Supersymmetric Tensor Hierarchies from Superspace Cohomology

In this set of lectures we give a pedagogical introduction to the way in which the nilpotency of a super-de Rham operator can be exploited for the construction of gauge theories in superspace. We begin with a discussion of how the super-geometric closure conditions can be solved by simply computing the cocycles of the super-algebra. The next couple lectures are then devoted to applying this idea to extensions of the standard super-de Rham complex. This eventually results in a geometric "trivialization" of the consistency conditions required for non-abelian tensor hierarchies. Although this is a general conclusion, we focus specifically on the hierarchy obtained by compactifying the 3-form gauge field of 11D supergravity to 4D, $N = 1$ superspace. In the final lecture, we use the cohomological arguments developed herein to provide a geometric construction of the non-trivial Chern-Simons-type invariant in that tensor hierarchy and comment on generalizations. These lectures are based on a series of talks given at Texas A&M University from March 21-25.