On c-theorems in arbitrary dimensions

TL;DR

This paper explores the existence and nature of c-theorems in arbitrary dimensions, using holographic methods to analyze trace anomalies and Weyl invariance in both even and odd dimensions.

Contribution

It introduces a novel approach to understanding c-theorems in odd dimensions via holography, extending the concept beyond traditional anomaly-based proofs.

Findings

In even dimensions, the cut-off promotion yields a Wess-Zumino-like action.

In three dimensions, an exact Hamilton-Jacobi counterterm leads to a Weyl-invariant action.

The work connects these actions to F-theorem conjectures.

Abstract

The dilaton action in 3+1 dimensions plays a crucial role in the proof of the a-theorem. This action arises using Wess-Zumino consistency conditions and crucially relies on the existence of the trace anomaly. Since there are no anomalies in odd dimensions, it is interesting to ask how such an action could arise otherwise. Motivated by this we use the AdS/CFT correspondence to examine both even and odd dimensional CFTs. We find that in even dimensions, by promoting the cut-off to a field, one can get an action for this field which coincides with the WZ action in flat space. In three dimensions, we observe that by finding an exact Hamilton-Jacobi counterterm, one can find a non-polynomial action which is invariant under global Weyl rescalings. We comment on how this finding is tied up with the F-theorem conjectures.