The Theory of Gauge Fields in Four Dimensions

Front Cover
American Mathematical Soc., 1985 - Mathematics - 101 pages
Lawson's expository lectures, presented at a CBMS Regional Conference held in Santa Barbara in August 1983, provide an in-depth examination of the recent work of Simon Donaldson, and is of special interest to both geometric topologists and differential geometers. This work has excited particular interest, in light of Mike Freedman's recent profound results: the complete classification, in the simply connected case, of compact topological 4-manifolds. Arguing from deep results in gauge field theory, Donaldson has proved the nonexistence of differentiable structures on certain compact 4-manifolds. Together with Freedman's results, Donaldson's work implies the existence of exotic differentiable structures in $\mathbb R^4$-a wonderful example of the results of one mathematical discipline yielding startling consequences in another. The lectures are aimed at mature mathematicians with some training in both geometry and topology, but they do not assume any expert knowledge. In addition to a close examination of Donaldson's arguments, Lawson also presents, as background material, the foundation work in gauge theory (Uhlenbeck, Taubes, Atiyah, Hitchin, Singer, et al.) which underlies Donaldson's work.
 

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Contents

Introduction
1
Simplyconnected 4manifolds
2
Differentiable 4manifolds
4
Exotica
5
An introduction to Donaldsons proof
10
The Geometry of Connections
17
Connections
19
Riemannian connections
20
The Selfdual YangMills Equations
39
Selfduality
40
The fundamental elliptic complex
43
Solutions on S
45
The Moduli Space
47
Reducible selfdual connections
50
Perturbations
51
The orientability of the moduli space
53

Spiconnections
21
Change of connections
22
Automorphisms the gauge group
23
Sobolev completions
25
Reductions
28
The action of on QPE
31
Equivalence classes of connections
33
Fundamental Results of K Uhlenbeck
59
The Taubes Existence Theorem
71
Final Arguments
85
The Sobolev Embedding Theorems
91
References
99
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