N ov 2 00 6 Abelian Conformal Field Theory and Determinant Bundles

Following [KNTY] we study a so-called bc-ghost system of zero conformal dimension from the viewpoint of [TUY] and [U2]. We show that the ghost vacua construction results in holomorphic line bundles with connections over holomorphic families of curves. We prove that the curvature of these connections are up to a scale the same as the curvature of the connections constructed in [TUY] and [U2]. We study the sewing construction for nodal curves and its explicit relation to the constructed connections. Finally we construct preferred holomorphic sections of these line bundles and analyze their behaviour near nodal curves. These results are used in [AU2] to construct modular functors form the conformal field theories given in [TUY] and [U2] by twisting with an appropriate factional power of this Abelian theory.