2002 Fiscal Year Final Research Report Summary
Gauge Theory and 3-Dimensional Manifolds
| Project/Area Number |
13640065
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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 |
Geometry
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
YOSHIDA Tomoyoshi Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (60055324)
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| Co-Investigator(Kenkyū-buntansha) |
TSUJI Hajime Graduate School of Science and Engineering, Assistant Professor, 大学院・理工学研究科, 助教授 (30172000)
SHIGA Hiroshige Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (10154189)
FUTAKI Akito Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (90143247)
ENDO Hisaaki Graduate School of Science and Engineering, Assistant, 大学院・理工学研究科, 助手 (20323777)
MURAKAMI Hitoshi Graduate School of Science and Engineering, Assistant Professor, 大学院・理工学研究科, 助教授 (70192771)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Gauge theory / 3-dimensional manfold / Witten invariant / Conformal field theory |
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| Research Abstract |
We obtained an explicit expression of a base of the conformal block of SU(2) conformal field theory by classical Riemann theta functions. As a result we can define a Hermitian product on the space of the conformal block which is invariant under the projectively flat connection. Using these result we defined Witten's invariant for oriented closed 3-manifolds with canonical framing in terms of the Hermitian product of the vacuum vectors in the conformed block of SU(2) conformal field theory. Such a definition of the invariant gives an algorithm to compute it for a 3-manifold with Heegaard splitting. It enables us to express the invariant as a Gauss sum which is a kind of Fourier transform of the similar expression obtained by the usual method using the quantum group and link expression of 3-manifolds. It means that the said way of the definition of the invariant is not merely a change of the definition but it brings a new insight about the geometric nature of the invariant.
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