Local BRST cohomology and covariance

TL;DR

This paper develops a general, algebraic framework for computing the local BRST cohomology across a wide class of gauge theories by recasting the problem in the jet-space setting and using the combined operator $\tilde{s}=s+d$. It identifies a gauge covariant algebra realized on tensor fields and generalized connections, enabling a compact formulation of the BRST algebra and a local reduction of cohomology computations to objects built from these geometric data. The approach yields universal insights into gauge-invariant actions, Noether currents, anomalies, and the covariance of the equations of motion, and it is illustrated on Yang–Mills theory, gravity in the metric formulation, D=4, N=1 supergravity, and two-dimensional sigma models. It also clarifies the role of $x$-dependence and diffeomorphism invariance in the cohomological analysis, and it provides a practical method to eliminate trivial pairs via contracting homotopies, simplifying the cohomology to a tractable, covariant core.

Abstract

The paper provides a framework for a systematic analysis of the local BRST cohomology in a large class of gauge theories. The approach is based on the cohomology of s+d in the jet space of fields and antifields, s and d being the BRST operator and exterior derivative respectively. It relates the BRST cohomology to an underlying gauge covariant algebra and reduces its computation to a compactly formulated problem involving only suitably defined generalized connections and tensor fields. The latter are shown to provide the building blocks of physically relevant quantities such as gauge invariant actions, Noether currents and gauge anomalies, as well as of the equations of motion.