Rational Vertex Operator Algebras

有理顶点算子代数

• 批准号: 1404741
• 负责人: Chongying Dong
• 金额: $ 15万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404741&HistoricalAwards=false
• 关键词: Rational; Vertex; Operator; Algebras

A vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and string theory in physics. In addition to physical applications, vertex operator algebras have connections with many branches of mathematics. Rational vertex operator algebras, which are closely related to classical objects of symmetry known as Lie algebras, form the most important class of vertex operator algebras. The proposed research will solve some fundamental problems concerning rational vertex operator algebras. The research will lead to important progress in the theory of vertex operator algebras and its connections with other branches of mathematics and should have applications in conformal field theory in physics.This proposal deals with various topics on rational vertex operator algebras. The proposal consists of four parts. The first part is dedicated to the study of quantum dimensions. The quantum dimensions, which are analogues of dimensions of vector spaces, are important invariants of vertex operator algebras. The PI will use the quantum dimensions and global dimensions to study orbifold theory and classify the irreducible modules for orbifold vertex operator algebras. The quantum dimensions will also be used to give a characterization of rational vertex operator algebras. The second part of the proposal investigates the relation between the modularity of trace functions and rationality for vertex operator algebras. The PI will establish that a vertex operator algebra is rational if and only if the q-characters of the irreducible modules are modular functions. The third part on mirror extensions suggests a new way to construct vertex operator algebras which are not simple current extensions in general. Finally, in the fourth part the PI will give characterizations of the E-series of the rational vertex operator algebras with central charge 1 and complete the classification of rational vertex operator algebras with central charge 1.

顶点算子代数是一种代数结构，在物理学中的共形场论和弦理论中发挥着重要作用。 除了物理应用之外，顶点算子代数还与数学的许多分支有联系。 有理顶点算子代数与称为李代数的经典对称对象密切相关，形成最重要的一类顶点算子代数。 所提出的研究将解决有关有理顶点算子代数的一些基本问题。 该研究将导致顶点算子代数理论及其与其他数学分支的联系取得重要进展，并且应该在物理学中的共形场论中得到应用。本提案涉及有理顶点算子代数的各种主题。 该提案由四个部分组成。 第一部分致力于量子维度的研究。 量子维数与向量空间维数类似，是顶点算子代数的重要不变量。 PI将使用量子维度和全局维度来研究轨道理论并对轨道折叠顶点算子代数的不可约模进行分类。量子维度还将用于给出有理顶点算子代数的表征。该提案的第二部分研究了迹函数的模块化与顶点算子代数的合理性之间的关系。当且仅当不可约模的 q 字符是模函数时，PI 将确定顶点算子代数是有理数。关于镜像扩展的第三部分提出了一种构造顶点算子代数的新方法，这通常不是简单的当前扩展。 最后，在第四部分中，PI将给出中心电荷为1的有理顶点算子代数的E级数的表征，并完成中心电荷为1的有理顶点算子代数的分类。