modifying the It^o method of defining the Feynman measure by using the Wiener measure. Concerning the Gauge theory including the Chern-Simons theory, we study the random surfaces from a viw point of set-valued stochastic processes. After defining the set-valued stochastic integral, we get the existence and uniqueness for the solutions of a set-valued stochastin differential equation undersome restriction. Metaphorically speaking the restriction in a word, we catch the motion of a jellyfish but not of an ameba.
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