Research on structural symmetries of vertex operator algebras

• Project/Area Number: 16K05073
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Woman's Christian University
• Principal Investigator: Yamauchi Hiroshi 東京女子大学, 現代教養学部, 准教授 (40452213)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2016: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 頂点作用素代数 / ヴィラソロ代数 / ３互換群 / テンソル圏 / 双対性 / 三互換群 / グライス代数 / 単純カレント / 散在型有限単純群 / 共形デザイン

Outline of Final Research Achievements

I have researched on structural symmetries of vertex operator algebras of OZ-type. The Griess algebra is a finite dimensional substructure of a vertex operator algebra which is an infinite dimensional object. I have proved the uniqueness of the structure of a simple vertex operator algebra generated by Ising vectors of sigma-type, where the Griess algebra for such a vertex operator algebra is described by Matsuo algebras. I have also researched on a vertex operator algebras which involves a subalgebra having group-like fusion. I have described a duality between Grothendieck rings of the representation categories of vertex operator overalgebra/subalgebra via quadratic forms on finite abelian groups arising from modular tensor category.

Academic Significance and Societal Importance of the Research Achievements

高い対称性を持った構造は数学および物理学で重要であり，また興味を引く研究対象である。本研究では二次元共形場理論の代数的定式化である頂点作用素代数における対称性について，構造論によるアプローチで研究を行い，３互換群に関連するクラスについて，内在的な特徴付けを与える成果を得た。また，このようなクラスに属する例を与える構成法について一般論を展開し，フュージョン代数間の双対性を有限可換群上の二次形式による双対性として記述する理論を確立した。

Research Products

• [Int'l Joint Research] 上海交通大学(中国)
• [Int'l Joint Research] 中央研究院数学研究所(台湾)
• [Journal Article] Vertex operator algebras generated by Ising vectors of $$\sigma $$σ-type (2018)
  • Author(s): Jiang Cuipo、Lam Ching Hung、Yamauchi Hiroshi
  • Journal Title: Mathematische Zeitschrift Volume: 印刷中 Issue: 1-2 Pages: 1-18
  • DOI: 10.1007/s00209-018-2184-0
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Vertex operator algebras associated with Z/kZ-codes (2017)
  • Author(s): Tomoyuki Arakawa, Hiromichi Yamada and Hiroshi Yamauchi
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 191 Pages: 513-521
  • DOI: 10.1007/978-981-10-2636-2_38
  • ISBN: 9789811026355, 9789811026362
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] 群的フュージョンを持つ VOA と二次形式 (2018)
  • Author(s): 山内博
  • Organizer: RIMS共同研究(公開型)「代数的組合せ論と関連する群と代数の研究」
  • Invited
• [Presentation] Simple current extensions and quadratic spaces (2018)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Vertex Operator Algebras and Related Topics, Sichuan University, Chengdu, China
  • Int'l Joint Research / Invited
• [Presentation] 群的フュージョンと二次形式 (2018)
  • Author(s): 山内博
  • Organizer: 第30回有限群論草津セミナー
• [Presentation] VOAs and Miyamoto involutions of σ-type (2018)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Finite Groups, VOAs, and Related Topics 2018
  • Int'l Joint Research / Invited
• [Presentation] Matsuo algebras and vertex operator algebras (2017)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Representation Theory XV
  • Int'l Joint Research / Invited
• [Presentation] Fi23 から M への VOA によるアプローチ (2017)
  • Author(s): 山内 博
  • Organizer: 第29回有限群論草津セミナー
• [Presentation] An approach to the moonshine VOA via the symmetric group of degree 24 (2017)
  • Author(s): Hiroshi Yamauchi
  • Organizer: One day Workshop on VOA
• [Presentation] On Conway-Miyamoto correspondences for Fischer 3-transposition groups (2016)
  • Author(s): Hiroshi Yamauchi
  • Organizer: The XXIVth International Conference on Integrable Systems and Quantum Symmetries
  • Place of Presentation: Cezch Technical University Prague
  • Int'l Joint Research
• [Funded Workshop] 10th Seminar on Conformal Field Theory (2018)
• [Funded Workshop] RIMS Gasshuku-style Seminar Vertex operator algebras and conformal field theory (2018)