  We review the Kaluza-Klein reduction of Type IIA string theory on Calabi-Yau
fourfolds and apply mirror symmetry to the resulting two-dimensional $
\mathcal{N}=(2,2) $ effective theories. In the course of the reduction we focus
especially on non-trivial three-form cohomology on these fourfolds and
investigate the couplings of the corresponding massless zero-modes. These show
a dependence on both complex structure as well as K\"ahler structure
deformations and we provide evidence that they are determined by two
holomorphic functions that get exchanged via mirror symmetry. Application of
the mirror map enables us to give an explicit description of these functions at
the large volume and large complex structure point of the moduli space.
