Scale without Conformal Invariance in bottom-up Holography

In holography, the isometry group of the bulk spacetime corresponds to the symmetries of the boundary theory. We thus approach the question of whether (and when) scale invariance in combination with Poincar\'e invariance implies full conformal invariance in quantum field theory from a holographic bulk perspective. To do so, we study bulk spacetimes that include a warped extra dimension and in which the isometry group corresponds to scale without conformal invariance. Firstly, we show that the bulk Weyl tensor plays a pivotal role in distinguishing those metrics exhibiting conformal invariance (Weyl=0) from those merely exhibiting scale invariance (Weyl$\neq$0). Based on this, we then prove the following theorem: For putative boundary theories with $n\geq2$ dimensions, the bulk metric can not exhibit scale without conformal invariance if its warped extra dimension is compact and the null energy condition is required to hold. For $n=1$, we discuss that a more general ansatz for the bulk metric must be made, a detailed analysis of which is left for future research.