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

TL;DR

This paper explores the modular properties of amplitudes and flux superpotentials on elliptic Calabi-Yau fourfolds, revealing monodromy actions, anomaly equations, and flux effects in F-theory compactifications.

Contribution

It introduces a general method to fix integral periods, analyzes monodromy and modularity, and studies flux superpotentials and K"ahler potentials in elliptic Calabi-Yau fourfolds.

Findings

Identified a subgroup acting as PSL(2,Z) on the fiber's K"ahler modulus.

Derived holomorphic anomaly equations relating amplitudes to quasi-modular forms.

Verified attractor behavior at the conifold for specific flux choices.

Abstract

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. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order K\"ahler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.