Rational Vertex Operator Algebras

顶点算子代数是一种代数结构，在物理学的共形场论和弦论中起着重要的作用。除了物理应用外，顶点算子代数还与许多数学分支有联系。有理顶点算子代数是最重要的一类顶点算子代数，它与经典对称对象李代数密切相关。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，并将在物理学的共形场论中得到应用。本文讨论了关于有理顶点算子代数的各种问题。该提案由四个部分组成。第一部分致力于量子维度的研究。量子维是向量空间维数的类似物，是顶点算子代数的重要不变量。PI将利用量子维数和全局维数研究轨道理论，并对轨道顶点算子代数的不可约模进行分类。量子维度也将用于给出有理顶点算子代数的表征。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将证明一个顶点算子代数是有理的当且仅当不可约模的q-字符是模函数。第三部分是镜像扩展，提出了一种构造顶点算子代数的新方法。最后，在第四部分给出了中心电荷为1的有理顶点算子代数的e级数的刻画，并完成了中心电荷为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.