2003 Fiscal Year Final Research Report Summary
De Rham Theory of Loop Spaces and Quantum Topological Invariants
| Project/Area Number |
12440014
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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 |
Geometry
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| Research Institution | The University of Tokyo |
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Principal Investigator |
KOHNO Toshitake The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (80144111)
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| Co-Investigator(Kenkyū-buntansha) |
MURAKAMI Jun Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90157751)
MORITA Shigeyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
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| Project Period (FY) |
2000 – 2003
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| Keywords | loop space / configuration space / iterated integral / braid group / mapping class group / moduli space / Poisson structure / conformal field theory |
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| Research Abstract |
We investigated the algebraic structure of the homology of the loop spaces of configuration spaces and clarified its relation to finite type topological invariants for braids. Especially, we studied the homology of the loop spaces of configuration spaces and finite type topological invariants. We focused on the homology of the loop spaces of orbit configuration spaces associated with the action of Fuchsian groups on the complex upper half plane. We showed that the total homology of such loop space is isomorphic to the algebra of horizontal chord diagrams on the quotient surface. We introduced a structure of a Poisson algebra for the homology of the iterated loop space of the orbit configuration space based on the Browder operation.
We gave a complete description of the space of conformal blocks for the conformal field theory on the Riemann sphere in terms of hypergeometric integrals. In particular, we clarified the integration cycles as the regularizable cycles in the homology of locally finite chains with coefficients in a certain local system defined over the complement a disrciminantal arrangement.
Morita investigated the structure of various moduli spaces as well as their associated modular groups, such as the moduli space of Riemann surface -mapping class groups and the moduli space of graphs -outer automorphism group of free groups. Murakami gave a new point of view on a conjecture concerning the asymptotics of the Jones invariants for knots, the hyperbolic volume of the knot complement, and the geometric structure of 3-manifolds.
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