0 Ja n 19 98 The Pfaffian Calabi – Yau , its Mirror , and their link to the Grassmannian G ( 2 , 7 ) Einar

The rank 4 locus of a general skew-symmetric 7 × 7 matrix gives the pfaffian variety in P 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 (Kähler structure related) moduli space, and, thus, represent different parts of the same conformal field theory moduli space. 1 The Pfaffian Variety Let E be a rank 7 vector space. For N ∈ E ∧ E non-zero, we look at the locus of ∧3 N = 0 ∈ ∧6 E: the rank 4 locus of N if viewed as a skew-symmetric matrix. This defines a degree 14 variety of codimension 3 in P(E ∧E) ∼= P20. As N is skewsymmetric, this variety is defined by the pfaffians, ie. square roots of the determinants, of the 6×6 diagonal minors of the matrix. Intersecting this with a general 6-plane in P(E ∧E) ∼= P20 will give a 3-dimensional Calabi–Yau. In coordinates xi on P 6, the matrix N can be written NA = ∑6 i=0 xiAi where the Ai ∈ E ∧E are skew-symmetric matrices spanning the P6. Denote this variety XA ⊂ P 6. The pfaffian variety in P20 is smooth away from the rank 2 locus which has dimension 10. Hence, by Bertini’s theorem, the variety XA is smooth for general A. Definition 1 Let NA = ∑6 i=0 xiAi where Ai are 7×7 skew-symmetric matrices. Let XA ⊂ P 6 denote the zero-locus of the pfaffians of the 6 × 6 diagonal minors of NA: ie., the rank 4 locus of the matrix. ∗Dept. of Mathematics, University of Oslo, Box 1053 Blindern, 0316 Oslo, Norway; E-mail: [email protected]; WWW: http://www.math.uio.no/∼einara.