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

SIGMA 8 (2012), 056, 10 pages      arXiv:1206.1787
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

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.

pdf (289 kb)   tex (34 kb)


  1. Albano A., Katz S., Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture, Trans. Amer. Math. Soc. 324 (1991), 353-368.
  2. Candelas P., de la Ossa X.C., Green P.S., Parkes L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21-74.
  3. Dimofte T., Gukov S., Lenells J., Zagier D., Exact results for perturbative Chern-Simons theory with complex gauge group, Commun. Number Theory Phys. 3 (2009), 363-443, arXiv:0903.2472.
  4. Donagi R., Markman E., Cubics, integrable systems, and Calabi-Yau threefolds, in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, 199-221, alg-geom/9408004.
  5. Doran C.F., Morgan J.W., Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi-Yau threefolds, in Mirror Symmetry. V, AMS/IP Stud. Adv. Math., Vol. 38, Amer. Math. Soc., Providence, RI, 2006, 517-537.
  6. Green M., Griffiths P., Kerr M., Néron models and limits of Abel-Jacobi mappings, Compos. Math. 146 (2010), 288-366.
  7. Musta ta A., Degree 1 curves in the Dwork pencil and the mirror quintic, Math. Ann., to appear, math.AG/0311252.
  8. Walcher J., On the arithmetics of D-brane superpotentials, arXiv:1201.6427.
  9. Walcher J., Opening mirror symmetry on the quintic, Comm. Math. Phys. 276 (2007), 671-689, hep-th/0605162.
  10. Zagier D., The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 3-65.

Previous article  Next article   Contents of Volume 8 (2012)