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Tables of Contents for Seiberg-Witten Gauge Theory
Chapter/Section Title
Page #
Page Count
Introduction
1
8
Seiberg--Witten on four-manifolds
9
48
Preliminary Notions
10
19
Clifford Algebras and Dirac Operators
10
3
Spin and Spinc Structures
13
1
Spinor Bundles
14
1
Topology of the gauge group
15
1
Symplectic and kahler Manifolds
15
4
The Index theorem
19
2
Equivariant cohomology
21
3
Sobolev norms
24
2
Fredholm properties
26
1
Exercises
27
2
The Functional and the Equations
29
12
The Equations
29
1
The Gauge Group
29
1
The Seiberg--Witten Functional and the Variational Problem
30
4
Analytic properties of the functional
34
5
Exercises
39
2
Invariants of 4-manifolds
41
16
The Moduli Space
41
8
The Invariants
49
2
Finiteness
51
1
A Cobordism Argument
52
5
Seiberg--Witten on three-manifolds
57
58
Three-manifolds
58
2
A three-manifolds invariant
60
17
Dimensional reduction
60
2
The moduli space and the invariant
62
4
Cobordism and wall crossing formulae
66
3
Casson invariant and Alexander polynomial
69
8
Seiberg--Witten Floer homology
77
38
The Chern-Simons-Dirac functional
78
2
Hessian and relative index
80
3
Flow lines: asymptotics
83
3
Flow lines: moduli spaces
86
4
Homology
90
4
The cobordism argument
94
4
Equivariant Floer homology and wall crossing
98
5
Exact triangles
103
3
The relation to instanton Floer homology
106
2
Summary
108
1
Exercises
108
7
Topology and Geometry
115
38
Computing Seiberg--Witten invariants
116
19
Connected Sum theorem
116
1
The blowup formula
116
2
Kahler Surfaces
118
4
Symplectic Manifolds
122
5
Pseudo--holomorphic curves
127
3
Beyond the symplectic world
130
2
Algebraic Surfaces
132
3
Topology of embedded surfaces
135
9
The Thom Conjecture
136
4
Contact structures
140
3
Three-manifolds: Thurston norm
143
1
Further applications
144
9
Exercises
145
8
Seiberg--Witten and Physics
153
48
Mathai-Quillen formalism
154
25
The finite dimensional case
154
5
The infinite dimensional case
159
4
Euler numbers in Seiberg--Witten theory
163
7
N = 2 symmetry and the Euler characteristic
170
5
Quantum Field Theory and Floer homology
175
2
Exercises
177
2
Seiberg--Witten and Donaldson theory
179
22
The Physics way: S-duality
181
7
The Mathematics way
188
13
Appendix: a bibliographical guide
201