Lorentz covariant spin two superspaces

TL;DR

This paper explores Lorentz covariant superalgebras with generators up to spin two, deriving their structure constants and constructing explicit classes of solutions within a framework reminiscent of supergravity configuration spaces.

Contribution

It provides a comprehensive analysis of spin two superalgebras, deriving their defining equations and explicitly constructing several classes of solutions, advancing understanding of Lorentz covariant superspaces.

Findings

Derived the complete set of quadratic equations for structure constants.

Found non-trivial solutions to highly overdetermined superalgebra equations.

Constructed explicit classes of Lorentz covariant spin two superalgebras.

Abstract

Superalgebras including generators having spins up to two and realisable as tangent vector fields on Lorentz covariant generalised superspaces are considered. The latter have a representation content reminiscent of configuration spaces of (super)gravity theories. The most general canonical supercommutation relations for the corresponding phase space coordinates allowed by Lorentz covariance are discussed. By including generators transforming according to every Lorentz representation having spin up to two, we obtain, from the super Jacobi identities, the complete set of quadratic equations for the Lorentz covariant structure constants. These defining equations for spin two Heisenberg superalgebras are highly overdetermined. Nevertheless, non-trivial solutions can indeed be found. By making some simplifying assumptions, we explicitly construct several classes of these superalgebras.