W-代数及相关顶点算子代数的研究

我们在本项目中主要对某些W-顶点算子代数及相关的coset和orbifold顶点算子代数的表示、结构、分类、共形扩张及张量范畴等问题开展研究。具体地，我们将首先对一类由Ising向量生成的moonshine型顶点算子代数进行研究，通过对这一类顶点代数的单性、正定性与表示的研究希望给出其唯一性刻画及分类，并给出相应的群与顶点代数的一一对应，这一类代数可看成由一类特殊的W-代数生成的顶点算子代数；其次我们将对某些coset顶点算子代数和某些orbifold顶点算子代数的结构与表示及张量范畴进行研究，这些顶点算子代数大部分都是由满足某些关系的W-代数所生成，或某些W-代数的张量积的共形扩张。我们将首先研究这些代数的表示、fusion rule, 进一步研究其唯一性刻画和张量范畴；本项目的第三部分将着重研究顶点算子代数理论中的各种对偶、张量范畴等价及与其它分支如有限群、量子群等的联系。

This project will focus on the study of W-algebras and related vertex operator algebras, including representations, characterization, classification, conformal extension and associated tensor categories. We will first study a class of moon-shine type vertex operator algebras generated by Ising vetctors of $\sigma$-type. By studying simplicity, positivity, and representations, we expect to give a complete classification ans characterization of such a class of vertex operator algebras. Secondly, we will study some coset or orbifold vertex operator alegbras. Most of these vertex operator alegbras are generated by W-algebras or conformal extensions of tensor products of W-algebras. We will focus on their representations, fusion rules and associated tensor categories. Finally, we will study various dualities in the theory of vertes operator algebras, equivalence of tnesor categories, and relations to other fields.