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 $$ mathfrak{g} $$ . the resulting non-abelian tensor hierarchy can be interpreted via a $$ mathfrak{g} $$ -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.