Conformal Field Theory and Cohomologies of the Lie Algebra of Holomorphic Vector Fields on a Complex Curve

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Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990. (C) The Mathematical Society of Japan, 1991 . — [(ICM) 1990. (No 1). — P.: 0071-0086], eBook, English. (Boris L. Feigin. Institute of Solid State Physics, USSR Academy of Sciences, Chernogolovka 142432 Moscow Region, USSR). Introduction. In this text we shall deal with only one aspect of the conformai field theory - the concept of the modular functor. Recall the main structure of the conformai field theory. Let M be a complex curve. First of all the theory associates to M a finite dimensional vector space H(M). The construction of H(M) depends on the choice of a complex structure on M, but really the correspondence M - H (M) is topological by nature. Our approach to the definition of H(M) is the following. Let Dif(JVf) be the group of all C°°-diffeomorphisms of M. The group Dif(M) acts on the space of all complex structures on M.Contents. Introduction. Definition of the Lie Algebra of Vector Fields on a Complex Manifold. Continuous Cohomologies of Lie(M). Description of a 1-Cocycle of the Cosimplicial Lie Algebra Lie A(M). Homologies of the Lie Algebra of Holomorphic Vector Fields on a Compact One-dimensional Manifold M. An Analytic Version of the Homology of the Lie Algebra of Holomorphic Vector Fields on M with Coefficients in the One-dimensional Representation Qc. Integrable Homologies of the Lie Algebra Lie^(M).