**Mathematical Surveys and Monographs**

Volume: 68;
1999;
469 pp;
Softcover

MSC: Primary 14;
Secondary 81

Print ISBN: 978-0-8218-2127-5

Product Code: SURV/68.S

List Price: $60.00

AMS Member Price: $48.00

MAA member Price: $54.00

**Electronic ISBN: 978-1-4704-1295-1
Product Code: SURV/68.S.E**

List Price: $60.00

AMS Member Price: $48.00

MAA member Price: $54.00

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#### Supplemental Materials

# Mirror Symmetry and Algebraic Geometry

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*David A. Cox; Sheldon Katz*

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 & Endorsements

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

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

# Table of Contents

## Mirror Symmetry and Algebraic Geometry

- Contents ix10
- Preface xiii14
- Notation xix20
- Chapter 1. Introduction 124
- Chapter 2. The Quintic Threefold 1538
- Chapter 3. Toric Geometry 3154
- Chapter 4. Mirror Symmetry Constructions 5376
- Chapter 5. Hodge Theory and Yukawa Couplings 7396
- Chapter 6. Moduli Spaces 113136
- Chapter 7. Gromov-Witten Invariants 167190
- Chapter 8. Quantum Cohomology 217240
- Chapter 9. Localization 275298
- Chapter 10. Quantum Differential Equations 301324
- Chapter 11. The Mirror Theorem 331354
- Chapter 12. Conclusion 397420
- Appendix A. Singular Varieties 407430
- Appendix B. Physical Theories 411434
- Bibiliography 437460
- Index 453476