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

## TL;DR

This paper explores mirror symmetry in Calabi-Yau fourfolds with three-form cohomology, analyzing how zero-mode couplings depend on moduli and are exchanged under mirror symmetry, with explicit descriptions at special points.

## Contribution

It provides a detailed analysis of three-form cohomology couplings in Calabi-Yau fourfolds and demonstrates their exchange via mirror symmetry, including explicit functional descriptions at key moduli points.

## Key findings

- Couplings depend on both complex and Kähler structure deformations.
- Mirror symmetry exchanges two holomorphic functions determining these couplings.
- Explicit functions are described at large volume and large complex structure limits.

## Abstract

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.