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Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory. (English) Zbl 0899.32006
Summary: We analyze the moduli spaces of Calabi-Yau three-folds and their associated conformally invariant nonlinear \(\sigma{}\)-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kähler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kähler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular Calabi-Yau spaces in which the spacetime metric has degenerated in certain regions. We show that the union of these domains is isomorphic to the complex structure moduli space of a single topological Calabi-Yau space – the mirror. In this way we resolve a puzzle for mirror symmetry raised by the apparent asymmetry between the Kähler and complex structure moduli spaces of a Calabi-Yau manifold. Furthermore, using mirror symmetry, we show that we can interpolate in a physically smooth manner between any two theories represented by distinct points in the Kähler moduli space, even if such points correspond to topologically distinct spaces. Spacetime topology change in string theory, therefore, is realized for the most basic operation of deformation by a truly marginal operator. Finally, this work also yields some important insights on the nature of orbifolds in string theory.

32G20 Period matrices, variation of Hodge structure; degenerations
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
32G81 Applications of deformations of analytic structures to the sciences
14J30 \(3\)-folds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
32J81 Applications of compact analytic spaces to the sciences
14J10 Families, moduli, classification: algebraic theory
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