Modular invariance of vertex operator algebras and its application to representation theory

• Project/Area Number: 25800003
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: University of Tsukuba
• Principal Investigator: ARIKE Yusuke 筑波大学, 数理物質系, 助教 (50583770)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 頂点作用素代数 / モジュラー不変性 / モジュラー微分方程式 / ヴィラソロ代数 / 大域次元 / モジュラー形式 / モジュラーな微分方程式 / アフィンリー環 / 共形場理論 / リーマン面 / 共形ブロック

Outline of Final Research Achievements

We study several relations between characters of simple modules of vertex operator algebras and modular linear differential equations and obtain the following results. (i) We prove that if the dimension of characters of simple affine vertex operator algebras is less than 6 and greater than 1, then the space of characters is a solution space of a modular linear differential equation. (ii) We give a characterization of Virasoro vertex operator algebras by means of modular linear differential equations. (iii) We show that there are no good vertex operator algebras with central charges 164/5 and 236/7 whose characters are solutions of modular linear differential equations of order 3.

Research Products

• [Journal Article] Characterization of the simple Virasoro vertex operator algebras with 2 and 3-dimensional space of characters (2017)
  • Author(s): Yusuke Arike, Kiyokazu Nagatomo, Yuichi Sakai
  • Journal Title: Contemporary Mathematics Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Affine vertex operator algebras and modular linear differential equations (2016)
  • Author(s): Yusuke Arike, Masanobu Kaneko, Kiyokazu Nagatomo, Yuichi Sakai
  • Journal Title: Letters in Mathematical Physics Volume: - Issue: 5 Pages: 1-26
  • DOI: 10.1007/s11005-016-0837-7
  • NAID: 120007187839
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] Characters and modular linear differential equations of order 3 and 4 (2016)
  • Author(s): Yusuke Arike
  • Organizer: Workshop on Finite Groups, VOA, and algebraic combinatorics
  • Place of Presentation: Foguang University, Yilan, Taiwan
  • Year and Date: 2016-03-21
  • Int'l Joint Research / Invited
• [Presentation] Affine vertex operator algebras and modular linear differential equations (2016)
  • Author(s): 有家雄介，金子昌信，永友清和，境優一
  • Organizer: 日本数学会２０１６年度年会
  • Place of Presentation: 筑波大学（茨城県つくば市）
  • Year and Date: 2016-03-16
• [Presentation] Vertex operator algebras, minimal models, and modular linear differential equations of order 4 (2016)
  • Author(s): 有家雄介，永友清和，境優一
  • Organizer: 日本数学会２０１６年度年会
  • Place of Presentation: 筑波大学（茨城県つくば市）
  • Year and Date: 2016-03-16
• [Presentation] Minimal models and modular linear differential equations of order 4 (2016)
  • Author(s): 有家雄介
  • Organizer: 有限群とその表現，頂点作用素代数，代数的組合せ論の研究
  • Place of Presentation: 京都大学数理解析研究所（京都府京都市）
  • Year and Date: 2016-01-05
  • Invited
• [Presentation] Minimal models and modular linear differential equations of 4th order (2015)
  • Author(s): Yusuke Arike
  • Organizer: Vertex Operator Algebra and Related Topics
  • Place of Presentation: Sichuan University, Chengdu, China
  • Year and Date: 2015-09-07
  • Int'l Joint Research
• [Presentation] アフィン頂点作用素代数の指標のみたすモジュラー微分方程式 (2015)
  • Author(s): 有家雄介
  • Organizer: 有限群論・駒場セミナー
  • Place of Presentation: 東京大学（東京都目黒区）
  • Year and Date: 2015-05-25
• [Presentation] Characters of affine VOAs and modular differential equations (2015)
  • Author(s): Yusuke Arike
  • Organizer: Taitung Workshop on finite groups, VOA and algebraic combinatorics
  • Place of Presentation: National Taitung University (台湾)
  • Year and Date: 2015-03-07 – 2015-03-10
  • Invited
• [Presentation] Modular invariance of vertex operator algebras (2014)
  • Author(s): Yusuke Arike
  • Organizer: Around Mathematical Physics and Geometry
  • Place of Presentation: 神戸大学
  • Year and Date: 2014-09-20 – 2014-09-21
  • Invited
• [Presentation] Fusion products for symplectic fermionic VOSA
  • Author(s): Yusuke Arike
  • Organizer: Beyond the Moonshine
  • Place of Presentation: ホテル華乃湯(宮城県仙台市)
  • Invited
• [Presentation] Fusion rules for irreducible modules for symplectic fermionic VOAs
  • Author(s): Yusuke Arike
  • Organizer: Hualien workshop on finite groups, VOA, algebraic combinatorics, and related topics
  • Place of Presentation: National Dong Hua university, Taiwan
  • Invited