Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 5

Calabi–Yau manifolds realizing symplectically rigid monodromy tuples

Pages: 1271 – 1359



Charles F. Doran (Department of Mathematics, University of Alberta, Edmonton, AB, Canada)

Andreas Malmendier (Department of Mathematics and Statistics, Utah State University, Logan, Ut., U.S.A.)


We define an iterative construction that produces a family of elliptically fibered Calabi–Yau $n$-folds with section from a family of elliptic Calabi–Yau varieties of one dimension lower. Parallel to the geometric construction, we iteratively obtain for each family with a point of maximal unipotent monodromy, normalized to be at $t = 0$, its Picard–Fuchs operator and a closed-form expression for the period holomorphic at $t = 0$, through a generalization of the classical Euler transform for hypergeometric functions. In particular, our construction yields one-parameter families of elliptically fibered Calabi–Yau manifolds with section whose Picard–Fuchs operators realize all symplectically rigid Calabi–Yau differential operators with three regular singular points classified by Bogner and Reiter, but also non-rigid operators with four singular points.

Published 12 February 2020