The Structure of Superforms

In this thesis we examine a set of foundational questions concerning closed forms in superspace. By reformulating a number of definitions through the use of a new ring of (anti-)commuting variables and the concept of an exact Bianchi form, we demonstrate a significantly streamlined method for analyzing superforms. We also study the dimensional reduction of superforms and how the relative cohomology of the superspaces involved allows for the construction of additional closed forms not in the main complex. In particular, the entire de Rham complex of closed superforms in five-dimensional superspace with eight supercharges $(N = 1)$ is derived from the complex in the corresponding six-dimensional superspace. As a concluding effort, we work out the component formulation for the matter multiplets defined by five-dimensional $p$-form field-strengths for $p = 2, 3, 4$. The first and last of these come directly from the de Rham complex and coincide with multiplets that are already well-known, while the 3-form field-strength multiplet happens to require additional effort to find. This leads to the conclusion that, in general, the super-de Rham complex is not the result of supersymmetrizing the bosonic de Rham complex.