Scale without Conformal Invariance in bottom-up Holography

TL;DR

This paper investigates holographic models of quantum field theories, demonstrating that under certain conditions, scale invariance without conformal invariance cannot occur in the bulk spacetime, especially when the extra dimension is compact.

Contribution

It proves a theorem linking bulk Weyl tensor properties to the impossibility of scale without conformal invariance in boundary theories with compact extra dimensions.

Findings

Bulk Weyl tensor distinguishes conformal from scale invariance.

Scale without conformal invariance is impossible with compact extra dimension under null energy condition.

The result applies to boundary theories with two or more dimensions.

Abstract

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.