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Using the fact that flat space with a boundary is related by a Weyltransformation to anti-de Sitter (AdS) space, one may study observables inboundary conformal field theory (BCFT) by placing a CFT in AdS. In addition tocorrelation functions of local operators, a quantity of interest is the freeenergy 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 canbe used to define a quantity that decreases under boundary renormalizationgroup (RG) flows. We test this idea by discussing in detail the case of thelarge $N$ critical $O(N)$ model in general dimension $d$, as well as itsperturbative descriptions in the epsilon-expansion. Using the AdS approach, werecover the various known boundary critical behaviors of the model, and wecompute the free energy for each boundary fixed point, finding results whichare consistent with the conjectured $F$-theorem in a continuous range ofdimensions. Finally, we also use the AdS setup to compute correlation functionsand extract some of the BCFT data. In particular, we show that using the bulkequations of motion, in conjunction with crossing symmetry, gives an efficientway to constrain bulk two-point functions and extract anomalous dimensions ofboundary operators.

CFT in AdS and boundary RG flows