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A step to the perfect classification of 24dimensional meromorphic vertex operator algebras

Research Project

Project/Area Number 09440004
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionUniversity of Tsukuba

Principal Investigator

MIYAMOTO Masahiko  Univ.of Tsukuba, Inst.of Mathematics Professor, 数学系, 教授 (30125356)

Co-Investigator(Kenkyū-buntansha) MORITA Jun  Univ.of Tsukuba, Inst.of Mathematics, Professor, 数学系, 教授 (20166416)
NAITOU Satoshi  Univ.of Tsukuba, Inst.of Mathematics, Associated Professor, 数学系, 助教授 (60252160)
KIMURA Hiroshi  Jobu University, Dept.of Information, Professor, 情報学部, 教授 (70023570)
KOGISO Takeyoshi  Jyosai University, Dept of Mathematics, Lecturer, 理学部, 講師 (20282296)
KITAZUME Masaaki  Chiba University, Dept of Mathematics, Associated Professor, 理学部, 助教授 (60204898)
Project Period (FY) 1997 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥9,700,000 (Direct Cost: ¥9,700,000)
Fiscal Year 2000: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1999: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 1998: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1997: ¥3,600,000 (Direct Cost: ¥3,600,000)
KeywordsIsing models / Lattices / Meromorphic / Virasoro algebra / Automorphism groups / Monster simple group / Vertex operator algebra / Moonshine conjecture / ムーンシャン予想
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. It is now understand to be a rigorous and mathematical concept of 2 dimensional conformal field theory in physics. Namely, it offers axioms on such a theory. Although 24 dimensional meromorphic conformal field theory have a special important value in physics, the examples we know at present are only Moonshine VOA, VOAs constructed from the Niemeier lattices and their orbifold VOAs. Recently, Dong and Mason found that all of them contain a tensor product of 48 Ising modules.
With the support of this Grant, M.Miyamoto (Head of this research) has been studying VOAs containing a tensor product of Ising modules, he found new VOAs called "code VOAs", which are easy to hand … More le compared with the other VOAs in 1997. We also determined their representations (all modules) and introduced a concept of "induced modules." In 1998, we found a special property of Hamming code VOAs (constructed from an extended [8, 4, 4]-Hamming code) and determined its fusion rules among its irreducible modules. Using this special property, we found a new construction of the famous moonshine VOA and then a new construction of Monster simple group. It is very easier than the original construction and obtained a lot of properties of Monster simple group. In 2000, we applied the new method of construction to the known VOAs (lattice VOAs, etc.) and found that it is possible to construct all known 24 dimensional holomorphic VOAs by this way. We also succeed to construct twisted modules of code VOAs. For twisted modules, the existence was proved theoretically, but we don't know examples except very easy one. So we hope that this new construction have many applications, especially we expect to construct twisted modules for the moonshine VOAs, which are the essential parts of the moonshine conjectures. Less

Report

(5 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] 北詰,宮本雅彦,山田: "Ternary codes and vertex operator algebras"Journal of Algebra. 223. 379-395 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 宮本雅彦: "A modular invariance on the theta functions defined on vertex operator algebras"Duke Mathematical Journal. 101. 221-236 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 宮本雅彦: "Codes and the construction of vertex operator algebras"Sugaku. 52. 159-171 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 北詰,宮本雅彦,山田: "Borwein identity and vertex operator algebras"Journal of Number Theory. 82. 100-108 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Kitazume, M.Miyamoto, H.Yamada: "Ternary codes and vertex operator algebras"Journal of Algebra. Vol.223. 379-395 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Miyamoto: "A modular invariance on the theta functions defined on vertex operator algebras"Duke Mathematical Journal. Vol.101. 221-236 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Miyamoto: "Codes and the construction of vertex operator algebras"Sugaku. Vol.52. 159-171 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Kitazume, M.Miyamoto, H.Yamada: "Borwein identity and vertex operator algebras"Journal of Number theory. Vol 82. 100-108 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 北詰,宮本雅彦,山田: "Ternary codes and vertex operator algebras"Journal of Algebra. 223. 379-395 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 宮本雅彦: "A modular invariance on the theta functions defined on vertex operator algebras"Duke Mathematical Journal. 101. 221-236 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 宮本雅彦: "Codes and the construction of vertex operator algebras"Sugaku. 52. 159-171 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 北詰,宮本雅彦,山田: "Borwein identity and vertex operator algebras"Journal of Number Theory. 82. 100-108 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 宮本雅彦: "A Hamming code vertex operator algebra and construction of vertex operator algebras"Journal of Algebra. 215. 509-530 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 花木、丹原、宮本雅彦: "Quantum Galois theory for finite groups"Duke Math.J.. 97. 541-544 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 江川、宮本雅彦: "Graph labelings in Boolean lattices"Ars Combin.. 52. 13-19 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 北詰、宮本雅彦、山田: "Ternary codes and vertex operator algebras"Journal of Algebra. 223. 379-395 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 宮本雅彦: "Representation theory of code vertex operator algebra" Journal of Algebra. 201巻. 115-150 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 宮本雅彦: "The moonshine VOA and a tensor product of Ising models" “The monste and Lie algebras" OSU M.R.I.P.7巻. 99-110 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 宮本雅彦: "Quantum Galois theory for finite groups" Duke Mathematical Journal.

    • Related Report
      1997 Annual Research Report
  • [Publications] 宮本雅彦: "Representation theory of Code VOA" Journal of Algebra.

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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