The Maurer-Cartan structure of BRST differential

TL;DR

This paper reformulates the BRST differential using Maurer-Cartan structures, revealing its Lie algebraic nature and enabling extensions of nonintegrable systems to integrable ones in a geometric context.

Contribution

It introduces a new Maurer-Cartan-based formulation of the BRST differential, connecting it to Lie algebra cohomology and providing a geometric extension method.

Findings

BRST differential is analogous to Maurer-Cartan differential

For certain theories, BRST differential is a Chevalley-Eilenberg differential

Enables extension of nonintegrable systems to integrable systems

Abstract

In this paper, we construct a new sequence of generators of the BRST complex and reformulate the BRST differential so that it acts on elements of the complex much like the Maurer-Cartan differential acts on left-invariant forms. Thus our BRST differential is formally analogous to the differential defined on the BRST formulation of the Chevalley-Eilenberg cochain complex of a Lie algebra. Moreover, for an important class of physical theories, we show that in fact the differential is a Chevalley-Eilenberg differential. As one of the applications of our formalism, we show that the BRST differential provides a mechanism which permits us to extend a nonintegrable system of vector fields on a manifold to an integrable system on an extended manifold.