Exterior differentials in superspace and Poisson brackets

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.

Hodge Duality Operation and Its Physical Applications on Supermanifolds

“…The ensuing expression for (⋆Ã (1) ), due to the Hodge duality operation given in (3.8) and definition (2.9), is 26) which is nothing but the 5-form defined on the six (4 + 2)-dimensional supermanifold. Applying now the super exterior derivatived = dZ M ∂ M on the above 5-form, we obtain the following 6-form…”

“…In fact, the existence of the totally symmetric metric tensor and the totally antisymmetric Levi-Civita tensor on the spacetime manifold plays a crucial role in such a consistent and systematic definition of the duality operation ( * ). However, such a consistent, precise and elaborate definition of the Hodge duality ⋆ operation on a supermanifold, to the best of our knowledge, is not well-known in the literature (see, e.g., [18][19][20][21][22][23][24][25][26] for details). The purpose of our present paper is to provide a working rule for the definition of the Hodge duality ⋆ operation on the four (2 + 2)-and six (4 + 2)-dimensional supermanifolds on which the 2D-and 4D free 1-form (A (1) = dx µ A µ ) Abelian gauge theories are defined for the derivation of the nilpotent (anti-)co-BRST symmetry transformations in the framework of superfield approach to BRST formalism.…”

Generalized Wigner-Inönü contractions and Maxwell (super)algebras

“…4 In this note we shall apply such a contraction scheme to the derivation in Sect. 2 of Maxwell algebra [10]- [12] and simple nonstandard [13] as well as standard [14] Maxwell superalgebras. In Sect.…”