2004 Fiscal Year Final Research Report Summary
Toward to a generalization of modular invariance of vertex operator algebras into Hilbert type and Siegel type.
| Project/Area Number |
13440002
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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 | University of Tsukuba |
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Principal Investigator |
MIYAMOTO Masahiko University of Tsukuba, Graduate School of Pure and Applied Silences, Professor, 大学院・数理物質科学研究科, 教授 (30125356)
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| Co-Investigator(Kenkyū-buntansha) |
MORITA Jun University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (20166416)
KIMURA Tatsuo University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (30022726)
NAITOU Satoshi University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate professor, 大学院・数理物質科学研究科, 助教授 (60252160)
TAKEUCHI Kiyoshi University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor, 大学院・数理物質科学研究科, 助教授 (70281160)
KITAZUME Masaaki Chiba University, Dept of Mathematics, Professor, 理学部・数学科, 教授 (60204898)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Vertex Operator Algebra / The moonshine module / Monster simple group / C2-condition / Modular forms / Modular invariance / Trace funditons / Pseudo-trace functions |
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| Research Abstract |
A concept of vertex operator algebras (VOA shortly) has originated from the moonshine vertex operator algebra, which was constructed in order to explain a mysterious relation (the moonshine conjecture) between Monster simple finite group (the largest sporadic finite simple group) and the classical elliptic modular function. Our purpose of this project is to clarify the modular invariance property of VOAs and extend it in multivarables. (1)We found a new construction of the moonshine vertex operator algebra by using Ising models, which offers a new modular invariance in multivariables. Compared with the original construction, our construction is easy and we can apply our construction for many other VOAs. (2)We have shown that C2-condition is enough to get, a modular invariance. Classically, the rationality (completely reducibility of modules) was considered to be more important than C2-condition, but our research has shown that we don't need rationality. (3)We construct an infinitely many VOAs with Euclidian Jordan Algebras as Griess algebras for any complex central charge c. So we construct a candidate of Siegel modular invariance. (4)We found an order formula to determine the automorphism group of VOAs. In our construction, Miyamoto involution plays an essential role and so we can easily get information about the centralizer of Involution in the full automorphism group. The question is if we can determine the automorphism group from it.
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