Algebraic and Number Theoretic Aspects of Vertex Algebra Theory

目前正在提出三个项目。在第一部分PI将追求一个新的方向表示理论的顶点代数，通过研究（新）家庭的顶点代数有限表示类型，其中涉及不可分解和对数模块。 可以预期，以这种方式得到的模的范畴等价于在单位根上的某些量子（超）群的模的范畴。 此外，将这个项目扩展到顶点超代数的建立。在这一部分中研究的结构是建立对数共形场论的基本成分，在物理学家中越来越受欢迎。第二部分研究了顶点代数模的特征标、与其密切相关的伪特征标和模微分方程。从这些考虑出发，PI提出了一系列数论家感兴趣的结果，包括常项恒等式和模恒等式。 最后，PI与他的合作者将继续致力于顶点代数理论的组合方面。特别是，相当多的注意力将致力于“主子空间”的标准模仿射李代数，主要是使用理论的交织算子。共形场论和弦论对现代数学产生了重大影响。二维共形场论在凝聚态物理和统计力学中也有重要的应用。 本研究的目的是使用物理理论中的对称性（通过表示论的语言）来研究相关函数和配分函数的分析和组合性质。这一方面将促进我们对自然过程的理解，另一方面将导致数学的新结果和新结构。

Three projects are being proposed. In the first part the PI will pursue a new direction in representation theory of vertex algebras, by studying(new) families of vertex algebras of finite representation type, which involve both indecomposable and logarithmic modules. It is expected that categories of modules obtained in this way are equivalent to categories of modules for certain quantum (super)groups at root of unity. In addition, an extension of this project to the setup of vertex superalgebras will be obtained. The structures investigated in this part are basic ingredients for building logarithmic conformal field theories, increasingly popular among physicists. The second part involves studies of characters of vertex algebra modules, closely related pseudocharacters and modular differential equations. From these considerations the PI proposes a whole array of results of interest to number theorists, including constant term identities and modular identities. Finally, the PI, jointly with his collaborators, will continue to work on combinatorial aspects of vertex algebra theory. In particular, considerable attention will be devoted to "principal subspaces" of standard modules for affine Lie algebras, by using primarily the theory of intertwining operators. It is expected that this will lead to new combinatorial bases of standard modules.Conformal field theory and string theory have had major impact on modern mathematics. Two-dimensional conformal field theory has also important applications in condensed mater physics and statistical mechanics. The aim of this research is to use symmetries in physical theories (through the language of representation theory) to study analytic and combinatorial properties of correlation functions and partition functions. This will, on one hand, advance our understanding of processes in nature and, on the other, lead to new results and structures in mathematics.