The Pfaffian Calabi-Yau, its Mirror, and their link to the Grassmannian G(2,7)

Abstract

The rank 4 locus of a general skew-symmetric 7 × 7 matrix gives the Pfaffian variety in P20 which is not defined as a complete intersection. Intersecting this with a general P6 gives a Calabi–Yau manifold. An orbifold construction seems to give the 1-parameter mirror-family of this. However, corresponding to two points in the 1-parameter family of complex structures, both with maximally unipotent monodromy, are two different mirror-maps: one corresponding to the general Pfaffian section, the other to a general intersection of G(2,7) ⊂ P20 with a P13. Apparently, the Pfaffian and G(2,7) sections constitute different parts of the A-model (Kahler structure related) moduli space, and, thus, represent different parts of the same conformal field theory moduli space.