Moduli space of pointecl autrves and conismal field theory

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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.