Conformal gravity with totally antisymmetric torsion

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