Aspects of $(d+D)$-dimensional Anisotropic Conformal Gravity

We discuss various aspects of anisotropic gravity in $(d+D)$-dimensional spacetime where $D$ dimensions are treated as extra dimensions. It is based on the foliation preserving diffeomorphism invariance and anisotropic conformal invariance. The anisotropy is embodied by introducing a factor $z$ which discriminates the scaling degree of the extra $D$ dimensions against the $d$-dimensional base spacetime and Weyl scalar field which mediates the anisotropic scaling symmetry. There is no intrinsic scale but a physical scale $M_*$ emerges as a consequence of spontaneous conformal symmetry breaking. Some vacuum solutions are obtained and we discuss an issue of `size separation' between the base spacetime and the extra dimensions. The size separation means large hierarchy between the scales appearing in the base spacetime and the extra dimensions respectively. We also discuss interesting theories obtained from our model. In the case of (4,1), we propose a resolution of hierarchy problem and discuss comparison with the results of the brane-world model. In a $(d,D)=(2,2)$ case, we suggest a UV-complete unitary quantum gravity which might become Einstein gravity in IR. In a certain (2,1) case, we obtain CGHS-model.

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