The Pfaffian Calabi-Yau, its Mirror, and their link to the Grassmannian   G(2,7)
  The rank 4 locus of a general skew-symmetric 7x7 matrix gives the pfaffian variety in P^20 which is not defined as a complete intersection. Intersecting this with a general P^6 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) in P^20 with a P^13. 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. 
Einar Andreas Rodland