Higher-derivative couplings and torsional Riemann curvature

Using the most general higher-derivative field redefinitions for the closed spacetime manifolds, we show that the tree-level couplings of the metric, B-field and dilaton at orders α′ and α′ that have been recently found by the T-duality, can be written in a particular scheme in terms of the torsional Riemann curvature R and the torsion tensor H. The couplings at order α′ have structures R, HR, H, and the couplings at order α′ have only structures R, HR. Replacing R with the ordinary Riemann curvature, the couplings in the structure HR reproduce the couplings found in the literature by the S-matrix method.