CONNECTIONS ON THE STATE-SPACE OVER CONFORMAL FIELD-THEORIES

Motivated by the problem of background independence of closed string f ield theory we study geometry on the infinite vector bundle of local f ields over the space of conformal field theories (CFTs). With any conn ection we can associate an excluded domain D for the integral of margi nal operators, and an operator one-form omega(mu). The pair (D, omega( mu)) determines the covariant derivative of any correlator of local fi elds. We obtain interesting classes of connections in which omega(mu)' s can be written in terms of CFT data. For these connections we comput e their curvatures in terms of four-point correlators, D, and omega(mu ). Among these connections three are of particular interest. A flat, m etric compatible connection F, and connections c and cBAR with non-van ishing curvature, with the latter metric compatible. The flat connecti on cannot be used to do parallel transport over a finite distance. Par allel transport with either c or cBAR, however, allows us to construct a CFT in the state-space of another CFT a finite distance away. The c onstruction is given in the form of perturbation theory manifestly fre e of divergences.