On c-theorems in arbitrary dimensions

TL;DR

The paper shows that holographic counterterms, obtained by promoting the AdS radial cutoff to a dynamical field σ, reproduce the dilaton effective action associated with broken conformal symmetry. In even dimensions, the resulting action matches the flat-space Wess-Zumino form and its coefficient aligns with the Euler anomaly a^{*}, as demonstrated explicitly in d=2,4,6 using GaussBonnet gravity; in d=3, where no anomaly exists, a DBI-type counterterm yields a global Weyl-invariant, nonpolynomial σ-action, hinting at a universal F-theorem structure. The odd-d case is explored via exact Hamiltonian-constraint solutions for specific boundaries (S^3, S^1×S^2), revealing scale-invariant counterterms tied to Cotton tensor contributions and suggesting a holographic route to a potential odd-dimensional a/theorem. Together, these results reinforce the consistency of the AdS/CFT framework across dimensions and point to new avenues for proving dilation-based monotonicity theorems in both even and odd dimensions through holographic RG flows and entanglement analyses.

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.