Lectures on Seiberg-Witten Invariants (Lecture Notes in Mathematics)

The author gives a brief introduction to the use of Seiberg-Witten equations in chapter 1, along with a review of the background needed from the theory of vector bundles, differential geometry, and algebraic topology needed to read the book. All of this background is pretty standard, although the appearance of spin structures may not be as familiar to the mathematician-reader, but completely familiar to the physicist reader. Detailed proofs of the main results are not given, but reference to these are quoted. Also, the theory of characteristic classes is outlined only briefly so no insight is given as to why they work so well. In particular, the reason for the vanishing of the second Stiefel-Whitney class as a precondition for the manifold having a spin structure is not given.

In chapter 2, the author goes into the spin geometry of 4-manifolds in more detail. After discussing the role of quaternions in this regard, spin structures are defined. A spin structure on a manifold M, via its cocycle condition, give two complex vector bundles of rank two over M. The complexified tangent bundle can thus be represented in terms of these vector bundles, which are themselves quaternionic line bundles over M. The author also defines spin(c) structures, and shows how, using an almost complex structure, to obtain a canonical spin(c) structure on a complex manifold of complex dimension two. The spin(c) structure also allows a construction of the "virtual vector bundles" W+, W-, and L, for manifolds that do not have a spin structure. These bundles play a central role in the book. Clifford algebra becomes meaningful on the direct sum W of W+ and W-, and spin connections can be defined on W. In particular given a unitary connection on a complex line bundle L over a spin manifold M, one can obtain a connection on the tensor product of W and L. When M is not a spin manifold, this is still possible but one must use the "square" L^2 of L. One can then define the Dirac operator over the sections of this tensor product, which the author does and extends it to one with coefficients in a general vector bundle. The author then discusses, but does not prove, the Atiyah-Singer index theorem and the Hirzebruch signature theorem. These theorems, the author emphasizes, are proved in the context of linear partial differential equations, and give invariants of 4-manifolds.

This sets up the discussion in chapter 3, which deals with the problem of how to find invariants of 4-manifolds if one works in the context of nonlinear partial differential equations. Those familiar with the Donaldson theory, which was done using the (nonlinear!) Yang-Mills equations, will understand the difficulties of this approach. The strategy of the nonlinear approach as outlined by the author is to show that the solution set of a nonlinear PDE is compact and a finite-dimensional compact manifold. The solution set depends on the Riemannian metric, but its cobordism class does not, and this may give a topological invariant. The fact that it is defined in terms of a PDE might give a way of distinguishing smooth structures.

The Seiberg-Witten theory is one method for doing this. The Seiberg-Witten equations are nonlinear, but the nonlinearity is "soft" enough that it can be dealt with. They arise in the context of oriented 4-dimensional Riemannian manifolds with a spin(c) structure and a positive spinor bundle W+ tensored with L. A connection on L^2 and a section of this spinor bundle are chosen to satisfy these equations, which involve the self-dual part of the connection. One also needs to work with the "perturbed" Seiberg Witten equations, where a self-dual two-form is added. The moduli space of the solutions to the perturbed Seiberg-Witten equations is shown to form a compact finite-dimensional manifold. The proof follows essentially from the Weitzenbock formula, the Sobolev embedding theorem, and Rellich's theorem. Sard's theorem shows that the moduli space is smooth and the Fredholm theory shows it is oriented. The Seiberg-Witten invariants are associated to virtual complex line bundles over the 4-manifold, and when the dimension of the self-dual harmonic two-forms is greater than or equal to 2, and the dimension of the moduli space is even. Their definition does involve the Riemannian metric, but changing this metric only alters the moduli space by a cobordism. It is proved that oriented Riemannian manifolds with positive scalar curvature have vanishing Seiberg-Witten invariants. Kahler surfaces are shown to have positive Seiberg-Witten invariants, and the author proves that there is a compact topological manifold with infinitely many distinct smooth structures. Unfortunately though, an explicit example of one of these is not given. Such an example may be very important from the standpoint of physics, for the behavior of dynamical systems or quantum field theories might be very different for different smooth structures.