Modular Amplitudes and Flux-Superpotentials on elliptic Calabi-Yau fourfolds

Understand

We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces.

• In particular, we describe a general procedure to fix integral periods.
• Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds.
• The full monodromy group contains a subgroup that acts as PSL(2,Z) on the K\"ahler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants.