Moduli space of pointecl autrves and conismal field theory

在2005年和2007年期间，我们研究了点黎曼曲面的模空间及其上的共形场论。我们首先研究了顶点算子代数，当VOA具有Zhut的有限条件时，我们建立了表示理论。这允许我们在投影线上构造保形线。给定点黎曼曲面，我们可以用顶点算子代数定义当前李代数。进一步利用李代数引入了协不变束的概念，在协不变束的朱纤维为有限维的条件下，得到了相干束。此外，由于维拉索罗代数的作用，还有一些麦捆带有平面连接。这意味着这些轮轴在局部是自由的。其次，对于具有朱条件的有理点算子代数，我们证明了射影线的易分解性，即。，当n-s大于4时，任意n个协变量的点轴表示为3个点轴。另一方面，我们研究了非理性顶点算子代数。例如，w -代数就是这样一个对象，由于w -代数不是完全可约的，我们必须对不可分解模块进行分类。现在提出了不可分解模块列表，并期望这些模块为投影模块。如果这些被证明，Ext*组将完全确定。w代数是由正整数p>1决定的。当P=2时，w代数的模的范畴等价于受限量子群Uq(sl(2))q=1的有限维模的范畴。在这个研究项目中，我们构造了结点的不变量。这一成功的背后是有限维受限量子群的范畴是带状范畴。

During 2005 and 2007 we studied moduli space of pointed Riemann surfaces and conformal field theory over them. We first studied vertex operator algebras and when VOA has Zhut's finiteness condition we established representation theory. This allows is construct conformal filed over the projective line. Given pointed Riemann surface we are able to define current Lie algebra from vertex operator algebra. Further using current Lie algebra we introduce a notion of sheaf of coinvariants, Under the condition of Zhu fibers of sheaf of coinvariants are finite dimensional This means that we obtain coherent sheaves. Moreover there sheaves axe equipped with flat connections which come from the action of Virasoro algebra. These imply that these sheaves are locally free. Next for rational vertex operator algebra with Zhu's condition case we are able to prove so-called factorization property for he ease of projective line ,i.e., any n point sheaves of coinvariants are expressed by 3 point sheave if n-s greater than 4.On the other hand we have studied non-rational vertex operator algebra. For instance, W-algebra is an example such object Since W-algebm is not completely reducible, we have to classify indecomposable module. Now the list of indecomposable module is proposed and these modules are expected to be projective modules. If these are proved Ext* group will be completely determined. W-algebra is determined by positive integer p>1. When P=2 the category of modules of W-algebra is equivalent to the category of finite dimensional module for the restricted quantum group Uq(sl(2))q=1. In this research project we have constructed invariants of knots. Behind this successful work there is the fact that the category of finite dimensional restricted quantum group is ribbon category.