Scale without Conformal Invariance in bottom-up Holography

TL;DR

The paper addresses whether scale invariance plus Poincaré invariance necessarily implies conformal invariance in holographic duals, using a bottom-up warped-bulk model with a compact extra dimension. The authors identify a diagnostic Weyl-scalar $\phi$ that governs the transition between SwCI and full conformal invariance, proving that for boundary dimensions $n\ge2$ a physical bulk with NEC and a single warped extra dimension cannot realize SwCI; conformal invariance then follows if $\phi=0$, and SwCI is incompatible otherwise. They provide a thorough analysis of the warped-product geometry, Killing algebras, base-space topology, and the null energy condition, culminating in a general no-go argument complemented by special-case proofs. The work clarifies when holographic SwCI models are or are not allowed and outlines future directions, including topology relaxations and NEC relaxations, to explore potential exceptions or extensions. Overall, the Weyl tensor criterion and NEC-based analysis offer a robust framework to distinguish scale from conformal invariance in holographic contexts and guide subsequent model-building efforts.

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 Poincaré 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.

Figures (2)

• Figure 1: Embedding of the pseudosphere into $\mathbb{R}^3$ as a surface of revolution. The black lines are integral lines of the only globally defined Killing vector field.
• Figure 2: Phase diagrams in the $\beta-\theta-$plane for $f_S(\theta)=f_0(\theta)$ (left) and $f_S(\theta)=f_0(\theta)-\frac{1}{400} \cos ^2(2 \theta)$ (right). These diagrams show domains where $t_{ij}$ is negative-definite (black), indefinite (grey), or positive definite (white). The boundaries of the domains are highlited by dotted grey lines and correspond to the zeroes of $\det[t_{ij}]$\ref{['det_with_beta_and_q']} which is a third order polynomial in $\beta$. The dashed blue line signifies the location where $\text{tr}[t_{ij}]$\ref{['tij_with_beta_and_q']} vanishes. Many of the lessons of this subsection can be understood quite intuitively from such phase diagrams. For instance, wherever $\bar{R}<0$, a negative-definite domain (black) will appear attached to the $\theta$-axis, contained within a roughly oval shaped contour where $\text{tr}[t_{ij}]=0$. Hence the NEC will obviously be violated for values of $|\beta|$ too close to zero unless $\bar{R}=0$ everywhere. We also see in the left picture that $\det{[t_{ij}]}=0$ at $\beta=1/2$ where $\phi=0$, as discussed in section \ref{['sec::det_phi']}.