Monodromy of an Inhomogeneous Picard-Fuchs Equation

Abstract

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.

The two-loop sunrise graph with arbitrary masses

“…As a side remark, we would like to mention that inhomogeneous Picard-Fuchs equations have also been discussed recently in the context of mirror symmetry. 26 This paper is organised as follows: In Sec. II we discuss the relation between (factorised) differential equations and iterated integrals.…”

Monodromy of inhomogeneous Picard-Fuchs equations

“…As we have seen in the previous sections, the extended moduli spaces associated with lines on the octic are rather similar to that of the van Geemen lines on the quintic, studied in [2]. Let's label the lines with this calculated inhomogeneity on the positive real axis as C z,1 and C z,2 for the two choices of square-root respectively, and the associated truncated normal functions τ 1 , τ 2 .…”

“…What we study in the present paper is how these local extensions fit together into the global structure of (1.1). This is a generalization of the work [2].…”

“…As is well-known, the generic number of lines on a weighted sextic Calabi-Yau threefold is 7884. 2 Since this is divisible by 3, it is possible for the cyclic permutations of the homogeneous variables, such as…”

Open mirror symmetry for higher dimensional Calabi-Yau hypersurfaces

“…is a local coordinate around the large complex structure point. The disk partition function (2.10) satisfies the inhomogeneous Picard-Fuchs equation associated with C + − C − given by [2,12,14] (see also [30,31,32,33])…”