In the preset paper we shall report our recent results with J. E. Andersen([1-3]) on Modular Functor and Topological Quantum Field Theory . Conformal field theory (CFT) is a functor from the category of pointedRiemann surfaces with coordinates to the category of finite dimensionalcomplex vector spaces, which satisfies several basic properties (see §3 be-low). Modular functor (MF) is a functor from the category of pointed ori-ented surfaces with tangent vectors and Lagrangian subspace to the cat-egory of finite dimensional complex vector spaces, which satisfies similarproperties to those of conformal field theory (see §2). The difference is thatCFT does depend on a complex structure of a surface but MF only dependson a differentiable structure of a surface. Nevertheless we can construct amodular functor by using non-abelian and abelian conformal filed theories([1,2]), and the naturally defined projectively flat connections.
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