On representations of a chiral alternative to vierbein

In an attempt to facilitate the construction of a quantum theory of gravity, 't Hooft has considered a chiral alternative to the vierbein field in general theory of relativity. These objects, $f^a{}_{\mu\nu}$, behave like the "cube root" of the metric tensor. We try to construct specific representations of these tensors in terms of Dirac $\gamma$ matrices in Euclidean and Minkowski space and promote these to curved space through Penrose-Newman formalism. We conjecture that these new objects, with physical significance, are the analog of Killing-Yano tensors.