Abstract
We examine the five-dimensional super-de Rham complex with $N = 1$ supersymmetry. The elements of this complex are presented explicitly and related to those of the six-dimensional complex in $N = (1, 0)$ superspace through a specific notion of dimensional reduction. This reduction also gives rise to a second source of five-dimensional super-cocycles that is based on the relative cohomology of the two superspaces. In the process of investigating these complices, we discover various new features including branching and fusion (loops) in the super-de Rham complex, a natural interpretation of "Weil triviality", $p$-cocycles that are not supersymmetric versions of closed bosonic $p$-forms, and the opening of a "gap" in the complex for $D > 4$ in which we find a multiplet of superconformal gauge parameters.
Highlights
During this period some authors turned their attention to the problem of establishing a theory of integration for super p-forms on supermanifolds and significant formal progress was made [7,8,9,10,11]
We examine the five-dimensional super-de Rham complex with N = 1 supersymmetry
In this article we have constructed the super-de Rham complex in five-dimensional, N = 1 superspace and related it to the complex of six-dimensional, N = (1, 0) superspace via dimensional reduction. This turned out to be only one part of the reduced complex, with the remaining part serving as an additional source of closed superforms coming from the relative cohomology of the two superspaces
Summary
This collection is graded by increasing engineering dimension with the component ωα1 ...αr a1 ...as having dimension r 2 This allows the determination of the higher-dimension components of the cocycle in terms of the lowest non-vanishing one(s). This lowest nonvanishing component will be a superfield, possibly in a non-trivial (iso-)spin representation. In addition to determining the components of the cocycle in terms of this defining superfield, the Bianchi identities generally impose a series of constraints on it, again organized by engineering dimension. The complex can branch if it happens that there is more than one constraint on the defining superfield in the highest dimension (as we will see explicitly when passing from the 1-cocycle to the 2-cocycle) and we work out the components of each of the resulting cocycles
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