Abstract
Starting from the σ-model conformal invariance conditions we find all four dimensional metrics when the dilaton field satisfies the condition ▿ μ ▿ νφ =0. Such configurations have been previously employed in a flat D dimensional target space and naturally come out as solutions consistent with conformal invariance of the supersymmetric σ-model beyond the two loop order. The four dimensional metrics emerging out are either of the Robertson-Walker type or a special kind of a generalized pp-wave metric. Of these solutions only the Robertson-Walker metric can be related to a noncritical string theory.
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