2006 Fiscal Year Final Research Report Summary
Recent development of special functins … an approach from the representation thery and the complex integrals
| 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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| Keywords | twisted cycles / intersection numbers / Selberg type integral / conformal field theory / configuration space / Jones polynomials / zeta regularized products / resonant condition |
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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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[Book] 群論の進化2004
Author(s)
堀田良之, 渡辺敬一, 庄司俊明, 三町勝久
Total Pages
445
Publisher
朝倉書店
Description
「研究成果報告書概要(和文)」より
