The Euler characteristic correction to the Kahler potential - revisited

We confirm the leading alpha'(3) correction to the 4d, N = 1 Kahler 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 Kahler deformations of the internal Calabi-Yau threefold for arbitrary h(1,1). We analyze these kinetic terms in the 4d, N = 2 un-orientifolded theory, confirming the expected correction to the Kahler moduli space prepotential, as well as in the 4d, N = 1 orientifolded theory, thus determining the corrections to the Kahler potential and Kahler coordinates.