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 the first 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, Kähler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.

Features:

• Numerous examples worked out in detail
• An appendix on mathematical physics
• An exposition of the algebraic theory of Gromov-Witten invariants and quantum cohomology
• A proof of the mirror theorem for the quintic threefold

Readership

Graduate students, research mathematicians interested in the relations between mathematics and physics; algebraic geometers, symplectic geometers, and theoretical physicists.

Reviews

"As the authors observed, the greatest obstacle facing a mathematician who wants to learn about mirror symmetry is knowing where to start. Another problem is the scattering of many mathematical ideas throughout the physics literature, which is difficult for mathematicians to read. The present book seems to be a successful attempt to collect all these ideas. It could also be used as a starting reference for mathematicians interested in learning about mirror symmetry. It is especially very helpful for the reader that the authors have summarized in Appendix B some of the key points of physical theories mentioned in the book."

-- Bulletin of the AMS

"Mathematicians wanting to get into the field will find it an essential book. They 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. We can be grateful to the authors for a book which not only summarizes current knowledge, but also points to the future."

-- Bulletin of the LMS

Featured Review:

"The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics. It can serve as an introduction both for a mathematician who wants to learn about mirror symmetry, and for a physicist who knows about mirror symmetry and wants to understand the mathematics behind it. It even contains enough details to also be useful for the mathematician who actively wants to do research in the subject."

-- Mathematical Reviews

"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

Table of Contents

• Introduction
• The quintic threefold
• Toric geometry
• Mirror symmetry constructions
• Hodge theory and Yukawa couplings
• Moduli spaces
• Gromov-Witten invariants
• Quantum cohomology
• Localization
• Quantum differential equations
• The mirror theorem
• Conclusion
• Singular varieties
• Physical theories
• Bibliography
• Index