Exterior Differentials in Superspace and Poisson Brackets of Diverse Grassmann Parities

TL;DR

This paper explores two different definitions of exterior differentials in superspace and their impact on Poisson brackets, illustrating their connection with BRST charges through examples involving Lie groups.

Contribution

It demonstrates that two definitions of exterior differential in superspace yield different results when applied to Poisson brackets, highlighting their relevance to BRST structures.

Findings

Different exterior differential definitions lead to distinct Poisson bracket results.

Examples include even and odd brackets related to semi-simple Lie groups.

Connections to BRST and anti-BRST charges are discussed.

Abstract

It is shown that two definitions for the exterior differential in superspace, giving the same exterior calculus, when applied to the Poisson bracket lead to the different results. Examples of the even and odd linear brackets, corresponding to semi-simple Lie groups, are given and their natural connection with BRST and anti-BRST charges is indicated.