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

TL;DR

This paper computes five-point string amplitudes to determine the complete form of second-order curvature corrections to the D-brane action, resolving ambiguities and applicable to arbitrary geometries.

Contribution

It provides the full gravitational correction terms at order ^2 in the D-brane effective action, clarifying previous ambiguities involving the second fundamental form.

Findings

Derived the complete '^2 correction terms for D-brane action.

Resolved ambiguities related to the second fundamental form.

Validated results for arbitrary ambient geometries and 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 $\alpha'$ 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 ${\calO}(\alpha^{' 2})$. Our results are valid for arbitrary ambient geometries and world-volume embeddings.