  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.
