Hori-Vafa mirror periods, Picard-Fuchs equations, and Berglund-Hübsch-Krawitz duality

Abstract

This paper discusses the overlap of the Hori-Vafa formulation of mirror symmetry with some other constructions. We focus on compact Calabi-Yau hypersurfaces \( {\mathcal{M}_{\,G}} = \left\{ {G = 0} \right\} \) in weighted complex projective spaces. The Hori-Vafa formalism relates a family \( \left\{ {{M_G} \in WCP_{{Q_1}, \ldots, {Q_m}}^{m - 1}\left[ s \right]\left| {\sum\nolimits_{i = 1}^m {{Q_i} = s} } \right.} \right\} \) Ginzburg mirror theory. A technique suggested by Hori and Vafa allows the Picard-Fuchs equations satisfied by the corresponding mirror periods to be determined. Some examples in which the variety \( {\mathcal{M}_G} \) is crepantly resolved are considered. The resulting Picard-Fuchs equations agree with those found elsewhere working in the Batyrev-Borisov framework. When G is an invertible nondegenerate quasihomogeneous polynomial, the Chiodo-Ruan geometrical interpretation of Berglund-Hübsch-Krawitz duality can be used to associate a particular complex structure for \( {\mathcal{M}_G} \) with a particular Kähler structure for the mirror \( {\tilde{\mathcal{M}}_G} \). We make this association for such G when the ambient space of \( {\mathcal{M}_G} \) is CP 2, CP 3, and CP 4. Finally, we probe some of the resulting mirror Kähler structures by determining corresponding Picard-Fuchs equations.