Pfaffian Calabi-Yau threefolds and mirror symmetry

The aim of this paper is to report on recent progress in understanding mirror symmetry for some non-complete intersection Calabi- Yau threefolds. We first construct four new smooth non-complete intersection 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.