THE DILATON THEOREM AND CLOSED STRING BACKGROUNDS

The zero-momentum ghost-dilaton is a non-primary BRST physical state p resent in every bosonic closed string background. It is given by the a ction of the BRST operator on another state \chi], but remains nontriv ial in the semirelative BRST cohomology. When local coordinates arise from metrics we show that dilaton and \chi] insertions compute Riemann ian curvature and geodesic curvature respectively. A proper definition of a CFT deformation induced by the dilaton requires surface integral s of the dilaton and line integrals of \chi]. Surprisingly, the ghost number anomaly makes this a trivial deformation. While dilatons cannot deform conformal theories, they actually deform conformal string back grounds, showing in a simple context that a string background is not n ecessarily the same as a CFT. We generalize the earlier proof of quant um background independence of string theory to show that a dilaton shi ft amounts to a shift of the string coupling in the field-dependent pa rt of the quantum string action. Thus the ''dilaton theorem'', familia r for on-shell string amplitudes, holds off-shell as a consequence of an exact symmetry of the string action.