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 ωμ. The pair (D,ωμ) determines the covariant derivative of any correlator of local fields. We obtain interesting classes of connections in which ωμ’s can be written in terms of CFT data. For these connections we compute their curvatures in terms of four-point correlators, D, and ωμ. Among these connections three are of particular interest. A flat, metric compatible connection Γ̂, and connections c and c̄ 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 or c̄, 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. ⋆ Supported in part D.O.E. contract DE-AC02-76ER03069 and NSF grant PHY91-06210. † Permanent address: Department of Physics, UCLA, Los Angeles, CA 90024-1547, USA. Supported in part by D.O.E. contract DE-AT03-88ER 40384 Mod A006 Task C.