EAPSI: Understanding the Integral Forms of Vertex Operator Algebras and their Applications in Theoretical Physics

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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 attempts 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. Integral forms are known to bridge this gap and are precisely the primary interest of this project. 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. The project will be conducted at the School of Mathematics at Sichuan University under the mentorship of Prof. Li Ren. Prof. Ren, who specializes in vertex operator algebras. Prof. Ren is a coauthor of a manuscript regarded as the first paper dealing with modular vertex operator algebras.This project aims to answer the question: When does a vertex operator algebra have an integral form? In this branch of mathematics, rationality and C2-cofiniteness are among the most important concepts. The research will investigate the conjecture that if V is a rational, C2-cofinite, and self-dual simple vertex operator algebra, then V has an integral form. Professor Ren's recent works include new results on the representations of vertex operator algebras over an arbitrary field, not just fields of characteristic zero. This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and China's Ministry of Science and Technology.