On $({\alpha }^{\prime }{)}^2$ corrections to the D-brane action for non-geodesic world-volume embeddings

Abstract

In hep-th/9903210 (curvature)${}^2$ 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 ${\alpha }^{\prime }$ 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 ${\calO}(α^{' 2})$. Our results are valid for arbitrary ambient geometries and world-volume embeddings.

34 pages, LaTeX, 4 Postscript figures, expanded introduction and conclusions, typos corrected, references added, final version to appear in JHEP09(2001)005

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