In search of almost generic Calabi-Yau 3-folds

We call a projective Calabi-Yau (CY) 3-fold almost generic if it has only isolated nodes as singularities and the homology classes of all of the exceptional curves in an analytic small resolution are non-trivial but torsion. Such a Calabi-Yau supports a topologically non-trivial flat B-field and the corresponding A-model topological string partition function encodes a torsion refinement of the Gopakumar-Vafa invariants of the smooth deformation. Our goal in this paper is to find new examples of almost generic CY 3-folds, using both conifold transitions as well as the integral structure of the periods of the mirrors. In this way we explicitly construct two quintic CY 3-folds with $\mathbb{Z}_2$-torsion, two octics with $\mathbb{Z}_3$-torsion and deduce the existence of a complete intersection $X_{(6,6)}\subset\mathbb{P}^5_{1,1,2,2,3,3}$ with $\mathbb{Z}_5$-torsion. Via mirror symmetry, the examples give new geometric interpretations to several AESZ Calabi-Yau operators. The mirror periods of the almost generic $X_{(6,6)}$ with non-trivial B-field topology are annihilated by an irrational Picard-Fuchs operator. We describe how the usual integral structure of the periods has to be modified and in all of the cases we calculate the monodromies around the singular points to verify integrality. Additional points of maximally unipotent monodromy in the moduli spaces lead us to find several more examples of smooth or almost generic CY 3-folds and to conjecture new twisted derived equivalences. We integrate the holomorphic anomaly equations and extract the torsion refined Gopakumar-Vafa invariants up to varying genus. For our construction of the almost generic octic CY 3-folds, we also give a short introduction to the subject of hypermatrices and hyperdeterminants.