T-duality and hints of generalized geometry in string $\alpha'$ corrections

We examine the structure of higher-derivative string corrections under a cosmological reduction and make connection to generalized geometry and T-duality. We observe that, while the curvature $R^\mu{}_{\nu\rho\sigma}(\Omega_+)$ of the generalized connection with torsion, $\Omega_+=\Omega+\frac{1}{2}H$, is an important component in forming T-duality invariants, it is necessarily incomplete by itself. We revisit the tree-level $\alpha'R^2$ corrections to the bosonic and heterotic string in the language of generalized geometry and explicitly demonstrate that additional $H$-field couplings are needed to restore T-duality invariance. We also comment on the structure of the T-duality completion of tree-level $\alpha'^3R^4$ in the type II string.