Higher-derivative couplings and torsional Riemann curvature

Using the most general higher-derivative field redefinition for the closed spacetime manifolds, we show that the tree-level couplings of the metric, $B$-field and dilaton at orders ${\alpha }^{\mathrm{\prime }2}$ and ${\alpha }^{\mathrm{\prime }3}$ that have been recently found by the T-duality, can be written in a particular scheme in terms of the torsional Riemann curvature $\mathcal{R}$ and the torsion tensor $H$. The couplings at order ${\alpha }^{\mathrm{\prime }2}$ have structures ${\mathcal{R}}^3,H^2{\mathcal{R}}^2$, $H^6$, and the couplings at order ${\alpha }^{\mathrm{\prime }3}$ have only structures ${\mathcal{R}}^4$, $H^2{\mathcal{R}}^3$. Replacing $\mathcal{R}$ with the ordinary Riemann curvature, the couplings in the structure $H^2{\mathcal{R}}^3$ reproduce the couplings found in the literature by the S-matrix method.