Exterior Differentials in Superspace and Poisson Brackets

TL;DR

This paper explores two definitions of exterior differentials in superspace, their impact on Poisson brackets, and introduces a transition method that generalizes the Schouten-Nijenhuis bracket, linking to Lie superalgebras and BRST cohomology.

Contribution

It introduces a transition prescription between Poisson brackets using exterior differentials, generalizes the Schouten-Nijenhuis bracket for superspace, and connects these structures to Lie superalgebras and BRST cohomology.

Findings

Different exterior differentials lead to distinct results in Poisson brackets.

The transition method generalizes the Schouten-Nijenhuis bracket to superspace.

Resulting brackets relate to Lie superalgebras and BRST charges.

Abstract

It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity to another bracket is introduced. It is also indicated that the resulting bracket leads to generalization of the Schouten-Nijenhuis bracket for the cases of superspace and brackets of diverse Grassmann parities. It is shown that in the case of the Grassmann-odd exterior differential the resulting bracket is the bracket given on exterior forms. The above-mentioned transition with the use of the odd exterior differential applied to the linear even/odd Poisson brackets, that correspond to semi-simple Lie groups, results, respectively, in also linear odd/even brackets which are naturally connected with the Lie superalgebra. The latter contains the BRST and anti-BRST charges and can be used for calculation of the BRST operator cohomology.