Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 056, 10 pages      arXiv:1206.1787      https://doi.org/10.3842/SIGMA.2012.056
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

Monodromy of an Inhomogeneous Picard-Fuchs Equation

Guillaume Laporte a and Johannes Walcher a, b
a) Department of Physics, McGill University, Montréal, Québec, Canada
b) Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada

Received June 08, 2012, in final form August 20, 2012; Published online August 22, 2012

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.

Key words: algebraic cycles; mirror symmetry; quintic threefold.

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