Abelian Conformal Field theories and Determinant Bundles

The present paper is the first in a series of papers, in which we shall construct modular functors and Topological Quantum Field Theories from the conformal field theory developed in [TUY]. The basic idea is that the covariant constant sections of the sheaf of vacua associated to a simple Lie algebra over Teichmüller space of an oriented pointed surface gives the vector space the modular functor associates to the oriented pointed surface. However the connection on the sheaf of vacua is only projectively flat, so we need to find a suitable line bundle with a connection, such that the tensor product of the two has a flat connection. We shall construct a line bundle with a connection on any family of N pointed curves with formal coordinates. By computing the curvature of this line bundle, we conclude that we actually need a fractional power of this line bundle so as to obtain a flat connection after tensoring. In order to functorially extract this fractional power, we need to construct a preferred section of the line bundle. We shall construct the line bundle by the use of the so-called bc-ghost systems (Faddeev-Popov ghosts) first introduced in covariant quantization [FP]. The bc system have two anticommuting fields b(z), c(z) of conformal dimension j, 1− j, respectively, where j is an integer or half integer. In the case j = 1/2 a mathematically rigorous treatment was given in the paper [KNTY]. The case j = 1/2 corresponds to the study of the determinant bundle of half-canonical line bundles on smooth curves, i.e. on compact Riemann surfaces. Since we cannot define the half-canonical line bundles for curves with nodes, whose normalization has at least two components, the boundary behavior of the sheaf of vacua is complicated [KSUU]. Therefore, in the present paper we shall consider the case j = 0, following the ideas of