  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.
