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 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.
Highlights
The correlation functions of the boundary conformal field theory (BCFT) can be translated to correlation functions in anti-de Sitter (AdS) by performing the required Weyl rescaling of the operators
When the CFT is placed in AdS space, such a free energy can be defined by using the hyperbolic ball coordinates of AdS, so that the boundary is a sphere and the problem is conformally related to the BCFT on the round ball
The quantity Fis related to one of the boundary anomaly coefficient, as we review in section 2.3 below in the d = 3 case, and the inequality for the AdS free energy is in accordance with what was proved in [17]
Summary
In preparation to the calculations in the interacting O(N ) model, we compute the AdS free energy in simple free field theory examples, and check consistency with the conjectured boundary F -theorem in terms of the quantity defined in (1.3). We briefly discuss the case of weakly relevant boundary flows, and elaborate on the relation of the free energy to the trace anomaly coefficients, focusing on the d = 3 case. As explained in the introduction, to calculate the free energy we consider the case in which the boundary of AdS is a round sphere, in other words we will be computing the free energy on a hyperbolic ball. Ωd) are the coordinates on the d − 1 sphere with |Ωi|2 = 1 This gives the hyperbolic ball metric ds. Whenever we write F or Fit should be understood to be computed on AdS
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