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

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考虑在复数域或特征为0的域上，但对素数特征为p的域上的顶点算子代数知之甚少。已知积分形式可以弥补这一差距，并且正是本项目的主要兴趣所在。更好地理解素数特征域上的顶点算子代数对于有限群的模表示、有理顶点算子代数和有理共形场论的研究具有重要的意义。该项目将在四川大学数学学院进行，由李仁教授指导。任教授，专攻顶点算子代数。任教授是一篇论文的合著者，该论文被认为是处理模顶点算子代数的第一篇论文。这个项目旨在回答这个问题：顶点算子代数何时具有积分形式？在这个数学分支中，合理性和c2 -有限性是最重要的概念之一。研究了如果V是一个有理的、c2 -有限的、自对偶的简单顶点算子代数，则V具有积分形式的猜想。任教授最近的工作包括在任意域上的顶点算子代数表示的新结果，而不仅仅是特征为零的域。该奖项隶属于东亚和太平洋暑期研究所项目，由美国国家科学基金会和中国科技部共同资助，支持美国研究生的暑期研究。

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.