Algebraic and Number Theoretic Aspects of Vertex Algebra Theory

目前提出了三个项目。在第一部分中，PI将在顶点代数的表示理论中探索一个新的方向，通过研究（新的）有限表示型顶点代数族，它涉及不可分解和对数模。期望用这种方法得到的模的范畴等价于某量子（超）群在单位根处的模的范畴。此外，将本课题推广到顶点超代数的建立。本部分所研究的结构是建立对数共形场理论的基本成分，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性质、密切相关的伪性质和模微分方程。从这些考虑出发，PI提出了一系列数论学家感兴趣的结果，包括常数项恒等式和模恒等式。最后，PI和他的合作者将继续研究顶点代数理论的组合方面。特别地，相当多的注意力将投入到仿射李代数标准模的“主子空间”，主要是通过使用缠结算子的理论。预计这将导致标准模块的新组合基。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。本研究的目的是利用物理理论中的对称性（通过表征理论的语言）来研究相关函数和配分函数的解析性和组合性。一方面，这将促进我们对自然界过程的理解，另一方面，这将导致数学中的新结果和新结构。

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.