Motivated by the problem of background independence of closed string field theory we study geometry on the infinite vector bundle of local fields over the space of conformal field theories (CFT's). With any connection we can associate an excluded domain D for the integral of marginal operators, and an operator one-form ω µ . The pair (D, ω µ ) determines the covariant derivative of any correlator of local fields. We obtain interesting classes of connections in which ω µ 's can be written in terms of CFT data. For these connections we compute their curvatures in terms of four-point correlators, D, and ω µ . Among these connections three are of particular interest. A flat, metric compatible connection Γ, and connections c andc with non-vanishing curvature, with the latter metric compatible. The flat connection cannot be used to do parallel transport over a finite distance. Parallel transport with either c orc, however, allows us to construct a CFT in the state space of another CFT a finite distance away. The construction is given in the form of perturbation theory manifestly free of divergences.
There are two methods to study families of conformal theories in the operator formalism.In the first method we begin with a theory and a family of deformed theories is defined in the state space of the original theory. In the other there is a distinct state space for each theory in the family, with the collection of spaces forming a vector bundle. This paper establishes the equivalence of a deformed theory with that in a nearby state space in the bundle via a connection that defines maps between nearby state spaces. We find that an appropriate connection for establishing equivalence is one that arose in a recent paper by Kugo and Zwiebach. We discuss the affine geometry induced on the space of backgrounds by this connection. This geometry is the same as the one obtained from the Zamolodchikov metric.
It has been proposed that the string diagrams of closed string field theory be defined by a minimal area problem that requires that all nontrivial homotopy curves have length greater than or equal to 2π. Consistency requires that the minimal area metric be flat in a neighbourhood of the punctures. The theorem proven in this paper, yields a criterion which if satisfied, will ensure this requirement. The theorem states roughly that the metric is flat in an open set, U if there is a unique closed curve of length 2π through every point in U and all of these closed curves are in the same free homotopy class.
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