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Topology, geometry and physics: background for the Witten conjecture. II. (English) Zbl 1080.58014

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.

MSC:

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
81T13 Yang-Mills and other gauge theories in quantum field theory
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
57R57 Applications of global analysis to structures on manifolds

Citations:

Zbl 1079.58010
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