这是第一本系统阐述量子上同调各种相关论题的专著。该学科最初起源于理论物理学（量子弦理论），并在过去十年中继续广泛发展。特别地，本书为研究镜像猜想提供了不可或缺的数学背景，镜像猜想是物理学家最近发现的量子弦理论的对偶性之一。
作者对量子上同调的研究基于Frobeni,Js流形的概念。本书的第一部分将全面阐述这一概念及其与操作（operad）、微分方程、扰动和几何的代数形式的广泛联系。在本书的第二部分，作者描述了量子上同调的构造，并回顾了这种构造中涉及的代数几何机制（Deligne-Artin和Mumford叠层的相交和形变理论）。
作者Yuri I.Manin因为本书获得了匈牙利科学院的Bolyai奖。一百多年来，只有五人获此殊荣！
本书可供代数几何、微分几何、可积系统理论和数学物理领域的研究人员和研究生使用。

Preface
Chapter 0.Introduction: What Is Quantum Cohomology?
Chapter I.Introduction to Frobenius Manifolds
1.Definition of Frobenius manifolds and the structure connection
2.Identity, Euler field, and the extended structure connection
3.Semisimple Frobenius manifolds
4.Examples
5.Weak Frobenius manifolds
Chapter II.Frobenius Manifolds and Isomonodromic Deformations
1.The second structure connection
2.Isomonodromic deformations
3.Semisimple Frobenius manifolds as special solutions to the Schlesinger equations
4.Quantum cohomology of projective spaces
5.Dimension three and Painleve VI
Chapter III.Frobenius Manifolds and Moduli Spaces of Curves
1.Formal Frobenius manifolds and Commoo-algebras
2.Pointed curves and their graphs
3.Moduli spaces of genus 0
4.Formal Frobenius manifolds and Cohomological Field Theories
5.Gromov-Witten invariants and quantum cohomology: Axiomatic theory
6.Formal Frobenius manifolds of rank one and Weil-Petersson volumes of moduli spaces
7.Tensor product of analytic Frobenius manifolds
8.K.Salto's frameworks and singularities
9.Maurer-Cartan equations and Gerstenhaber-Batalin-Vilkovyski algebras
10.From dGBV-algebras to Frobenius manifolds
Chapter IV.Operads, Graphs, and Perturbation Series
1.Classical linear opera(is
2.Operads and graphs
3.Sums over graphs
4.Generating functions
Chapter V.Stable Maps, Stacks, and Chow Groups
1.Prestable curves and prestable maps
2.Flat families of curves and maps
3.Groupoids and moduli groupoids
4.Morphisms of groupoids and moduli groupoids
5.Stacks
6.Homological Chow groups of schemes
7.Homological Chow groups of DM--stacks
8.Operational Chow groups of schemes and DM-stacks
Chapter VI.Algebraic Geometric introduction to the Gravitational Quantum Cohomology
1.Virtual fundamental classes
2.Gravitational descendants and Virasoro constraints
3.Correlators and forgetful maps
4.Correlators and boundary maps
5.The simplest Virasoro constraints
6.Generalized correlators
7.Generating functions on the large phase space
Bibliography
Subject Index