Monodromy of an Inhomogeneous Picard-Fuchs Equation

TL;DR

This paper investigates the monodromy and global behavior of a normal function linked to a family of lines on the mirror quintic, using inhomogeneous Picard-Fuchs equations and series expansions.

Contribution

It provides explicit calculations of monodromies and series expansions around key points, revealing the irrational nature of the normal function's limit at large complex structure.

Findings

Monodromies are explicitly calculated and confirmed to be integral.

Series expansions are obtained around large complex structure, conifold, and open string discriminant.

The normal function's limit at large complex structure is an irrational number involving the di-logarithm.

Abstract

The global behaviour of the normal function associated with van Geemen's family of lines on the mirror quintic is studied. Based on the associated inhomogeneous Picard-Fuchs equation, the series expansions around large complex structure, conifold, and around the open string discriminant are obtained. The monodromies are explicitly calculated from this data and checked to be integral. The limiting value of the normal function at large complex structure is an irrational number expressible in terms of the di-logarithm.