Chern-Simons actions and their gaugings in 4D, N=1 superspace

We gauge the abelian hierarchy of tensor fields in 4D by a Lie algebra g. The resulting non-abelian tensor hierarchy can be interpreted via a g-equivariant chain complex. We lift this structure to N = 1 superspace by constructing superfield analogs for the tensor fields, along with covariant superfield strengths. Next we construct Chern-Simons actions, for both the bosonic and N = 1 cases, and note that the condition of gauge invariance can be presented cohomologically. Finally, we provide an explicit realization of these structures by dimensional reduction, for example by reducing the three-form of eleven-dimensional supergravity into a superspace with manifest 4D, N = 1 supersymmetry.