**University Lecture Series**

Volume: 6;
1994;
209 pp;
Softcover

MSC: Primary 53;
Secondary 58; 57

Print ISBN: 978-0-8218-0332-5

Product Code: ULECT/6

List Price: $30.00

AMS Member Price: $24.00

MAA Member Price: $27.00

**Electronic ISBN: 978-0-8218-3439-8
Product Code: ULECT/6.E**

List Price: $30.00

AMS Member Price: $24.00

MAA Member Price: $27.00

# \(J\)-Holomorphic Curves and Quantum Cohomology

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*Dusa McDuff; Dietmar Salamon*

\(J\)-holomorphic curves revolutionized the study of symplectic geometry when Gromov first introduced them in 1985. Through quantum cohomology, these curves are now linked to many of the most exciting new ideas in mathematical physics. This book presents the first coherent and full account of the theory of \(J\)-holomorphic curves, the details of which are presently scattered in various research papers. The first half of the book is an expository account of the field, explaining the main technical aspects. McDuff and Salamon give complete proofs of Gromov's compactness theorem for spheres and of the existence of the Gromov-Witten invariants. The second half of the book focuses on the definition of quantum cohomology. The authors establish that this multiplication exists, and give a new proof of the Ruan-Tian result that is associative on appropriate manifolds. They then describe the Givental-Kim calculation of the quantum cohomology of flag manifolds, leading to quantum Chern classes and Witten's calculation for Grassmannians, which relates to the Verlinde algebra. The Dubrovin connection, Gromov-Witten potential on quantum cohomology, and curve counting formulas are also discussed. The book closes with an outline of connections to Floer theory.

#### Readership

Advanced graduate students, research mathematicians, and mathematical physicists.

#### Reviews & Endorsements

All in all it is rewarding to read this book, as many delicate points are first explained in easy-to-understand terms before the authors dive into the proofs … this book will certainly remain a standard for background on quantum cohomology for many years to come.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## $J$-Holomorphic Curves and Quantum Cohomology

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- 1 Introduction 110 free
- 2 Local Behaviour 1322
- 3 Moduli Spaces and Transversality 2332
- 4 Compactness 4150
- 5 Compactification of Moduli Spaces 5968
- 6 Evaluation Maps and Transversality 7180
- 7 Gromov-Witten Invariants 8998
- 8 Quantum Cohomology 107116
- 9 Novikov Rings and Calabi-Yau Manifolds 141150
- 10 Floer Homology 153162
- A: Gluing 167176
- B: Elliptic Regularity 181190
- Bibliography 197206
- Indexes 203212
- Index of Notations 209218 free
- Back Cover Back Cover1219