Exercises in Equivariant Cohomology and Topological Theories

Before going into the subject of this talk, I would like to describe some concrete exercises done by Claude and I which represent a very small portion of the numerous discussions we had, mostly by exchange of letters. We happened to be both guests of the CERN theory division during the academic year 19721973. The perturbative renormalization of gauge theories was still a hot subject, and, whereas most of our colleagues considered the problem as solved we were both still very innocent. I happened to be scheduled for a set of lectures for the ”Troisième cycle de la Suisse Romande” in the spring 1973, on the subject ”Models with renormalizable Lagrangians: Perturbative approach to symmetry breaking”, and I decided to conclude those lectures with a summary of the known constructions related to gauge theories, mostly at the classical level, except for a heuristic derivation of the now called 1 Slavnov Taylor identities, taking seriously the Faddeev Popov ghost and antighost as local fields. What had to be done was indicated in A. Slavnov’s preprint which I had remarked: perform a gauge transformation of parameter mξ̄ where m is the Faddeev Popov operator and ξ̄ the source of the antighost field. That strange trick was due to E.S. Fradkin and I.V. Tyutin as indicated in Slavnov’s preprint. At the time, I was not aware of J.C. Taylor’s paper which came to my attention much later. Anyway, Claude and I carried out that calculation whose result is reported in the notes, with details in an appendix for which the authors (A. Rouet and I) thank Claude Itzykson for generous help . It is that form of the identity which, a few months later drew Carlo Becchi and Alain Rouet’s atten-