CFT in AdS and boundary RG flows

Abstract

Using the fact that flat space with a boundary is related by a Weyl transformation to anti-de Sitter (AdS) space, one may study observables in boundary conformal field theory (BCFT) by placing a CFT in AdS. In addition to correlation functions of local operators, a quantity of interest is the free energy of the CFT computed on the AdS space with hyperbolic ball metric, i.e. with a spherical boundary. It is natural to expect that the AdS free energy can be used to define a quantity that decreases under boundary renormalization group (RG) flows. We test this idea by discussing in detail the case of the large $N$ critical $O(N)$ model in general dimension $d$, as well as its perturbative descriptions in the epsilon-expansion. Using the AdS approach, we recover the various known boundary critical behaviors of the model, and we compute the free energy for each boundary fixed point, finding results which are consistent with the conjectured $F$-theorem in a continuous range of dimensions. Finally, we also use the AdS setup to compute correlation functions and extract some of the BCFT data. In particular, we show that using the bulk equations of motion, in conjunction with crossing symmetry, gives an efficient way to constrain bulk two-point functions and extract anomalous dimensions of boundary operators.

Figures (6)

• Figure 1: Phase diagram of the $O(N)$ model as we vary the boundary interaction strength $c$Die97, defined such that $c=0$ corresponds to tuning to the special transition. The blue line describes the boundary critical temperature when it is above the bulk critical temperature, and immediately below this line one has a phase where there is ordering only on the boundary and not in the bulk.
• Figure 2: Surface RG flow in the large $N$$O(N)$ model between $2 < d < 4$.
• Figure 3: Free energy difference between the Neumann and Dirichlet boundary fixed points of the free bulk theory.
• Figure 4: Large $N$ free energy between $2<d<6$ for the different boundary fixed points. We are plotting $\tilde{F}_{\textrm{subtracted}} = (\tilde{F} - N \tilde{F} (d/2))/N$ on $y-$ axis. On right, we zoom in to the region between $4$ and $6$ dimensions for clarity.
• Figure 5: Large $N$ free energy between $2<d<4$ for special(S), ordinary(O) and extraordinary(E) transitions. We are plotting $\tilde{F}_{\textrm{subtracted}} = (\tilde{F} - N \tilde{F} (d/2))/N$ on $y-$ axis.