## Calabi-yau Manifolds: A Bestiary For PhysicistsCalabi-Yau spaces are complex spaces with a vanishing first Chern class, or equivalently, with trivial canonical bundle (canonical class). They are used to construct possibly realistic (super)string models and are thus being studied vigorously in the recent physics literature.In the main part of the Book, collected and reviewed are relevant results on (1) several major techniques of constructing such spaces and (2) computation of physically relevant quantities such as massless field spectra and their Yukawa interactions. Issues of (3) stringy corrections and (4) moduli space and its geometry are still in the stage of rapid and continuing development, whence there is more emphasis on open problems here. Also is included a preliminary discussion of the conjectured universal moduli space and related open problems. Finally, several detailed models and sample computations are included throughout the Book to exemplify the techniques and the general discussion.The Book also contains a Lexicon (28 pages) of 150 assorted terms, key-words and main results and theorems, well suited for a handy reference. Although cross-referenced with the main part of the Book, the Lexicon can also be used independently.The level of mathematics is guided and developed between that of the popular Physics Reports of Eguchi, Gilkey and Hanson and the book Superstrings (Vol. 2) by Green, Schwarz and Witten on one end and Principles of Algebraic Geometry of Griffiths and Harris on the other.This is the first systematic exposition in book form of the material on Calabi-Yau spaces, related mathematics and the physics application, otherwise scattered through research articles in journals and conference proceedings. |

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### Contents

1 | |

Part I Constructions | 25 |

Part II Cohomology | 153 |

Part III Changelings | 257 |

Part IV Concordance | 295 |

LEXICON | 315 |

343 | |

359 | |

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### Common terms and phrases

algebraic ample anti-canonical bundle blow-up Calabi-Yau 3-folds Calabi-Yau manifolds Calabi-Yau spaces canonical bundle Chern class choice coefficients cohomology groups compactification complete intersection complex manifold complex projective spaces complex structure computation cone configuration conifold consider constraints constructions corresponding cr-model curves defining equations defining polynomials del Pezzo surfaces denote diagram dimension divisor dual elements embedding space Euler characteristic example Fano 3-folds fibre field theory finite flag spaces formula holonomy homology hypersurface integral intersections in products invariant isomorphic Kahler class Koszul Lefschetz hyperplane theorem Lett line bundle linear massless matrix metric monomials nodes non-trivial non-zero Nucl obtained parameter space Pezzo surfaces Phys polynomial polynomial deformations precisely quadric quintic quotient relation represented respectively restriction Ricci-flat sections simple singular points small resolution smooth spacetime spanned spectral sequence String Theory submanifold subspace superfields Superstring supersymmetry symmetry tangent bundle tensor vanish vector bundle Yukawa couplings