SU(3) x SU(2) x U(1): The residual symmetry of extended conformal gravity

Abstract

Within the 4-dimensional conformal algebra, the presence of two translation operators implies the existence of 3 distinct metrics of definite Weyl weight constructible from the translational gauge fields. If we demand that each of these metrics give rise to a gauge theory of gravity, we are led to extend the symmetry so that each of these three metrics has a corresponding translation operator. Assigning a vierbein to each of these three translations, a different spacetime metric arises for every choice of inner product of the vierbeins. The covering group of the compact part of the minimal transitive group classifying these inner products is $SU(4)$. An additional $SU(2)$ symmetry classifies the antisymmetric parts of the vierbein product. If the metric is chosen as the gauge field of the translations in the standard way, the SU(4) part of this symmetry is broken to the semidirect product of $SU(3)$ with $U(1)$.