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.


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)


Zbl 0721.58056
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