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，其重要性远远超出了数学。它在理论物理学中很普遍，特别是共形场论和弦理论，它们试图提供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领域的复数或领域的特征零，很少有人知道那些领域的素数特征p.更好地了解顶点算子代数领域的素数特征起着重要的作用，在研究模表示的有限群，理性顶点算子代数，和理性共形场论。本课题将在大坂大学研究生院情报科学研究科进行，导师是物理学中与共形场论对应的顶点算子代数理论的长友清和教授。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。本课题的主要目的是研究Virasoro代数的极小级数所对应的模顶点算子代数的合理性。目的是将已有的一些结果从特征为零的域推广到特征满足某种约束关系的域。我们的计划是研究这类域上Verma模的奇异向量。这个NSF EAPSI奖为美国研究生提供了一个国际研究机会，并与日本科学促进协会（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).