EAPSI: Modular Vertex Operator Algebras Associated with the Virasoro Algebra

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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).