The Euler characteristic correction to the Kaehler potential - revisited

We confirm the leading $\alpha'^3$ correction to the 4d, $\mathcal N = 1$ K\"{a}hler potential of type IIB orientifold compactifications, proportional to the Euler characteristic of the Calabi-Yau threefold (BBHL correction). We present the explicit solution for the $\alpha'^3$-modified internal background metric in terms of the non-harmonic part of the third Chern form of the leading order Calabi-Yau manifold. The corrected internal manifold is almost Calabi-Yau and admits an $SU(3)$ structure with non-vanishing torsion. We also find that the full ten-dimensional Einstein frame background metric is multiplied by a non-trivial Weyl factor. Performing a Kaluza-Klein reduction on the modified background we derive the $\alpha'^3$-corrected kinetic terms for the dilaton and the K\"{a}hler deformations of the internal Calabi-Yau threefold for arbitrary $h^{1,1}$. We analyze these kinetic terms in the 4d, $\mathcal N = 2$ un-orientifolded theory, confirming the expected correction to the K\"ahler moduli space prepotential, as well as in the 4d, $\mathcal N = 1$ orientifolded theory, thus determining the corrections to the K\"ahler potential and K\"ahler coordinates.