2004 Fiscal Year Final Research Report Summary
Conformal Field Theory and Witten Invariant
| Project/Area Number |
15540064
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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 Tokyo Institute of Technology, 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) |
FUTAKI Akito Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (90143247)
SHIGA Hiroshige Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (10154189)
MURAKAMI Hitoshi Tokyo Institute of Technology, Graduate School of Science and Engineering, Assistant Professor, 大学院・理工学研究科, 助教授 (70192771)
KOJIMA Sadayoshi Tokyo Institute of Technology, Graduate School of Information Science, Professor, 大学院・情報理工学研究科, 教授 (90117705)
ENDO Hisaaki Osaka University, Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (20323777)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Conformal field theory / 3-dimensional manfold / Witten invariant / Gauge theory |
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| Research Abstract |
We obtained an explicit expression of a base of the conformal block of $SU(2)$ WZW model in terms of the classical Riemann theta functions and theta constants as well as their transformation law. From these expression we can obtain the main results in the conformal field theory systematically, including the existence of a projectively flat connection on the bundle of the conformal blocks and a Hermitian product on the space of the conformal block which is compatible with the connection. With those results we carried out the abelianization of $SU(2)$ WZW model proposed by Atiyah and Hitchin around 1980. Using these result we defined vacuum vector for 3-dimensional handlebodies in the conformal block of $SU(2)$ WZW model and a series of topological invariants for oriented closed 3-manifolds with canonical framing in terms of the Hermitian product of the vacuum vectors. We can calcculate the asymptotic behavior of the invariants and we proved that the leading term of the asymptotic expansion can be expressed in terms of Chern-Simons invariant and Reidemeister torsion of 3-manifolds associated with $SU(2)$ representations of the fundamental groups.
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