F-theory Vacua and $alpha$-Corrections

In this work we analyze F-theory and Type IIB orientifold compactifications to study $alpha $-corrections to the four-dimensional, $mathcal{N} = 1$ effective actions. In particular, we obtain corrections to the Kahlermoduli space metric and its complex structure for generic dimension originating from eight-derivative corrections to eleven-dimensional supergravity. We propose a completion of the $G^ 2 R^3$ and $( abla G)^2R^2$-sector in eleven-dimensions relevant in Calabi--Yau fourfold reductions. We suggest that the three-dimensional, $mathcal{N}=2$ Kahler coordinates may be expressed as topological integrals depending on the first, second, and third Chern-forms of the divisors of the internal Calabi--Yau fourfold. The divisor integral Ansatz for the Kahler potential and Kahler coordinates may be lifted to four-dimensional, $mathcal{N} = 1$ F-theory vacua. We identify a novel correction to the Kahler potential and coordinates at order $ alpha^2$, which is leading compared to other known corrections in the literature. At weak string coupling the correction arises from the intersection of $D7$-branes and $O7$-planes with base divisors and the volume of self-intersection curves of divisors in the base. In the presence of the conjectured novel $alpha$-correction resulting from the divisor interpretation the no-scale structure may be broken. Furthermore, we propose a model independent scenario to achieve non-supersymmetric AdS vacua for Calabi-Yau orientifold backgrounds with negative Euler-characteristic.