Abstract
We introduce a covariant non-commutative deformation of 3+1-dimensional conformal field theory. The deformation introduces a short-distance scale ℓp, 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.
Highlights
One of the basic ways to generalize the classical notion of space-time is non-commutative geometry [1]
It would be of special interest to find examples of covariant non-commutative space-times, that preserve all global symmetries of the underlying classical space-time [2,3,4]
In this paper we propose a covariant non-commutative (CNC) deformation of a general 3+1-D conformal field theory (CFT) defined a homogeneous space-time with constant positive or negative curvature
Summary
One of the basic ways to generalize the classical notion of space-time is non-commutative geometry [1]. In this paper we propose a covariant non-commutative (CNC) deformation of a general 3+1-D conformal field theory (CFT) defined a homogeneous space-time with constant positive or negative curvature. To obtain a covariant non-commutative deformation, the isometry group of 3+1-D de Sitter space-time would need to act both on the coordinates XA and the tensor SAB, via the infinitesimal SO(4, 1) generators MAB [MAB, XC] = i ηACXB − ηBCXA ,. Space-time isometries act on these correlators by transforming all positions and spin variables simultaneously, as in equation (3)-(4) This use of an extended space-time sheds new light on how to obtain covariant noncommutativity. Combined with equation (2), the total CNC algebra extends to so(5, 1), the Lie algebra of the Lorentz group in 6 dimensions, under which the space-time coordinates and spin variables transform as an anti-symmetric tensor ZIJ , defined via Z5A = XA and ZAB = SAB.
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