Lectures on Seiberg-Witten Invariants

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Overview

This book gives a streamlined introduction to the theory of Seiberg-Witten invariants suitable for second-year graduate students. These invariants can be used to prove that there are many compact topological four-manifolds which have more than one smooth structure, and that others have no smooth structure at all. This topic provides an excellent example of how global analysis techniques, which have been developed to study nonlinear partial differential equations, can be applied to the solution of interesting geometrical problems. In the second edition, some material has been expanded for better comprehension.

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Product Details

  • ISBN-13: 9783540614555
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 1/28/2008
  • Series: Lecture Notes in Mathematics
  • Pages: 124
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.26 (d)

Table of Contents

1 Preliminaries 1
1.1 Introduction 1
1.2 What is a vector bundle? 4
1.3 What is a connection? 9
1.4 The curvature of a connection 16
1.5 Characteristic classes 19
1.6 The Thom form 24
1.7 The universal bundle 27
1.8 Classification of connections 34
1.9 Hodge theory 40
2 Spin geometry on four-manifolds 45
2.1 Euclidean geometry and the spin groups 45
2.2 What is a spin structure? 49
2.3 Almost complex and spin[superscript c] structures 53
2.4 Clifford algebras 54
2.5 The spin connection 58
2.6 The Dirac operator 63
2.7 The Atiyah-Singer Index Theorem 67
3 Global analysis of the Seiberg-Witten equations 73
3.1 The Seiberg-Witten equations 73
3.2 The moduli space 75
3.3 Compactness of the moduli space 79
3.4 Transversality 82
3.5 The intersection form 91
3.6 Donaldson's Theorem 97
3.7 Seiberg-Witten invariants 98
3.8 Dirac operators on Kahler surfaces 101
3.9 Invariants of kahler surfaces 110
Bibliography 117
Index 120
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