Seiberg-Witten理论讲义

在此讲义中，作者使用许多实例，极其详细地讲述了规范场理论研究所必需的一系列原理、技巧，及其在几何与拓扑学中的应用。本书包含了大部分单连通代数曲面的Seiberg-Witten不变量的接近和独立的计算，其中仅仅使用了Witten的分解法。本书还给出了一个新的方法去剖分和粘贴Seiberg-Witten不变量，这些都通过实例进行了讲解，诸如连通和定理，爆发（blow-up）公式，以及对于Fintushel和Stern得到的消灭结果的证明。本书适合作为微分几何、代数拓扑、基础偏微分方程和泛函分析方面的高年级研究生课程教材。

Introduction
Chapter 1.Preliminaries
1.1.Bundles,connections and characteristic classes
1.1.1.Vector bundles and connections
1.1.2.Chern-Weil theory
1.2.Basic facts about elliptic equations
1.3.Clifford algebras and Dirac operators
1.3.1.Clifford algebras and their representations
1.3.2.The Spin and Spinc groups
1.3.3.Spin and spine structures
1.3.4.Dirac operators associated to spin and spinc structures
1.4.Complex differential geometry
1.4.1.Elementary complex differential geometry
1.4.2.Cauchy-Riemann operators
1.4.3.Dirac operators on almost Khler manifolds
1.5.Fredholm theory
1.5.1.Continuous families of elliptic operators
1.5.2 Genericity results
Chapter 2.The Seiberg-Witten Invariants
2.1.Seiberg-Witten monopoles
2.1.1.The Seiberg-Witten equations
2.1.2.The functional set-up
2.2.The structure of the Seiberg-Witten moduli spaces
2.2.1.The topology of the moduli spaces
2.2.2.The local structure of the moduli spaces
2.2.3.Generic smoothness
2.2.4.Orientability
2.3.The structure of the Seiberg-Witten invariants
2.3.1.The universal line bundle
2.3.2.The case b+>1
2.3.3.The case b+=1
2.3.4.Some examples
2.4.Applications
2.4.1.The Seiberg-Witten equations on cylinders
2.4.2.The Thom conjecture
2,4.3.Negative definite smooth 4-manifolds
Chapter 3.Seiberg-Witten Equations on Complex Surfaces
3.1.A short trip in complex geometry
3.1.1.Basic notions
3.1.2.Examples of complex surfaces
3.1.3.Kodaira classification of complex surfaces
3.2.Seiberg-Witten invariants of Khler surfaces
3.2.1.Seiberg-Witten equations on Kahler surfaces
3.2.2.Monopoles,vortices and divisors
3.2.3.Deformation theory
3.3.Applications
3.3.1.A nonvanishing result
3.3.2.Seiberg-Witten invariants of simply connected elliptic surfaces
3.3.3.The failure of the h-cobordism theorem in four dimensions
3.3.4.Seiberg-Witten equations on symplectic 4-manifolds
Chapter 4.Gluing Techniques
4.1.Elliptic equations on manifolds with cylindrical ends
4.1.1.Manifolds with cylindrical ends
4.1.2.The Atiyah-Patodi-Singer index theorem
4.1.3.Eta invariants and spectral flows
4.1.4.The Lockhart-McOwen theory
4.1.5.Abstract linear gluing results
4.1.6.Examples
4.2.Finite energy monopoles
4.2.1.Regularity
4.2.2.Three-dimensional monopoles
4.2.3.Asymptotic behavior.Part Ⅰ
4.2.4.Asymptotic behavior.Part Ⅱ
4.2.5.Proofs of some technical results
4.3.Moduli spaces of finite energy monopoles:Local aspects
4.3.1.Functional set-up
4.3.2.The Kuranishi picture
4.3.3.Virtual dimensions
4.3.4.Reducible finite energy monopoles
4.4.Moduli spaces of finite energy monopoles:Global aspects
4.4.1.Genericity results
4.4.2.Compactness properties
4.4.3.Orientability issues
4.5.Cutting and pasting of monopoles
4.5.1.Some basic gluing constructions
4.5.2.Gluing monopoles:Local theory
4.5.3.The local surjectivity of the gluing construction
4.5.4.Gluing monopoles:Global theory
4.6.Applications
4.6.1.Vanishing results
4.6.2.Blow-up formula
Epilogue
Bibliography
Index