Vertex Operator Algebras

本课题主要研究关于顶点算子代数的各种课题。特别地，以下主题将被研究：(1)诱导模：PI将定义一个从顶点算子代数的扭曲模范畴到更大顶点算子代数的扭曲模范畴的诱导函子。这将为研究顶点算子代数的表示理论建立一个基本的工具，并导致解决轨道共形场理论中众所周知的猜想。(2)模不变性：PI期望证明在有理顶点算子代数的模群的同余子群上，扭曲迹函数或配分函数是模函数。特别地，任何不可约的扭曲模或扭曲扇区的分级维数都是模函数。在满模群下，期望得到不可约模的梯度维的绝对值平方和是不变的。这将在顶点算子代数的分类中有直接的应用。(3)中心电荷小的有理顶点算子代数的分类：有理顶点算子代数是最重要的一类顶点算子代数，在有理共形场论中起着重要的作用。PI将对中心电荷小于或等于1的有理顶点算子代数进行分类。这可以看作是有理顶点算子代数分类的第一步。顶点算子代数理论是一个新的数学领域。它为物理学中的二维量子共形场论提供了代数基础，并与表示论、群论、模形式、拓扑不变量、C*-代数等许多重要的数学领域有着深刻的联系。本课题研究了该领域的一些基本问题，在数学和物理领域都有重要的应用。

The principal investigator proposes to study various topics on vertex operator algebras. In particular, the following topics will be studied: (1) Induced modules: The PI will define an induced functor from the twisted module category for a vertex operator algebra to the twisted module category for a bigger vertex operator algebra. This will establish a basic tool for studying the representation theory for vertex operator algebras and lead to solving well known conjectures in orbifold conformal field theory.(2) Modular invariance: The PI expects to prove that twisted trace functions or partition functions are modular functions over congruence subgroups of the modular group for a rational vertex operator algebra. In particular, the graded dimension of any irreducible twisted module or twisted sector is a modular function. It is also expected to obtain that the sum of the squares of the absolute values of the graded dimensions of the irreducible modules is invariant under the full modular group. This will have immediate applications in classification of vertex operator algebras. (3) Classification of rational vertex operator algebras with small central charges: Rational vertex operator algebras which play the fundamental roles in rational conformal field theory form the most important class of vertex operator algebras. The PI will classify the rational vertex operator algebras with the central charge less than or equal to 1. This can be regarded as the first step in classification of rational vertex operator algebras.Vertex operator algebra theory is a new area of mathematics. It provides an algebraic foundation for the 2 dimensional quantum conformal field theory in physics and is also deeply related to many important areas of mathematics such as representation theory, group theory, modular forms, topology invariants, and C*-algebras. The proposed research studies some fundamental problems in the field and has important applications in both mathematics and physics.