Gauge Theories: Geometry and cohomological invariants

We develop a geometrical structure of the manifolds $Gamma$ and $hatGamma$ associated respectively to the gauge symmetry and to the BRST symmetry. Then, we show that ($hatGamma,hatzeta,Gamma$), where $hatzeta$ is the group of BRST transformations, is endowed with the structure of a principle fiber bundle over the base manifold $Gamma$. Furthermore, in this geometrical set up due to the nilpotency of the BRST operator, we prove that the effective action of a gauge theory is a BRST-exact term up to the classical action. Then, we conclude that the effective action where only the gauge symmetry is fixed, is cohomologically equivalent to the action where the gauge and the BRST symmetries are fixed.