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

EAPSI：理解顶点算子代数的积分形式及其在理论物理中的应用

• 批准号: 1614336
• 负责人: Danquynh Nguyen
• 金额: $ 0.54万
• 依托单位: Nguyen Danquynh T
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-15 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1614336&HistoricalAwards=false
• 关键词: EAPSI; Understanding; Integral; Forms; Vertex

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.

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范畴。它在理论物理中很流行，特别是保形场理论和弦理论，它试图为自然力提供一个统一的描述。虽然顶点算子代数最常被考虑在复数或特征为零的域上，但对于特征为p的域上的顶点算子代数却知之甚少。积分形式是已知的弥合这一差距的形式，也正是本项目的主要兴趣所在。更好地理解素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，他专门研究顶点算子代数。任教授是第一篇关于模顶点算子代数的论文的合著者。这个项目旨在回答这样一个问题：一个顶点算子代数什么时候有积分形式？在这一数学分支中，理性和C2余有限性是最重要的概念之一。本文将研究这样一个猜想：如果V是有理的、C2余定的、自对偶单顶点算子代数，则V具有积分形式。任教授最近的工作包括关于任意域上顶点算子代数表示的新结果，而不仅仅是特征零域上的顶点算子代数的表示。该奖项由东亚和太平洋暑期学院项目资助一名美国研究生进行暑期研究，由美国国家科学基金会和中国领导的科技部联合资助。