Conformal (super)gravities with several gravitons

TL;DR

This paper constructs consistent interacting theories for multiple conformal massless spin-2 fields, called Weyl gravitons, with specific properties including no more than four derivatives and non-positive definite internal metrics, extending conformal algebra.

Contribution

It introduces a novel method to build interacting conformal gravities with multiple gravitons using non-semi-simple algebra extensions and supercurvature formulations.

Findings

Constructed theories with M interacting Weyl gravitons

Ensured no more than four derivatives in interactions

Achieved supersymmetrization leading to conformal supergravities

Abstract

We construct consistent interacting gauge theories for M conformal massless spin-2 fields ("Weyl gravitons") with the following properties: (i) in the free limit, each field fulfills the equation ${\cal B}^{\mu \nu} = 0$, where ${\cal B}^{\mu \nu}$ is the linearized Bach tensor, (ii) the interactions contain no more than four derivatives, just as the free action and (iii) the internal metric for the Weyl gravitons is not positive definite. The interacting theories are obtained by gauging appropriate non-semi-simple extensions of the conformal algebra $so(4,2)$ with commutative, associative algebras of dimension M. By writing the action in terms of squares of supercurvatures, supersymmetrization is immediate and leads to consistent conformal supergravities with M interacting gravitons.