Nearby CFT's in the operator formalism: The role of a connection

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.