本课题研究顶点算子代数及相关议题。主要内容有四个方面：1）拟建立完整的orbifold理论，特别要证明有理顶点算子代数的orbifold子代数是有理的；2）研究顶点算子代数的有理性与C_2余有限性的关系，证明有理性蕴含着C_2余有限性，从而彻底解决本领域长达二十年之久的猜想；3）研究 Virasoro模顶点算子代数，拟证明由minimal series所确定的Virasoro模顶点算子代数在特征大于零的代数闭域上仍是有理的，并对不可约模进行分类。寻找与affine模顶点算子代数的联系；4）研究月光顶点算子代数，拟证明Frenkel-Lepowsky-Meurman的唯一性猜想，这是中心等于24全纯顶点算子代数分类的一部分。本项目所研究的问题是顶点算子代数的基本问题，这些问题的顺利解决不但对顶点算子代数理论的发展至关重要，且对物理中的二维共形场论提供更为坚实的代数基础。

This is a proposal on vertex operator algebra and related topics. The proposal consists of four major parts: 1）Establish a complete orbifold theory. In particular, prove that the fixed point vertex operator subalgebra of a rational vertex operator algebra under a finite group action is also rational, this will solve a well known conjecture in the orbifold theory. 2）Study the relation between rationality and the C_2-cofiniteness for vertex operator algebra. Prove that rationality implies the C_2-cofiniteness, this will give a positive solution to a more than twenty years old conjecture in the field. 3）Investigate the Virasoro modular vertex operator algebras. Prove that the vertex operator algebras determined by the minimal series of the Virasoro algebra over any algebraically closed field are rational and classify their irreducible modules. Also seek the ration with the affine modular vertex operator algebra via coset construction. 4）Study the moonshine vertex operator algebra. Prove the Frenkel-Lepowsky-Meurman’s uniqueness conjecture on the moonshine vertex operator algebra. This proposal will study the key and fundamental problems in the field. Solving these problems successfully will not only important for the development of the theory of vertex operator algebra, but also provide further algebraic foundation to the 2-dimensional conformal field theory.