Softcover ISBN:  9780821821275 
Product Code:  SURV/68.S 
List Price:  $64.00 
MAA Member Price:  $57.60 
AMS Member Price:  $51.20 
Electronic ISBN:  9781470412951 
Product Code:  SURV/68.S.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $48.00 

Book DetailsMathematical Surveys and MonographsVolume: 68; 1999; 469 ppMSC: Primary 14; Secondary 81;
Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in fourdimensional 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, CalabiYau manifolds, quantum cohomology, GromovWitten invariants, and the mirror theorem.
Features: Numerous examples worked out in detail
 An appendix on mathematical physics
 An exposition of the algebraic theory of GromovWitten invariants and quantum cohomology
 A proof of the mirror theorem for the quintic threefold
ReadershipGraduate students, research mathematicians interested in the relations between mathematics and physics; algebraic geometers, symplectic geometers, and theoretical physicists.

Table of Contents

Chapters

1. Introduction

2. The quintic threefold

3. Toric geometry

4. Mirror symmetry constructions

5. Hodge theory and Yukawa couplings

6. Moduli spaces

7. GromovWitten invariants

8. Quantum cohomology

9. Localization

10. Quantum differential equations

11. The mirror theorem

12. Conclusion


Additional Material

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


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Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in fourdimensional 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, CalabiYau manifolds, quantum cohomology, GromovWitten invariants, and the mirror theorem.
Features:
 Numerous examples worked out in detail
 An appendix on mathematical physics
 An exposition of the algebraic theory of GromovWitten invariants and quantum cohomology
 A proof of the mirror theorem for the quintic threefold
Graduate students, research mathematicians interested in the relations between mathematics and physics; algebraic geometers, symplectic geometers, and theoretical physicists.

Chapters

1. Introduction

2. The quintic threefold

3. Toric geometry

4. Mirror symmetry constructions

5. Hodge theory and Yukawa couplings

6. Moduli spaces

7. GromovWitten invariants

8. Quantum cohomology

9. Localization

10. Quantum differential equations

11. The mirror theorem

12. Conclusion

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