From Symmetries to Conformal Minkowski Space-Time: Construction via \(P \backslash O(n,2)\) and Implications to Gauge Gravity

Abstract

Starting from conformal Minkowski space as the projective null quadric in \(\mathbb {R}^{n,2}\), we construct an explicit four-chart atlas for the compactified spacetime, in contrast with the usual two-chart descriptions found in the literature. We also use explicitly the homogeneous-coordinate system dictated by the homogeneous-space structure associated with a parabolic subgroup \(P\subset G=O(n,2)\); in the matrix conventions used below, this space is realized as \(P\backslash G\), so that the geometry is genuinely treated through the quotient structure rather than through a single affine patch. This global construction reveals geometric structures that are not visible in the standard local description and clarifies their implications for natural gauge-gravity theories on conformal Minkowski space. In particular, the Lorentz and dilation sectors arise canonically as local actions on the fiber of the induced bundle, whereas translations and special conformal transformations are encoded through the base geometry and transition maps; they do not appear as canonical independent local gauge actions on the fiber in this induced-bundle construction. Consequently, gauge-gravity models built in this framework without a soldering form or vielbein are naturally restricted to topological-type structures, as is generally expected when no soldering mechanism is present. Since the construction is canonical, it also reproduces the standard background of Minkowskian conformal field theory, including conformal weights, field transformations, compactification, and conformal generators. As a concrete test of the usefulness of this global description, we additionally develop a scalar holographic example. The normalized spherical vector is first realized as a globally compatible section in all four boundary charts and is then extended into the projective AdS bulk by the same A- and N-directions that organize the conformal representation. The resulting expression is the standard scalar boundary-to-bulk kernel in the present light-cone coordinates. In the explicit \(n=2\) case, the kernel also produces a consistent family of bulk transition factors whose radial limit recovers the scalar principal-series gluing data on \(P\backslash G\). The point of the calculation is not to introduce a new scalar propagator, but to exhibit explicitly how global conformal geometry, induced representations, and a familiar AdS/CFT quantity fit into one calculational framework.