2007 Fiscal Year Final Research Report Summary
New construction of vector bundles on Riemann surfaces and Verlinde's formula
| Project/Area Number |
18540039
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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 |
Algebra
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| Research Institution | Saga University |
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Principal Investigator |
ICHIKAWA Takashi Saga University, Faculty of Science and Engineering, Professor (20201923)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAHARA Toru Saga University, Faculty of Science and Engineering, Professor (50039278)
MITOMA Itaru Saga University, Faculty of Science and Engineering, Professor (40112289)
UEHARA Tsuyoshi Saga University, Faculty of Science and Engineering, Professor (80093970)
TERAI Naoki Saga University, Faculty of Culture and Education, Associated Professor (90259862)
HIROSE Susumu Saga University, Faculty of Science and Engineering, Associated Professor (10264144)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Riemann surface / Schottky group / Vector bundle / Moduli space / Abel-Jacobi's theorem / Verlinde's formula / Siegel modular form / p-adic Siegel modular form |
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| Research Abstract |
1. We showed that any stable vector bundle of degree 0 on a Riemann surface close to a maximally degenerate curve is obtained from a linear representation of the Schottky group uniformizing the Riemann surface. Further, we described the relationship between the representation space of the Schottky group and the moduli space of the vector bundles by using Abel-Jacobi's theorem and Verlinde's formula.
2. We described the ring structure of Siegel modular forms of degree 2 over a ring containing 1/6 extending Igusa's result. Further, we extended results of Swinnerton-Dyer, Serre and Katz on congruence and p-adic properties of elliptic modular forms to the case of Siegel modular forms.
3. We gave a mathematical rigorous model of the one loop approximation of the perturbative Chern-Simons integral in an abstract Wiener space setting, and by appealing to the Malliavin-Taniguchi formula of changing variables on the abstract Wiener space, we derived the asymptotic expansion for the Chern-Simons integral with respect to the charge parameter.
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