Renormalization of a tensorial field theory on the homogeneous space SU(2)/U(1)

We study the renormalization of a general field theory on the homogeneous space (SU(2)/U(1))(xd) with tensorial interaction and gauge invariance under the diagonal action of SU(2). We derive the power counting for arbitrary d. For the case d = 4, we prove perturbative renormalizability to all orders via multi-scale analysis, study both the renormalized and effective perturbation series, and establish the asymptotic freedom of the model. We also outline a general power counting for the homogeneous space (SO(D)/SO(D-1))(xd), of direct interest for quantum gravity models in arbitrary dimension, and point out the obstructions to the direct generalization of our results to these cases.