Covariant non-commutative space-time

We introduce a covariant non-commutative deformation of 3+1-dimensional conformal field theory. The deformation introduces a short-distance scale ℓ_p, and thus breaks scale invariance, but preserves all space-time isometries. The non-commutative algebra is defined on space-times with non-zero constant curvature, i.e. dS₄ or AdS₄. The construction makes essential use of the representation of CFT tensor operators as polynomials in an auxiliary polarization tensor. The polarization tensor takes active part in the non-commutative algebra, which for dS₄ takes the form of so(5, 1), while for AdS₄ it assembles into so(4, 2). The structure of the non-commutative correlation functions hints that the deformed theory contains gravitational interactions and a Regge-like trajectory of higher spin excitations.