The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. (MN-44) ：The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. (MN-44) ( Mathematical Notes )

Publication subTitle ：The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. (MN-44)

Publication series ：Mathematical Notes

Author： Morgan John W.;;;  

Publisher： Princeton University Press‎

Publication year： 2014

E-ISBN: 9781400865161

P-ISBN(Paperback): 9780691025971

Subject： O189.3 analytical topology

Keyword： 数学

Language： ENG

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Description

The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants.


The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space. The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.

Chapter

Cover

Title

Copyright

Contents

1 Introduction

2 Clifford Algebras and Spin Groups

2.1 The Clifford Algebras

2.2 The groups Pin(V) and Spin(V)

2.3 Splitting of the Clifford Algebra

2.4 The complexification of the Cl(V)

2.5 The Complex Spin Representation

2.6 The Group Spin^c(V)

3 Spin Bundles and the Dirac Operator

3.1 Spin Bundles and Clifford Bundles

3.2 Connections and Curvature

3.3 The Dirac Operator

3.4 The Case of Complex Manifolds

4 The Seiberg-Witten Moduli Space

4.1 The Equations

4.2 Space of Configurations

4.3 Group of Changes of Gauge

4.4 The Action

4.5 The Quotient Space

4.6 The Elliptic Complex

5 Curvature Identities and Bounds

5.1 Curvature Identities

5.2 A Priori bounds

5.3 The Compactness of the Moduli Space

6 The Seiberg-Witten Invariant

6.1 The Statement

6.2 The Parametrized Moduli Space

6.3 Reducible Solutions

6.4 Compactness of the Perturbed Moduli Space

6.5 Variation of the Metric and Self-dual Two-form

6.6 Orientability of the Moduli Space

6.7 The Case when b^+2(X) > 1

6.8 An Involution in the Theory

6.9 The Case when b^+2(X) = 1

7 Invariants of Kähler Surfaces

7.1 The Equations over a Kähler Manifold

7.2 Holomorphic Description of the Moduli Space

7.3 Evaluation for Kähler Surfaces

7.4 Computation for Kähler Surfaces

7.5 Final Remarks

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