| Project/Area Number |
15340003
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Tokyo Institute of Technoroly |
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Principal Investigator |
MIMACHI Katsuhisa Tokyo Institute of Technology, Grad.school of Sci. and Tech., Professor., 大学院理工学研究科, 教授 (40211594)
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| Co-Investigator(Kenkyū-buntansha) |
KUROKAWA Nobushige Tokyo Institute of Technology, Grad.school of Sci. and Tech., Professor., 大学院理工学研究科, 教授 (70114866)
OCHIAI Hiroyuki Nagoya Univ., Grad. school. of Math., Professor., 大学院多元数理科学研究科, 教授 (90214163)
TAKATA Toshie Niigata Univ., Inst. of Sci. and Tech., Associate Professor., 自然科学系, 助教授 (40253398)
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| Project Period (FY) |
2003 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥12,400,000 (Direct Cost: ¥12,400,000)
Fiscal Year 2006: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2005: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2004: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2003: ¥3,600,000 (Direct Cost: ¥3,600,000)
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| Keywords | twisted cycles / intersection numbers / Selberg type integral / conformal field theory / configuration space / Jones polynomials / zeta regularized products / resonant condition / 接続問題 / 一般超幾何函数 / モノドロミー / 共鳴状態 / 捩れサイクル / セルバーグ積分 / 表現論 / 代数解析 / ゼータ正規化 / 複素積分 / 交差数 / q超幾何級数 / 寺田模型 |
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| Research Abstract |
Mimachi realized an irreducible representation of the Iwahori-Hecke algebra on the twisted homology group associated with a Selberg type integral. It was first constructed in the context of conformal field theory by Tsuchiya-Kanie. Our construction is based on the study of the homology group under a resonant condition on the exponents of integrands. We stress the importance of the study of integrals under such a resonant condition to the study of hypergeometric type functions and spherical functions. Mimachi with H. Ochiai (Nagoya) and M.Yoshida (Kyushu) formulated the concept of visible cycles and invisible cycles, and determined the dimension of the spaces of visible cycles under a resonant condition in some examples.
Mimachi with M.Yoshida calculated intersection numbers of twisted cycles associated with a Selberg type integral. It gives a natural interpretation of the coefficients of the four-point correlation function, in conformal field theory, calculated by Dotsenko-Fateev. This is an answer to a long standing problem of clarifying the meaning of such coefficients appearing in correlation functions. In higher dimensional cases, the Terada model (nonsingular model arising from the point configuration) plays an important role.
Kurokawa with M.Wakayama (Kyushu) studied generalized zeta regularizations. It shows that a discrete version of intersection numbers of twisted cycles should be settled.
Takata studied a q-hypergeometric series which appears as a factor of the n-colored Jones polynomial associated with a twisted knot or a torus knot and derived the A-polynomials associated with them.
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