F-theorem for Quantum Field Theories in Anti-de Sitter Space

TL;DR

This work extends RG monotonicity principles to non-conformal bulks by formulating a finite AdS free energy $\mathcal{F}_{\text{AdS}}$ that captures boundary RG flows in bCFTs. It provides a concrete definition via holographic renormalization and sphere subtraction, proves the $d=2$ case using a dilaton/Weyl-invariance framework, and presents evidence and applications in higher dimensions. The results yield bounds and consistency checks for long-range CFTs and large-$N$ models, offering a unifying perspective on boundary phase structure and RG trajectories in AdS. The AdS F-theorem thus supplies a powerful organizing principle for boundary phenomena in massive QFTs and related non-local theories.

Abstract

We introduce a regularized free energy $\mathcal{F}_{\text{AdS}}$ for massive quantum field theories (QFTs) on Anti-de Sitter space (AdS). We conjecture this quantity to be monotonic under the renormalization group (RG) flow induced by boundary perturbations, generalizing the known boundary $F$-theorem to non-conformal setups. We test this conjecture in several examples and provide a proof in two dimensions. We also discuss applications to long-range critical points, obtaining bounds on the sphere free energy of long- and short-range Ising models in three dimensions.

Figures (9)

• Figure 1: (a) Global Euclidean AdS space. (b) Introducing a modulated dilaton on the AdS boundary by a PBH transformation.
• Figure 2: Free energy difference between $+$ and $-$ b.c. as a function of $x=\tfrac{4M^2L^2+d^2}{\text{Min}\left[d^2,4\right]}\in[0,1]$, parametrizing the range of masses in which both $\pm$ b.c. are consistent.
• Figure 3: The AdS F theorem for the LRI CFT. Dashed vertical lines mark the transition points between GFF-LRI and SRI-LRI, the LRI phase is the stable RG fixed point in the shaded region. We have rescaled $s_{d=3}$ so as to make the dashed transition lines coincide between $d=2$ and $d=3$. By the AdS F-theorem and \ref{['eq: ineq']} the plotted function must be negative for $s< s_-$ and positive for $s>s_+$. For the $d=3$ plots we have used the numerical estimates $\Delta_\sigma \sim 0.5181$Chang:2024whx and $F_{\text{Ising}} = 0.0612$Hu:2024pen. We restrict to $s\leq\min(d,2)$ to ensure unitarity of the $-$ boundary condition.
• Figure 4: Upper bounds on the LRI free energy in $d=2$ and $d=3$ for $s_-<s<s_+$. The shaded purple regions are allowed by the AdS F-theorem.
• Figure 5: The large $N$ free energy difference between symmetry-preserving and symmetry-breaking b.c. for the O(N) model.