The gravitino mass is expected not to be much smaller than the Planck scale for a large fraction of vacua in flux compactifications. There is no continuous parameter to tune even by hand, and it seems that the gravitino mass can be small only as a result of accidental cancellation among period integrals weighted by integer-valued flux quanta. DeWolfe et.al. (2005) proposed to pay close attention to vacua where the Hodge decomposition is possible within a number field, so that the precise cancellation takes place as a result of algebra. We focus on a subclass of those vacua-those with complex multiplications-and explore more on the idea in this article. It turns out, in Type IIB compactifications, that those vacua admit non-trivial supersymmetric flux configurations if and only if the reflex field of the Weil intermediate Jacobian is isomorphic to the quadratic imaginary field generated by the axidilaton vacuum expectation value. We also found that flux statistics is highly enriched on such vacua, as F-term conditions become linearly dependent.

It is an important question in string compactification whether complex structure moduli stabilization inevitably ends up with a vacuum expectation value of the superpotential W of the order of the Planck scale cubed. Any thoughts on volume stabilization and inflation in string theory, as well as on phenomenology of supersymmetric Standard Models, will be affected by the answer to this question. In this work, we follow an idea for making W ≃ 0 where the internal manifold has a vacuum complex structure with arithmetic characterization, and address Calabi-Yau fourfold compactification of F-theory. The moduli space of K3 × K3 orbifolds contain infinitely many such vacua. Arithmetic conditions for a W = 0 flux are worked out, and then all the K3 moduli have supersymmetric mass. Possible gauge groups, matter representations and discrete symmetries are studied for the case of Z 2 -orbifolds.

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