Communications in Number Theory and Physics

Volume 8 (2014)

Number 1

Monodromy of inhomogeneous Picard-Fuchs equations

Pages: 1 – 40



Robert A. Jefferson (Institute for Theoretical Physics, Universiteit van Amsterdam, The Netherlands)

Johannes Walcher (Department of Physics and Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)


We study low-degree curves on one-parameter Calabi-Yau hypersurfaces, and their contribution to the space-time superpotential in a superstring compactification with D-branes. We identify all lines that are invariant under at least one permutation of the homogeneous variables, and calculate the inhomogeneous Picard-Fuchs equation. The irrational large volume expansions satisfy the recently discovered algebraic integrality. The bulk of our work is a careful study of the topological integrality of monodromy under navigation around the complex structure moduli space. This is a powerful method to recover the single undetermined integration constant that is itself also of arithmetic significance. The examples feature a variety of residue fields, both abelian and non-abelian extensions of the rationals, thereby providing a glimpse of the arithmetic D-brane landscape.

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