Gauge Theories: Geometry and cohomological invariants

TL;DR

This paper develops a geometric framework for gauge and BRST symmetries, showing that the effective action in gauge theories is cohomologically equivalent when fixing gauge and BRST symmetries, using fiber bundle structures.

Contribution

It introduces a geometric structure of manifolds associated with gauge and BRST symmetries and proves the cohomological equivalence of different gauge fixing procedures.

Findings

Effective action is BRST-exact up to the classical action.

The structure of a principal fiber bundle over the gauge manifold is established.

Cohomological equivalence of gauge-fixed actions is demonstrated.

Abstract

We develop a geometrical structure of the manifolds $\Gamma$ and $\hat\Gamma$ associated respectively to the gauge symmetry and to the BRST symmetry. Then, we show that ($\hat\Gamma,\hat\zeta,\Gamma$), where $\hat\zeta$ 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.