The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds

John W. Morgan

Princeton University Press, Sep 8, 2014 - Mathematics - 130 pages

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.

Contents

• 1 Introduction: 1
• 2 Clifford Algebras and Spin Groups: 5
• 3 Spin Bundles and the Dirac Operator: 23
• 4 The SeibergWitten Moduli Space: 55
• 5 Curvature Identities and Bounds: 69
• 6 The SeibergWitten Invariant: 87
• 7 Invariants of Kähler Surfaces: 109
• Bibliography: 127

About the author

John W. Morgan is Professor of Mathematics at Columbia University.

Bibliographic information

• Title: The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds; Volume 44 of Mathematical Notes
• Author: John W. Morgan
• Edition: reprint
• Publisher: Princeton University Press, 2014
• ISBN: 1400865166, 9781400865161
• Length: 130 pages
• Subjects: Mathematics / Topology