Perturbative Deformations of Conformal Field Theories Revisited

Recently, there has been renewed interest in the mathematics of the moduli space of conformal field theories, in particular in connection with speculations about elliptic cohomology. The purpose of this paper is to investigate this space by perturbative methods from first principles and from a purely “worldsheet” point of view. It is conjectured that at least at generic points, the moduli space of CFT’s is a manifold, and in fact, its tangent space consists of marginal fields, i.e. primary fields of weight (1, 1) of the conformal field theory (that is in the bosonic case, in the supersymmetric case there are modifications which we will discuss later). This then means that there should exist an exponential map from the tangent space at a point to the moduli space, i.e. it should be possible to construct a continuous 1-parameter set of conformal field theories by “turning on” a given marginal field.