Two-Dimensional Conformal Geometry and Vertex Operator Algebras

The focus of this volume is to formulate and prove one main theorem, the equivalance between the algebraic and geometric formulations of the notion of vertex operator algebra. The author introduces a geomatric notion of vertex operator algebra in terms of complex powers of the determinant line bundles over certain moduli spaces (parameter spaces) of spheres (genus-zero Riemann surfaces) with punctures and local analytic co-ordinates, and seeks to prove that this notion is precisely equivalent to the algebraic notion of vertex operator algebra. In particular, a detailed algebraic and analytic study of the sewing operation in the moduli space is presented.