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 is currently the director of the Max-Planck-Institut für Mathematik in Bonn, Germany. 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.

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

Table of Contents

• Introduction: What is quantum cohomology?
• Introduction to Frobenius manifolds
• Frobenius manifolds and isomonodromic deformations
• Frobenius manifolds and moduli spaces of curves
• Operads, graphs, and perturbation series
• Stable maps, stacks, and Chow groups
• Algebraic geometric introduction to the gravitational quantum cohomology
• Bibliography
• Subject index