On Mirror Symmetry for Calabi-Yau Fourfolds with Three-Form Cohomology

We study the action of mirror symmetry on two-dimensional N=(2,2) effective theories obtained by compactifying Type IIA string theory on Calabi-Yau fourfolds. Our focus is on fourfold geometries with non-trivial three-form cohomology. The couplings of the massless zero-modes arising by expanding in these forms depend both on the complex structure deformations and the Kahler structure deformations of the Calabi-Yau fourfold. We argue that two holomorphic functions of the deformation moduli capture this information. These are exchanged under mirror symmetry, which allows us to derive them at the large complex structure and large volume point. We discuss the application of the resulting explicit expression to F-theory compactifications and their weak string coupling limit. In the latter orientifold settings we demonstrate compatibility with mirror symmetry of Calabi-Yau threefolds at large complex structure. As a byproduct we find an interesting relation of no-scale like conditions on Kahler potentials to the existence of chiral and twisted-chiral descriptions in two dimensions.