From vertex operator algebras to modular tensor categories II

This is the follow-up of October 6 (and still a beginner's talk). I shall describe what is the tensor structure on the categories of modules of vertex operator algebras (=VOA), and thus how the modular tensor category (=MTC) is built. Then, I will introduce the notion of modular functor (=MF). From any given MTC, one can construct a modular functor -- and in particular representations of (central extension) of SL(2,Z) and more general mapping class groups of surfaces -- and there is a partial converse to go from MF to MTC. In the case of categories of modules of VOA, the representation of SL(2,Z) is realized by vector valued modular forms.