On the background independence of string field theory: II. Analysis of on-shell S-matrix elements

Given a solution ψcl of the classical equations of motion in either closed or open string field theory based on a certain conformal field theory (CFT) we may define a shifted string field ψ^=ψ−ψcl. The string field theory action expressed interms of the shifted field is then given by S^(ψ^)=S(ψ^+ψcl−S(ψcl), where S(ψ) is the original action of the string field theory. In S(ψ^) the coefficient (Q^B) of the quadratic term ψ^ in has the property that (Q^B)2=0. It was shown in a previous paper that in the limit when the background ψcl is weak, Q^B can be identified with the BRST charge of a new conformal field theory (CFT"), which is obtained by perturbing the orginal conformal field theory (CTF) by a marginal operator. In this paper we compute various on-shell S-matrix elements using the action S^ (ψ^) and show that they are identical to the on-shell S-matrix in the string based on the new conformal field theory (CTF"). This in turn shows that S^(ψ^) describes string theory formulated around the background of this new conformal field theory