# Three-form periods on Calabi-Yau fourfolds: Toric hypersurfaces and F-theory applications

## TL;DR

This paper develops mathematical tools to compute the moduli dependence of three-form periods on Calabi-Yau fourfolds, crucial for string compactifications, by relating them to one-form periods of associated Riemann surfaces.

## Contribution

It introduces a method to derive the complete moduli dependence of three-form periods on toric hypersurface Calabi-Yau fourfolds, linking them to one-form periods of Riemann surfaces.

## Key findings

- Derived integral expressions for three-form periods.
- Connected three-form periods to one-form periods of Riemann surfaces.
- Applied methods to explicit F-theory compactification examples.

## Abstract

The study of the geometry of Calabi-Yau fourfolds is relevant for compactifications of string theory, M-theory, and F-theory to various dimensions. This work introduces the mathematical machinery to derive the complete moduli dependence of the periods of non-trivial three-forms for fourfolds realized as hypersurfaces in toric ambient spaces. It sets the stage to determine Picard-Fuchs-type differential equations and integral expressions for these forms. The key tool is the observation that non-trivial three-forms on hypersurfaces in toric ambient spaces always stem from divisors that are build out of toric resolution trees fibered over Riemann surfaces. The three-form periods are then non-trivially related to the one-form periods of these surfaces. In general, the three-form periods are known to vary holomorphically over the complex structure moduli space and play an important role in the effective actions arising in fourfold compactifications. We discuss two explicit example fourfolds for F-theory compactifications in which the three-form periods determine axion decay constants.