Representation Theoretic Study of Two Dimensional Quantum Field Theory

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We studied the construction of the conformal field theories associated to Vertex Operator Algebra(V.O.A.) satisfying Zhu's C_2-finiteness condition.In order to construct the conformal field theories we defined universal enveloping algebra associated to V.O.A.satisfying Zhu's C_2-finiteness condition, and analyzed structure of Abelian categories constituted of modules of this algebra in more detail. These Abelian categories are Nothern and Artin, but not semi-simple.We gave an expression of Zhu algebra associated V.O.A.using enveloping algebra of V.O.A., and gave necessary and sufficient condition of the semi-simplicity of the Abelian categories using Zhu algebra.Using above results, we constructed coherent sheaves constituting of conformal blocks over moduli spaces of genus 0 N-pointed stable curves, and showed that they have D-module structures with regular singularities along the boundary of the moduli spaces.By analyzing of bi-module structure of enveloping algebra in further detail, we succeeded the construction of the Dx-module constituted conformal blocks over moduli spaces of genus g N-pointed stable curves, the formulation of the factorization properties of the coherent sheaves along the boundary of the moduli spaces. We are now preparing the paper.We are studying the so-called W(p) algebra satisfying Zhu's C_2-finiteness condition. These algebra are typical examples such that the Abelian categories are not semi-simple. We are now making detailed study of the Abelian categories, and Fusion product of conformal blocks, modular properties of genus 1 conformal blocks. Further more there are some nice relationships between representation theories of quantum groups at root of unity.