Monodromy of Inhomogeneous Picard-Fuchs Equations

TL;DR

This paper investigates the monodromy of inhomogeneous Picard-Fuchs equations related to low-degree curves on Calabi-Yau hypersurfaces, revealing arithmetic properties and integrality conditions relevant for string compactifications with D-branes.

Contribution

It provides a detailed analysis of the monodromy and arithmetic properties of inhomogeneous Picard-Fuchs equations in the context of Calabi-Yau hypersurfaces with D-branes, including new methods for recovering integration constants.

Findings

Identification of invariant lines under permutations

Calculation of inhomogeneous Picard-Fuchs equations

Demonstration of algebraic integrality in large volume expansions

Abstract

We study low-degree curves on one-parameter Calabi-Yau hypersurfaces, and their contribution to the space-time superpotential in a superstring compactification with D-branes. We identify all lines that are invariant under at least one permutation of the homogeneous variables, and calculate the inhomogeneous Picard-Fuchs equation. The irrational large volume expansions satisfy the recently discovered algebraic integrality. The bulk of our work is a careful study of the topological integrality of monodromy under navigation around the complex structure moduli space. This is a powerful method to recover the single undetermined integration constant that is itself also of arithmetic significance. The examples feature a variety of residue fields, both abelian and non-abelian extensions of the rationals, thereby providing a glimpse of the arithmetic D-brane landscape.