All Chern-Simons Invariants of 4D, N = 1 Gauged Superform Hierarchies
Abstract
We give a geometric description of supersymmetric gravity/(non-)abelian p-form hierarchies in superspaces with 4D, N = 1 super-Poincare invariance. These hierarchies give rise to Chern-Simons-like invariants, such as those of the 5D, N = 1 graviphoton and the eleven-dimensional 3-form but also generalizations such as Green-Schwarz-like/BF-type couplings. Previous constructions based on prepotential superfields are reinterpreted in terms of p-forms in superspace thereby elucidating the underlying geometry. This vastly simplifies the calculations of superspace field-strengths, Bianchi identities, and Chern-Simons invariants. Using this, we prove the validity of a recursive formula for the conditions defining these actions for any such tensor hierarchy. Solving it at quadratic and cubic orders, we recover the known results for the BF-type and cubic Chern-Simons actions. As an application, we compute the quartic invariant A dA dA dA + … relevant, for example, to seven-dimensional supergravity compactifications.