Pfaffian Calabi–Yau threefolds and mirror symmetry

Abstract

The aim of this article is to report on recent progress in un- derstanding mirror symmetry for some non-complete intersection Calabi- Yau threefolds. We first construct four new smooth non-complete in- tersection Calabi-Yau threefolds with h 1,1 = 1, whose existence was previously conjectured by C. van Enckevort and D. van Straten in (19). We then compute the period integrals of candidate mirror families of F. Tonoli's degree 13 Calabi-Yau threefold and three of the new Calabi- Yau threefolds. The Picard-Fuchs equations coincide with the expected Calabi-Yau equations listed in (18, 19). Some of the mirror families turn out to have two maximally unipotent monodromy points.