# All Chern-Simons Invariants of 4D, N = 1 Gauged Superform Hierarchies

## TL;DR

This paper provides a geometric framework for supersymmetric p-form hierarchies in 4D, N=1 superspace, simplifying calculations of Chern-Simons invariants and deriving recursive formulas for their actions, including new quartic invariants.

## Contribution

It introduces a geometric approach to supersymmetric p-form hierarchies, simplifying the derivation of Chern-Simons invariants and establishing recursive formulas for their actions.

## Key findings

- Simplified calculations of superspace field-strengths and Bianchi identities.
- Derived recursive formulas for tensor hierarchy actions.
- Computed new quartic Chern-Simons invariants relevant to supergravity.

## 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 $\sim A dA dA dA + \ldots$ relevant, for example, to seven-dimensional supergravity compactifications.

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Source: https://tomesphere.com/paper/1702.00799