Mirror Symmetry and Algebraic Geometry

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American Mathematical Soc., 1999 - Mathematics - 469 pages
Mathematicians wanting to get into the field ... will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry. --Bulletin of the LMS The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics. --Mathematical Reviews Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is a completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.
 

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Contents

The Quintic Threefold
15
Toric Geometry
31
Mirror Symmetry Constructions
53
Hodge Theory and Yukawa Couplings
73
Moduli Spaces
113
GromovWitten Invariants
167
Quantum Cohomology
217
Localization
275
Conclusion
397
Appendix A Singular Varieties
407
Nonlinear Sigma Models
419
Conformal Field Theories
423
LandauGinzburg Models
426
Gauged Linear Sigma Models
428
Topological Quantum Field Theories
430
Bibliography
437

Quantum Differential Equations
301
The Mirror Theorem
331

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About the author (1999)

David A. Cox, Amherst College, MA

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