
SIGMA 8 (2012), 056, 10 pages arXiv:1206.1787
http://dx.doi.org/10.3842/SIGMA.2012.056
Contribution to the Special Issue “Mirror Symmetry and Related Topics”
Monodromy of an Inhomogeneous PicardFuchs 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 PicardFuchs 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 dilogarithm.
Key words:
algebraic cycles; mirror symmetry; quintic threefold.
pdf (289 kb)
tex (34 kb)
References
 Albano A., Katz S., Lines on the Fermat quintic threefold and the
infinitesimal generalized Hodge conjecture, Trans. Amer. Math.
Soc. 324 (1991), 353368.
 Candelas P., de la Ossa X.C., Green P.S., Parkes L., A pair of CalabiYau
manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 2174.
 Dimofte T., Gukov S., Lenells J., Zagier D., Exact results for perturbative
ChernSimons theory with complex gauge group, Commun. Number
Theory Phys. 3 (2009), 363443, arXiv:0903.2472.
 Donagi R., Markman E., Cubics, integrable systems, and CalabiYau
threefolds, in Proceedings of the Hirzebruch 65 Conference on Algebraic
Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9,
BarIlan Univ., Ramat Gan, 1996, 199221, alggeom/9408004.
 Doran C.F., Morgan J.W., Mirror symmetry and integral variations of Hodge
structure underlying oneparameter families of CalabiYau threefolds, in
Mirror Symmetry. V, AMS/IP Stud. Adv. Math., Vol. 38, Amer. Math.
Soc., Providence, RI, 2006, 517537.
 Green M., Griffiths P., Kerr M., Néron models and limits of AbelJacobi
mappings, Compos. Math. 146 (2010), 288366.
 Musta ta A., Degree 1 curves in the Dwork pencil and the mirror quintic,
Math. Ann., to appear, math.AG/0311252.
 Walcher J., On the arithmetics of Dbrane superpotentials, arXiv:1201.6427.
 Walcher J., Opening mirror symmetry on the quintic, Comm. Math. Phys.
276 (2007), 671689, hepth/0605162.
 Zagier D., The dilogarithm function, in Frontiers in Number Theory, Physics,
and Geometry. II, Springer, Berlin, 2007, 365.

