**Colloquium Publications**

Volume: 47;
1999;
303 pp;
Hardcover

MSC: Primary 14; 58;

Print ISBN: 978-0-8218-1917-3

Product Code: COLL/47

List Price: $68.00

Individual Member Price: $54.40

**Electronic ISBN: 978-1-4704-3193-8
Product Code: COLL/47.E**

List Price: $68.00

Individual Member Price: $54.40

# Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces

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*Yuri I. Manin*

Yuri Manin received the Bolyai Prize of the Hungarian Academy of
Sciences for this title. Only five people have received this award in more than
100 years!

This is the first monograph dedicated to the systematic
exposition of the whole variety of topics related to quantum
cohomology. The subject first originated in theoretical physics
(quantum string theory) and has continued to develop extensively over the
last decade.

The author's approach to quantum cohomology is based on the notion
of the Frobenius manifold. The first part of the book is devoted to
this notion and its extensive interconnections with algebraic
formalism of operads, differential equations, perturbations, and
geometry. In the second part of the book, the author describes the
construction of quantum cohomology and reviews the algebraic geometry
mechanisms involved in this construction (intersection and deformation
theory of Deligne-Artin and Mumford stacks).

Yuri Manin was once the director of the Max-Planck-Institut für
Mathematik in Bonn, Germany, one of the most
prestigious mathematics institutions in the world. He has authored and
coauthored 10 monographs and almost 200 research articles in algebraic
geometry, number theory, mathematical physics, history of culture, and
psycholinguistics. Manin's books, such as Cubic Forms: Algebra,
Geometry, and Arithmetic (1974), A Course in Mathematical
Logic (1977), Gauge Field Theory and Complex Geometry
(1988), Elementary Particles: Mathematics, Physics and
Philosophy (1989, with I. Yu. Kobzarev), Topics in
Non-commutative Geometry (1991), and Methods of Homological
Algebra
(1996, with S. I. Gelfand), secured for him solid recognition as an
excellent expositor. Undoubtedly the present book will serve
mathematicians for many years to come.

#### Table of Contents

# Table of Contents

## Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces

- Cover Cover11
- Title page v6
- Dedication vii8
- Contents ix10
- Preface xi12
- Introduction: What is quantum cohomology? 116
- Introduction to Frobenius manifolds 1732
- Frobenius manifolds and isomonodromic deformations 4964
- Frobenius manifolds and moduli spaces of curves 8398
- Operads, graphs, and perturbation series 175190
- Stable maps, stacks, and Chow groups 201216
- Algebraic geometric introduction to the gravitational quantum cohomology 245260
- Bibliography 285300
- Subject index 299314
- Back Cover Back Cover1321

#### Readership

Researchers and graduate students working in algebraic geometry, differential geometry, theory of integrable systems, and mathematical physics.

#### Reviews

A good introduction to the theory of Frobenius manifolds and quantum cohomology … of interest to a broad mathematical audience.

-- Mathematical Reviews

A beautiful survey of the theory of Gromov-Witten invariants … An especially attractive feature … is the large number of examples of Frobenius manifolds … as well as background essays on isomonodromy, operads and their generalizations and intersection theory on stacks, rendering the book accessible to a wider audience.

-- Bulletin of the AMS

The book is certainly one which should be in every mathematical library. I would also recommend it to any mathematician (from graduate student to professor) interested in trying to learn about these important and fascinating subjects.

-- Bulletin of the London Mathematical Society