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. "This book provides an excellent introduction to the recently discovered Seilberg-Witten invariants for smooth closed oriented 4-mainifolds." ― Mathematical Reviews Beginning with the groundbreaking work of Donaldson in about 1980 it became clear that gauge-theoretic invariants of principal bundles and connections were an important tool in the study of smooth four-dimensional manifolds. Donaldson showed the importance of the moduli space of antiself-dual connections. The next fifteen years saw an explosion of work in this area leading to computations of Donaldson polynomial invariants for a wide class of four-dimensional manifolds, especially algebraic surfaces. John W. Morgan is Professor of Mathematics at Columbia University. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds By John W. Morgan PRINCETON UNIVERSITY PRESS Copyright © 1996 Princeton University Press All rights reserved. ISBN: 978-0-691-02597-1 Contents 1 Introduction, 1, 2 Clifford Algebras and Spin Groups, 5, 3 Spin Bundles and the pirac Operator, 23, 4 The Seiberg-Witten Moduli Space, 55, 5 Curvature Identities and Bounds, 69, 6 The Seiberg-Witten Invariant, 87, 7 Invariants of Kähler Surfaces, 109, Bibliography, 127, CHAPTER 1 Introduction Beginning with the groundbreaking work of Donaldson in about 1980 it became clear that gauge-theoretic invariants of principal bundles and connections were an important tool in the study of smooth four-dimensional manifolds. Donaldson showed the importance of the moduli space of anti-self-dual connections. The next fifteen years saw an explosion of work in this area leading to computations of Donaldson polynomial invariants for a wide class of four-dimensional manifolds, especially algebraic surfaces. These computations yielded many powerful topological consequences including, for example, the diffeomorphism classification of elliptic surfaces. For some of the results obtained by using these techniques see [1], [6], and [2]. Last fall, motivated by new work in quantum field theory, Seiberg and Witten [9], introduced a different gauge-theoretic invariant which they claimed should be closely related to Donaldson's invariants. Indeed they gave an explicit formula for the relationship of their invariant to Donaldson's. While the link claimed by Seiberg-Witten between their invariants and Donaldson's has not yet been established mathematically, it is clearly true and can be shown to hold in all computed examples. Nevertheless, one can forget this supposed link and work directly with the new invariants as a substitute for the anti-self-dual invariants. This has been the approach during the last year or so. It was clear from the beginning that the new invariants would be easier to work with since they involved principal bundles with structure group the circle instead of the non-abelian groups such as SU (2) which arise in Donaldson theory. The surprise was that such simple invariants could capture the subtlety that Donaldson's invariants revealed. But in short order, Witten [17] and then Taubes, Kronheimer, and Mrowka showed that indeed these new invariants did capture these subtleties and that they were easier to compute, at least in many cases. They did this by explicitly solving the Seiberg-Witten equations over Kähler surfaces. (See [3] for one account of the results for Kähler surfaces.) There followed in quick succession a series of remarkable theorems, each extending in a different way partial results from Donaldson's anti-self-dual theory. In fact, conjectures which seeme