A Strange Family of Calabi-Yau 3-folds

TL;DR

This paper investigates a specific family of Calabi-Yau 3-folds, computing their Picard-Fuchs equations, monodromy, and instanton numbers, revealing unusual properties like vanishing instantons and a positive Euler characteristic for the mirror.

Contribution

It provides new calculations of Picard-Fuchs equations, monodromy actions, and instanton numbers for a particular Calabi-Yau family, including the construction of a new rigid Calabi-Yau 3-fold.

Findings

Strange vanishing of instanton numbers in odd degrees.

Monodromy predicts a positive Euler characteristic for the mirror.

Construction of a new rigid Calabi-Yau 3-fold from a degenerate fiber.

Abstract

We study the predictions of mirror symmetry for the 1-parameter family of Calabi-Yau 3-folds $\tilde{X}$ with hodge numbers $h^{11}=31,h^{21}=1$ constructed in \cite{BN}. We calculate the Picard-Fuchs differential equation associated to this family, and use it to predict the instanton numbers on the hypothetical mirror. These exhibit a strange vanishing in odd degrees. We also calculate the monodromy action on $H^3(\tilde{X},\QQ)$ and find that it strangely predicts a positive Euler characteristic for its mirror. From a degenerate fiber of our family we construct a new rigid Calabi-Yau 3-fold. In an appendix we prove the expansion of the conifold period conjectured in \cite{ES} to hold for all 1-parameter families.