Some mirror partners with complex multiplication

Abstract

In theoretical physics rational conformal field theories are considered as particularly interesting class of conformal field theories. Let (X ,Y ) be a pair of families of Calabi–Yau 3-manifolds, which are mirror partners, X be a fiber of X and Y be a fiber of Y . In [7] Gukov and Vafa explain that X and Y yield a rational conformal field theory, if and only if both fibers have complex multiplication (CM). A family of Calabi–Yau manifolds over a Shimura variety has a dense set of CM fibers, if the variation of Hodge structures (VHS) is related to the Shimura datum of the base space in a natural way as in [10]. At present several of such families of Calabi–Yau 3-manifolds over Shimura varieties are known [2, 6, 10, 11, 12]. In general, one does not know a Shimura subvariety of the base space on the mirror side.1 Here we give new examples of pairs of families of Calabi–Yau 3-manifolds over Shimura varieties, which are subfamilies of mirror partners. We start with a family C3 of degree 3 covers of P1 with six different ramification points over an open Shimura subvariety M3 ⊂ (P1)3. By using the Fermat curve of degree 3 and C3, one can construct a family of K3 surfaces with a non-symplectic involution over M3 as described in [10, Section 8]. The Borcea–Voisin construction yields a family W of Calabi–Yau 3-manifolds, which has a dense set of CM fibers. Garbagnati and van Geemen [5] have given a more general method to construct K3 surfaces, which yields the same K3 surfaces for the fibers of C3. The latter method allows one to