Strings on a curved spacetime 
*

*Suppose we are interested in in string on a specific metric G, is it necessary to include a Dilaton field on back ground in order to preserve the Weyl invariance? suppose the spacetime is not empty, for example consider a cosmological constant term too. 

*In Polchinski's book it is shown that the Einstein's equation can be emerged from string theory as a consequence of the Weyl invariance but, it is true only to the first order of "Alpha-Prime". Does it mean that in the framework of string theory, Einstein Eq. is credited only in weak gravitational interactions?
 A: *

*The world sheet Weyl invariance requires/implies that the equations of motion for the spacetime fields are satisfied. These equations of motion constrain the dilaton (a field in the spacetime) as well. However, there exist solutions (i.e. also allowed CFTs) that have a constant dilaton and there also exist solutions that have a non-constant dilaton. Whether a particular background metric with a constant or a non-constant dilaton is a solution (i.e. defines a CFT without a Weyl anomaly) could only be decided if you said what background you are actually talking about.

*The $\alpha'$ corrections become important in very strong gravitational fields and/or very high energies. In terms of curvature, for example, $\alpha' R^2$ becomes comparable to $R$ (Riemann or Ricci tensor) when $R\sim 1/\alpha'$ i.e. the curvature radius is comparable to $l_{\rm string}$, an ultramicroscopic distance scale. This has clearly no measurable impact on astronomical objects. However, it does become important when one studies the character of spacetime near the Planck/string length. Let me emphasize that general relativity implies a whole tower of nonlinear (relativistic) corrections to Newton's gravitational law and string theory reproduces this whole tower! The terms suppressed by $\alpha'$ are much smaller and smaller in a different sense than the nonlinear corrections resulting from GR are smaller than the leading terms in Newton's theory. The theory resulting from string theory obeys all the principles of general relativity – just to be sure. For example, large black holes (strong gravitational fields in GR) are described by the same equations in string theory as in "ordinary" general relativity; the corrections from string theory are negligible because the curvature radius is much longer than the string or Planck scale.
