Holographic Reconstruction of Spacetime¶and Renormalization in the AdS/CFT Correspondence

Abstract

We develop a systematic method for renormalizing the AdS/CFT prescription for computing correlation functions. This involves regularizing the bulk on-shell supergravity action in a covariant way, computing all divergences, adding counterterms to cancel them and then removing the regulator. We explicitly work out the case of pure gravity up to six dimensions and of gravity coupled to scalars. The method can also be viewed as providing a holographic reconstruction of the bulk spacetime metric and of bulk fields on this spacetime, out of conformal field theory data. Knowing which sources are turned on is sufficient in order to obtain an asymptotic expansion of the bulk metric and of bulk fields near the boundary to high enough order so that all infrared divergences of the on-shell action are obtained. To continue the holographic reconstruction of the bulk fields one needs new CFT data: the expectation value of the dual operator. In particular, in order to obtain the bulk metric one needs to know the expectation value of stress-energy tensor of the boundary theory. We provide completely explicit formulae for the holographic stress-energy tensors up to six dimensions. We show that both the gravitational and matter conformal anomalies of the boundary theory are correctly reproduced. We also obtain the conformal transformation properties of the boundary stress-energy tensors. 1

Energy and the AdS/CFT correspondence

“…Thus, the concept of canonical energy that we considered on performing the quantization of the scalar field leads us, in a natural way, to a generalization of the usual Dirichlet prescription Eq. (28). The analysis of this generalized prescription has been carried out in [7] for the particular minimally coupled case, and we show in this paper that the root of such a generalized formalism can be found on the concept of canonical energy.…”

SupersymmetricAdS5black holes

“…The same is true of supersymmetric rotating black holes in D = 3 [39] and D = 4 [20]. Conversely, if we take ∂/∂t as the generator of time translations then equation 4.43 shows that the angular velocity of the black hole in the φ direction is 44) which implies that the black hole rotates at the speed of light with respect to the frame at infinity. We are ultimately interested in the AdS/CFT description of these black holes.…”

“…Note that the choice of the sign fixes the sign of J. An alternative definition of conserved charges is provided by the "holographic stress tensor" approach [42,43,44,45]. In this method, the expectation value of the stress tensor of the dual CFT on R × S 3 , in a conformal frame with metric (4.42), is calculated using the formula…”

Instantons and conformal holography

“…This would no longer be true for any other value of the bulk mass [10]. 3 Hence, Classical Holography of the conformally coupled scalar is a classical field theory problem in a flat space with a boundary.…”