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Topology, geometry and physics: background for the Witten conjecture. I. (English) Zbl 1079.58010

The paper surveys some background for the Witten conjecture that relates Donaldson invariants with Seiberg-Witten invariants. Starting with an elementary discussion of connections, curvature, etc. and their physical interpretations, the paper reviews the definition of Donaldson invariants. In the case of zero dimensional anti-self-dual moduli spaces, Donaldson invariant is viewed as the Euler number of an infinite rank vector bundle in a familiar way. Following M. F. Atiyah and L. Jeffrey [J. Geom. Phys. 7, 119–136 (1990; Zbl 0721.58056)], the Euler number in turn is interpreted as a path integral using the Mathai-Quillen formalism. The path integral has been derived by Witten via a physical approach, and it is the starting point for the Witten conjecture, the detail of which will be given in the second part of the series.

MSC:

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
57R56 Topological quantum field theories (aspects of differential topology)
57R57 Applications of global analysis to structures on manifolds
81T13 Yang-Mills and other gauge theories in quantum field theory
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)

Citations:

Zbl 0721.58056
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