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Generalized Calabi-Yau manifolds and the mirror of a rigid manifold. (English) Zbl 0899.32011
Summary: The \(Z\) manifold is a Calabi-Yau manifold with \(b_{21}=0\). At first sight it seems to provide a counter-example to the mirror hypothesis since its mirror would have \(b_{11}=0\) and hence could not be Kähler. However, by identifying the \(Z\) manifold with the Gepner model \(1^9\) we are able to ascribe a geometrical interpretation to the mirror, \(\widetilde Z\), as a certain seven-dimensional manifold. The mirror manifold \(\widetilde Z\) 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 \(\widetilde Z\) and check that this reproduces the known results for the Yukawa couplings and metric appropriate to the Kähler 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.

32J17 Compact complex \(3\)-folds
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32J18 Compact complex \(n\)-folds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
11F06 Structure of modular groups and generalizations; arithmetic groups
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