Vertex Operator Algebras and the Representation and Cohomology Theory of Groups and Algebras

顶点算子代数以及群和代数的表示和上同调理论

• 批准号: 9700909
• 负责人: Geoffrey Mason
• 金额: $ 7.5万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700909&HistoricalAwards=false
• 关键词: Vertex; Operator; Algebras; Representation; Cohomology

ABSTRACT, MASON 97-00909 Abstract : PI will investigate the mathematical foundations of the theory of vertex operator algebras, their automorphism groups and representation theory, and related questions in the cohomology of groups and algebras and in the theory of modular forms. Techniques are based on past work of the PI and collaborators on the foundations of algebraic conformal field theory, including the theory of Zhu's associative algebras, quantum Galois theory, existence and modular-invariance of twisted sectors, and a cohomological theory of discrete torsion and Dijkgraaf-Witten cocycles. In recent years, mathematicians and physicists alike have been developing conformal field theory, which incorporates infinite-dimensional symmetry, in order to understand certain critical phenomena and also (via string theory) the first rigorous approach to quantum gravity. This effort is a highly intensive mathematical endeavour, and the PI will investigate a number of new phenomena in abstract algebra which are part of this program. So-called vertex operator algebras constitute a rigorous mathematical axiomatization of some of the main ideas of conformal field theory, and the PI will continue his research into the symmetries inherent in these very complicated objects in order to solidify the foundations of the subject and to uncover new phenomena.

摘要，MASON 97-00909 摘要：PI将研究顶点算子代数理论的数学基础，它们的自同构群和表示理论，以及群和代数的上同调和模形式理论中的相关问题。技术是基于过去的工作PI和合作者的基础上的代数共形场论，包括理论的朱的结合代数，量子伽罗瓦理论，存在性和模不变性的扭曲部门，和上同调理论的离散扭转和Dijkgraaf-Witten cocycles。 近年来，数学家和物理学家都在发展包含无限维对称性的共形场论，以理解某些临界现象，也是（通过弦理论）量子引力的第一个严格方法。这项工作是一项高度密集的数学努力，PI将调查抽象代数中的一些新现象，这些现象是该计划的一部分。 所谓的顶点算子代数构成了共形场论的一些主要思想的严格的数学公理化，PI将继续研究这些非常复杂的对象中固有的对称性，以巩固该主题的基础并发现新的现象。