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.
Highlights
In Ref. [1], it was shown that the locally scale-invariant Weyl theory of gravity is the gauge theory of the conformal group, where conformal transformations are gauged by a nonpropagating gauge field
We give some preliminary results in the context of N 1⁄4 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
They claimed that in order to produce a conformally invariant theory in this setup, it is necessary to set the torsion to zero.1. In contrast with this claim, we show that it is possible to construct a gauge theory of the conformal group in four spacetime dimensions with a nonvanishing torsion component where proper conformal transformations are gauged by a nonpropagating gauge field
Summary
In Ref. [1], it was shown that the locally scale-invariant Weyl theory of gravity is the gauge theory of the conformal group, where conformal transformations (conformal boosts) are gauged by a nonpropagating gauge field. [2,3] to construct a quadratic Lagrangian with the curvatures associated with the conformal group in four spacetime dimensions They claimed that in order to produce a conformally invariant theory in this setup, it is necessary to set the torsion to zero.. We will adopt the second-order formalism, which will allow us to end up with a fourth-order propagation equation for the graviton, the Lorentz connection being torsionful In this setup, invariance under conformal boosts ( known as proper, or special, conformal transformations) implies a Killing vector equation—namely, a differential condition on the axial vector torsion which, upon further differentiation, leads to a Maxwell-like equation in a four-dimensional curved background. Since we will prove in the sequel of this work that, at least at the purely bosonic level, one can still recover invariance under (special) conformal transformations allowing for a nonvanishing axial vector torsion, we argue that something similar should presumably happen in the superconformal case. In the Appendix, some useful formulas on gamma matrices in four dimensions are collected
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