Conformal Coupling and Invariance in Different Dimensions

TL;DR

This paper compares different types of conformal transformations across various geometries and dimensions, highlighting the special role of the conformal coupling constant in ensuring invariance and equivalence of models.

Contribution

It systematically analyzes conformal invariance in multidimensional and minisuperspace geometries, emphasizing the critical conformal coupling constant across different dimensions.

Findings

Critical conformal coupling constant xi_c identified for specific dimensions

Conformal invariance established for various geometric models

Simplified rational form of xi_c for dimensions 3, 4, 6, 10

Abstract

Conformal transformations of the following kinds are compared: (1) conformal coordinate transformations, (2) conformal transformations of Lagrangian models for a D-dimensional geometry, given by a Riemannian manifold M with metric g of arbitrary signature, and (3) conformal transformations of (mini-)superspace geometry. For conformal invariance under this transformations the following applications are given respectively: (1) Natural time gauges for multidimensional geometry, (2) conformally equivalent Lagrangian models for geometry coupled to a spacially homogeneous scalar field, and (3) the conformal Laplace operator on the $n$-dimensional manifold $M of minisuperspace for multidimensional geometry and the Wheeler de Witt equation. The conformal coupling constant xi_c is critically distinguished among arbitrary couplings xi, for both, the equivalence of Lagrangian models with D-dimensional geometry and the conformal geometry on n-dimensional minisuperspace. For dimension D=3,4,6 or 10, the critical number xi_c={D-2}/{4(D-1)} is especially simple as a rational fraction.