A chiral alternative to the vierbein field in general relativity

Abstract

An alternative to the usual vierbein field in a (3 + 1)-dimensional (euclidean) space-time is proposed such that the internal index takes only three values and the external is a double: ea = −ea. In flat space-time this field reduces to the self-dual generalized Levi-Civita symbol a. Like the vierbein field, our field determines the metric field g uniquely. It can be viewed upon as the 'cube root' of the metric field. In euclidean space the internal symmetry group is SL(3). In Minkowski space, in a sense to be explained, the internal symmetry group is SU(3).

The Einstein-Hilbert action takes an elegant form in terms of this new field.