2005 Fiscal Year Final Research Report Summary
Representation Theoretic Study of Two Dimensional Quantum Field Theory
| Project/Area Number |
14204003
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 | Nagoya University |
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Principal Investigator |
TSUCHIYA Akihiro Nagoya Uni., Graduate School of Mathematics, professor, 大学院・多元数理科学研究科, 教授 (90022673)
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| Co-Investigator(Kenkyū-buntansha) |
KIMURA Yoshifumi Nagoya Uni, Graduate School of Mathematics, professor, 大学院・多元数理科学研究科, 教授 (70169944)
AWATA Hidetoshi Nagoya Uni., Graduate School of Mathematics, assistant professor, 大学院・多元数理科学研究科, 助教授 (40314059)
OHTA Hiroshi Nagoya Uni., Graduate School of Mathematics, assistant professor, 大学院・多元数理科学研究科, 助教授 (50223839)
KANNO Hiroaki Nagoya Uni., Graduate School of Mathematics, professor, 大学院・多元数理科学研究科, 教授 (90211870)
NAKANISHI Tomoki Nagoya Uni., Graduate School of Mathematics, assistant professor, 大学院・多元数理科学研究科, 助教授 (80227842)
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| Project Period (FY) |
2002 – 2005
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| Keywords | conformal field theory / vertex operator algebra / conformal block / quantum field theory |
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| Research Abstract |
We studied the construction of the conformal field theories associated to Vertex Operator Algebra(V.O.A.) satisfying Zhu's C_2-finiteness condition.
In order to construct the conformal field theories we defined universal enveloping algebra associated to V.O.A.satisfying Zhu's C_2-finiteness condition, and analyzed structure of Abelian categories constituted of modules of this algebra in more detail. These Abelian categories are Nothern and Artin, but not semi-simple.
We gave an expression of Zhu algebra associated V.O.A.using enveloping algebra of V.O.A., and gave necessary and sufficient condition of the semi-simplicity of the Abelian categories using Zhu algebra.
Using above results, we constructed coherent sheaves constituting of conformal blocks over moduli spaces of genus 0 N-pointed stable curves, and showed that they have D-module structures with regular singularities along the boundary of the moduli spaces.
By analyzing of bi-module structure of enveloping algebra in further detail, we succeeded the construction of the Dx-module constituted conformal blocks over moduli spaces of genus g N-pointed stable curves, the formulation of the factorization properties of the coherent sheaves along the boundary of the moduli spaces. We are now preparing the paper.
We are studying the so-called W(p) algebra satisfying Zhu's C_2-finiteness condition. These algebra are typical examples such that the Abelian categories are not semi-simple. We are now making detailed study of the Abelian categories, and Fusion product of conformal blocks, modular properties of genus 1 conformal blocks. Further more there are some nice relationships between representation theories of quantum groups at root of unity.
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