EAPSI: Modular Vertex Operator Algebras Associated with the Virasoro Algebra

顶点算子代数理论是一个数学框架，其重要性远远超出数学范畴。它在理论物理学中很流行，特别是在共形场论和弦理论中，它们试图提供对自然界所有力的统一描述。虽然顶点算子代数最常被认为是在复数域或特征为0的域上，但对素数特征p域上的顶点算子代数知之甚少。更好地理解素数特征域上的顶点算子代数在有限群的模表示、有理顶点算子代数和有理共形场论的研究中起着重要作用。该项目将在大阪大学信息科学与技术研究生院进行，由Kiyokazu Nagatomo教授指导，他专门研究与物理共形场论相对应的顶点算子代数理论。在顶点算子代数理论中，合理性被认为是最重要的概念之一。本课题的主要兴趣是研究与Virasoro代数的最小级数相关的模顶点算子代数的合理性。目的是将一些已有的结果从特征为零的场推广到特征满足某种猜想关系的场。我们的计划是研究这些领域上Verma模块的奇异向量。美国国家科学基金会EAPSI奖为美国研究生提供了一个国际研究机会，由日本科学促进会（JSPS）合作资助。

The theory of vertex operator algebras is a mathematical framework whose importance reaches well outside of mathematics. It is prevalent in theoretical physics, in particular, conformal field theory and string theory, which attempt to provide a unified description of all the forces of nature. While a vertex operator algebra is most often considered over the field of complex numbers or fields of characteristic zero, little is known about those over a field of prime characteristic p. A better understanding of vertex operator algebras over fields of prime characteristic plays an important role in the study of modular representations of finite groups, rational vertex operator algebras, and rational conformal field theory. This project will be conducted at the Graduate School of Information Science and Technology at Osaka University under the mentorship of Prof. Kiyokazu Nagatomo, who specializes in the theory of vertex operator algebras which correspond to conformal field theory in physics.In vertex operator algebra theory, rationality is considered to be among the most important concepts. The primary interest of this project is to investigate the rationality of modular vertex operator algebras associated with the minimal series of the Virasoro algebra. The goal is to extend some existing results from the fields of characteristic zero to fields whose characteristic satisfies a certain conjectured relation. The plan is to investigate the singular vectors of the Verma modules over such fields. This NSF EAPSI award provides an international research opportunity to a U.S. graduate student and is funded in collaboration with the Japan Society for the Promotion of Science (JSPS).