Representations of chiral alternative to vierbein and construction of non-Abelian instanton solution in curved space-time

Abstract

A chiral alternative to the vierbein field in general relativity was considered by 't Hooft in an attempt to facilitate the construction of a quantum theory of gravity. 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 Newman-Penrose formalism. We conjecture that these are new objects with physical significance and are the analog of Killing-Yano tensors. As an application, we try to construct non-Abelian instanton like solutions in curved space from the flat space 't Hooft tensors. The space part of the tensors are decomposed to a product of Levi-Civita tensor and flat space Dirac gamma matrices. The gamma matrices are promoted to the curved space with the help of vierbeins, as in separability of Dirac equation by Chandrasekhar, using Newman-Penrose formalism. The curved space generalisation of the instanton solution $A_{\mu}^a $ is now constructed by substituting all objects to their general covariant form.