Local BRST cohomology in the antifield formalism: I. General theorems

TL;DR

This paper develops general theorems on the cohomology of the BRST differential in local forms, linking it to conserved quantities and providing explicit calculations for key gauge theories, thus advancing the mathematical understanding of gauge invariance.

Contribution

It establishes new theorems relating BRST cohomology to Koszul-Tate cohomology and computes specific cohomology groups for important gauge theories, offering a deeper algebraic insight.

Findings

$H^{-k}(s|d)$ is isomorphic to $H_k( abla |d)$ in negative ghost degree.

$H_1( abla |d)$ corresponds to conserved quantities, reformulating Noether's theorem.

$H_2( abla |d)$ is explicitly calculated for electromagnetism, Yang-Mills, and Einstein gravity.

Abstract

We establish general theorems on the cohomology $H^*(s|d)$ of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of local $p$-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown that $H^{-k}(s|d)$ is isomorphic to $H_k(\delta |d)$ in negative ghost degree $-k\ (k>0)$, where $\delta$ is the Koszul-Tate differential associated with the stationary surface. The cohomological group $H_1(\delta |d)$ in form degree $n$ is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether theorem. More generally, the group $H_k(\delta|d)$ in form degree $n$ is isomorphic to the space of $n-k$ forms that are closed when the equations of motion hold. The groups $H_k(\delta|d)$ $(k>2)$ are shown to vanish for standard irreducible gauge theories. The group $H_2(\delta|d)$ is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groups $H^{k}(s|d)$ under the introduction of non minimal variables and of auxiliary