Topology, geometry and physics: background for the Witten conjecture. II (Q2483804)

This is the second part of the paper and begins with the section 6. The first part was published in the same journal [J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010)]. The author discusses the equivariant localization in order to describe the finite dimensional analog of Witten's partition function. He presents the generalized Duistermaat-Heckman theorem, and the equivariant localization theorem. Next he is concerned with duality and Seiberg-Witten. The notion of duality symmetries has its roots in classical electromagnetic theory and the symmetry might interchange strong and weak coupling. This notion is studied by using the Clifford algebra and the spin structures on a manifold. The coupled Dirac operator acts on the sections in certain spinor bundles \(\mathcal{S(L)}\) and the Seiberg-Witten equations fulfilled by this operator are the Dirac equation and the curvature equation. To any solution \((A,\psi)\) of the Seiberg-Witten map one can associate the fundamental elliptic complex and its cohomology groups have nice interpretations. The author presents an outline of a construction with no details and scarcely a word of explanation for the Witten conjecture asserting that, for certain four-manifolds, the zero-dimensional Seiberg-Witten invariants contain all the information available in all the Donaldson invariants.