Rational Vertex Operator Algebras

顶点算符代数是一种代数结构，在物理中的共形场论和弦理论中起着重要的作用。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经典的对称对象李代数密切相关，构成了最重要的一类顶点算子代数。本文的研究将解决有理顶点算子代数的一些基本问题。这项研究将在顶点算子代数理论及其与其他数学分支的联系方面取得重要进展，并将在物理中的保形场理论中得到应用。该提案由四个部分组成。第一部分致力于量子维度的研究。量子维度是向量空间维度的类比，是顶点算子代数的重要不变量。PI将利用量子维和整体维来研究orbillold理论，并对orb orbold顶点算子代数的不可约模进行分类。量子维度也将被用来刻画有理顶点算子代数。第二部分研究了迹函数的模性与顶点算子代数的合理性之间的关系。PI将证明顶点算子代数是有理的当且仅当不可约模的q-特征标是模函数。关于镜像扩张的第三部分提出了一种构造顶点算子代数的新方法，它通常不是简单的当前扩张。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-系列的刻划，并完成中心电荷为1的有理顶点算子代数的分类。

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.