A CHIRAL ALTERNATIVE TO THE VIERBEIN FIELD IN GENERAL-RELATIVITY

An alternative to the usual vierbein field in a (3 + l)-dimensional (euclidean) space-time is proposed such that the internal index takes only three values and the external is a double: e(a)mu-nu = -e(a)nu-mu. In flat space-time this field reduces to the self-dual generalized Levi-Civita symbol eta-a-mu-nu. Like the vierbein field, our field determines the metric field g-mu-nu 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.