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Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties. (English) Zbl 0899.32007
Summary: The moduli dependence of \((2,2)\) superstring compactifications based on Calabi-Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg orbifolds with \(c=9\) whose potential is a sum of \(A\)-type singularities. Here we consider the generalization to arbitrary quasi-homogeneous singularities at \(c=9\). We use mirror symmetry to derive the dependence of the models on the complexified Kähler moduli and check the expansions of some topological correlation functions against explicit genus zero and genus one instanton calculations. As an important application we give examples of how non-algebraic (“twisted”) deformations can be mapped to algebraic ones, hence allowing us to study the full moduli space. We also study how moduli spaces can be nested in each other, thus enabling a (singular) transition from one theory to another. Following the recent work of Greene, Morrison and Strominger we show that this corresponds to black hole condensation in type II string theories compactified on Calabi-Yau manifolds.

32G20 Period matrices, variation of Hodge structure; degenerations
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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