GENERALIZED CALABI-YAU MANIFOLDS AND THE MIRROR OF A RIGID MANIFOLD *

Abstract The Z manifold is a Calabi-Yau manifold with b21 = 0. At first sight it seems to provide a counter-example to the mirror hypothesis since its mirror would have b11 = 0 and hence could not be Kahler. However, by identifying the Z manifold with the Gepner model 19 we are able to ascribe a geometrical interpretation to the mirror, Z , as a certain seven-dimensional manifold. The mirror manifold Z is a representative of a class of generalized Calabi-Yau manifolds, which we describe, that can be realized as manifolds of dimension five and seven. Despite their dimension these generalized Calabi-Yau manifolds correspond to superconformal theories with c = 9 and so are perfectly good for compactifying the heterotic string to the four dimensions of space-time. As a check of mirror symmetry we compute the structure of the space of complex structures of the mirror Z and check that this reproduces the known results for the Yukawa couplings and metric appropriate to the Kahler class parameters on the Z orbifold together with their instanton corrections. In addition to reproducing known results we can calculate the periods of the manifold to arbitrary order in the blowing up parameters. This provides a means of calculating the Yukawa couplings and metric as functions also to arbitrary order in the blowing up parameters, which is difficult to do by traditional methods.