Superspace de Rham Complex and Relative Cohomology

TL;DR

This paper explores the super-de Rham complex in five-dimensional supersymmetry, revealing new cohomological features and simplifying the derivation of superspace Bianchi identities through algebraic methods.

Contribution

It introduces a supercommutative algebra approach to analyze the super-de Rham complex, uncovering novel features like splitting/joining and cocycles not linked to irreducible supermultiplets.

Findings

Reduction of Bianchi identities to linear algebra problem

Discovery of cocycles not corresponding to irreducible supermultiplets

Identification of a second complex from dimensional reduction

Abstract

We investigate the super-de Rham complex of five-dimensional superforms with $N=1$ supersymmetry. By introducing a free supercommutative algebra of auxiliary variables, we show that this complex is equivalent to the Chevalley-Eilenberg complex of the translation supergroup with values in superfields. Each cocycle of this complex is defined by a Lorentz- and iso-spin-irreducible superfield subject to a set of constraints. Restricting to constant coefficients results in a subcomplex in which components of the cocycles are coboundaries while the constraints on the defining superfields span the cohomology. This reduces the computation of all of the superspace Bianchi identities to a single linear algebra problem the solution of which implies new features not present in the standard four-dimensional, $N=1$ complex. These include splitting/joining in the complex and the existence of cocycles that do not correspond to irreducible supermultiplets of closed differential forms. Interpreting the five-dimensional de Rham complex as arising from dimensional reduction from the six-dimensional complex, we find a second five-dimensional complex associated to the relative de Rham complex of the embedding of the latter in the former. This gives rise to a second source of closed differential forms previously attributed to the phenomenon called "Weyl triviality".