Conformal gravity with totally antisymmetric torsion

TL;DR

This paper develops a conformal gauge theory in four dimensions incorporating totally antisymmetric torsion, demonstrating its conformal invariance and deriving related Maxwell-like equations, with preliminary insights into conformal supergravity.

Contribution

It introduces a conformal gauge theory with antisymmetric torsion and explores its implications, including a Maxwell-like equation and initial steps towards conformal supergravity with non-zero supertorsion.

Findings

Antisymmetric torsion does not break conformal invariance.

Derivation of a Maxwell-like equation in curved spacetime.

Preliminary results on conformal supergravity with non-zero supertorsion.

Abstract

We present a gauge theory of the conformal group in four spacetime dimensions with a non-vanishing torsion. In particular, we allow for a completely antisymmetric torsion, equivalent by Hodge duality to an axial vector whose presence does not spoil the conformal invariance of the theory, in contrast with claims of antecedent literature. The requirement of conformal invariance implies a differential condition (in particular, a Killing equation) on the aforementioned axial vector which leads to a Maxwell-like equation in a four-dimensional curved background. We also give some preliminary results in the context of $\mathcal{N}=1$ four-dimensional conformal supergravity in the geometric approach, showing that if we only allow for the constraint of vanishing supertorsion all the other constraints imposed in the spacetime approach are a consequence of the closure of the Bianchi identities in superspace. This paves the way towards a future complete investigation of the conformal supergravity using the Bianchi identities in the presence a non-vanishing (super)torsion.