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

The F manifold is a Calabi-Yau manifold with b21 = 0. At first sight i t seems to provide a counter-example to the mirror hypothesis since it s mirror would have b11 = 0 and hence could not be Yahler. However, by identifying the F manifold with the Gepner model 1(9) we are able to ascribe a geometrical interpretation to the mirror, F, as a certain se ven-dimensional manifold. The mirror manifold F 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 thei r dimension these generalized Calabi-Yau manifolds correspond to super conformal theories with c = 9 and so are perfectly good for compactify ing the heterotic string to the four dimensions of space-time. As a ch eck of mirror symmetry we compute the structure of the space of comple x structures of the mirror F and check that this reproduces the known results for the Yukawa couplings and metric appropriate to the Kahler class parameters on the F orbifold together with their instanton corre ctions. In addition to reproducing known results we can calculate the periods of the manifold to arbitrary order in the blowing up parameter s. This provides a means of calculating the Yukawa couplings and metri c as functions also to arbitrary order in the blowing up parameters, w hich is difficult to do by traditional methods.