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

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，以及群与代数的上同调和模形式理论中的相关问题。技术是基于PI和合作者过去在代数共形场论基础上的工作，包括朱的关联代数理论，量子伽罗瓦理论，扭曲扇区的存在性和模不变性，以及离散扭转和Dijkgraaf-Witten共环的上同调理论。近年来，数学家和物理学家都在发展共形场论，它包含了无限维对称性，以理解某些关键现象，并（通过弦理论）第一个严谨的方法来研究量子引力。这项工作是一项高度密集的数学努力，PI将研究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将继续他对这些非常复杂的物体固有的对称性的研究，以巩固这一学科的基础并发现新的现象。

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.