Connections on the State-Space over Conformal Field Theories

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 $\omega_\mu$. The pair $(D, \omega_\mu)$ determines the covariant derivative of any correlator of local fields. We obtain interesting classes of connections in which $\omega_\mu$'s can be written in terms of CFT data. For these connections we compute their curvatures in terms of four-point correlators, $D$, and $\omega_\mu$. Among these connections three are of particular interest. A flat, metric compatible connection $\HG$, and connections $c$ and $\bar c$ having non-vanishing curvature, with $\bar c$ being metric compatible. The flat connection cannot be used to do parallel transport over a finite distance. Parallel transport with either $c$ or $\bar c$, 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.