On (α′)2 corrections to the D-brane action for non-geodesic world-volume embeddings

In hep-th/9903210 (curvature) terms of the effective D-brane action were derived to lowest order in the string coupling. Their results are correct up to ambiguous terms which involve the second fundamental form of the D-brane. We compute five point string amplitudes on the disk. We compare the subleading order in α of the string amplitudes with the proposed lagrangian of hep-th/9903210 supplemented by the ambiguous terms. The comparison determines the complete form of the gravitational terms in the effective D-brane action to order O(α′ ). Our results are valid for arbitrary ambient geometries and world-volume embeddings.