顶点算子代数的扭正则表示理论

本项目主要系统地研究顶点算子代数的扭正则表示理论，研究内容主要分成如下三个方面: (1)引入和研究顶点算子代数的扭正则表示，给出扭正则表示的Peter-Weyl分解. 进一步地, 将研究扭正则表示与Huang-Lepowsky的张量积理论之间的关系. (2)利用顶点算子代数扭表示中的工具和方法研究扭正则表示，并研究相关的诱导模；研究将顶点算子代数扭正则表示与对于扭模的Frenkel-Zhu关于fusion rule定理之间建立起联系；用扭正则表示研究迹函数和零亏格以及高亏格的相关函数；(3)研究Möbius量子顶点代数的扭正则表示，给出Möbius量子顶点代数扭正则表示的Peter-Weyl分解；研究Möbius量子顶点代数的扭模的扭张量积理论.

In this project we mainly study twisted regular representations of vertex operator algebras systematically and the contents can be divided into the following three parts: (1)We will introduce and study the twisted regular representations of vertex operator algebras, give a Peter-Weyl type decomposition for twisted regular representations. Furthermore, we will study the relationship between twisted regular representations and Huang-Lepowsky’s tensor product theory. (2)We shall study twisted regular representations and related induced modules by using the tools and methods in the theory of twisted representations of vertex operator algebras. We also will study Frenkel-Zhu’s theorem on fusion rules for twisted modules in terms of twisted regular representations. In a sequel we shall associate the twisted regular representations with trace functions, genus-zero and higher genus correlation functions. (3)We will introduce and study the twisted regular representations of Möbius quantum vertex algebras, give a Peter-Weyl type decomposition for twisted regular representations of Möbius quantum vertex algebras. Furthermore, we will study twisted tensor product theory of twisted modules for Möbius quantum vertex algebras.