Topology, geometry and physics: background for the Witten conjecture. I (Q1773010)

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 \textit{M. F. Atiyah} and \textit{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.