Mirror Symmetry, N=1 Superpotentials and Tensionless Strings on Calabi-Yau Four-Folds

Abstract

We study aspects of Calabi-Yau four-folds as compactification manifolds of F-theory, using mirror symmetry of toric hypersurfaces. Correlation functions of the topological field theory are determined directly in terms of a natural ring structure of divisors and the period integrals, and subsequently used to extract invariants of moduli spaces of rational curves subject to certain conditions. We then turn to the discussion of physical properties of the space-time theories, for a number of examples which are dual to $E_8\times E_8$ heterotic N=1 theories. Non-critical strings of various kinds, with low tension for special values of the moduli, lead to interesting physical effects. We give a complete classification of those divisors in toric manifolds that contribute to the non-perturbative four-dimensional superpotential; the physical singularities associated to it are related to the apppearance of tensionless strings. In some cases non-perturbative effects generate an everywhere non-zero quantum tension leading to a combination of a conventional field theory with light strings hiding at a low energy scale related to supersymmetry breaking.

68 pages, harvmac.tex, references added, minor corrections, Comments on phase transitions in sect. 5. added, two figures added

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• Five-Brane Superpotentials and Heterotic/F-theory Duality
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• Moduli Stabilization in Asymptotic Flux Compactifications
• Duality in Heterotic Vacua With Four Supercharges
• Landau-Ginzburg Vacua of String, M- and F-Theory at c=12
• Relative periods and open-string integer invariants for a compact Calabi-Yau hypersurface
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• The Geometry of D-Brane Superpotentials
• F theory Vacua in Four Dimensions and Toric Threefolds
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• Codimension-Three Bundle Singularities in F-Theory
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• Special Geometry and Mirror Symmetry for Open String Backgrounds with N=1 Supersymmetry
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• Holomorphic Couplings In Non-Perturbative String Compactifications
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• Mirror Symmetry and the Web of Landau-Ginzburg String Vacua
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