Vertex operator algebras and modular differential equations

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K05171
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Nagatomo Kiyokazu 大阪大学, 情報科学研究科, 招へい准教授 (90172543)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: 頂点作用素代数 / モジュラー微分方程式 / 指標 / 分類理論 / 保型形式

Outline of Final Research Achievements

We had worked on the classification problem of vertex operator algebras (simply VOAs). The sets of characters were expected to characterize VOAs. However, there are examples of VOAs whose have the same set of characters but they are not isomorphic. Therefore we intended the classification by using modular linear differential equations (MLDEs for short) since any set of characters of a VOA is a subspace of the space of solutions of an MLDE. We achieved the classification of the Virasoro VOA (the so-called the minimal model) under very mild conditions. Moreover, since it is very important to have a recipe to obtain MLDEs of higher order. We found that the Rankin-Cohen brackets give a general description of MLDEs. This result contributes to the theory of modular forms by giving a new differential operator which generalizes the Serre operation.

Free Research Field

頂点作用素代数の理論

Academic Significance and Societal Importance of the Research Achievements

頂点作用素代数のモジュラー微分方程式を用いた分類は，国内外で前例がなく，その内容は高く評価されている。これは頂点作用素代数の理論において新しい研究分野を与え，多くの研究者が興味を持ち始めている。新しい研究分野を開拓することは困難を伴い，我々の研究成果がそれを与えたことは，国内外で広く認めらた事実であり，今後，この分野に重要な影響を与えると考えらられる。