CFT descriptions of bulk local states in the AdS black holes

Abstract

We present a new method for reconstructing CFT duals of states excited by the bulk local operators in the three dimensional AdS black holes in the AdS/CFT context. As an important procedure for this, we introduce a map between the bulk points in AdS and those on the boundary where CFT lives. This gives a systematic and universal way to express bulk local states even inside black hole interiors. Our construction allows us to probe the interior structures of black holes purely from the CFT calculations. We analyze bulk local states in the single-sided black holes as well as the double-sided black holes.

An exact operator that knows its location

“…Here we will show how to recover the global boundary operator expansion (BOE) for a scalar operator [58] …”

Localized states in global AdS space

“…Thus, the cut-off ǫ responsible for taming the UV divergence in the amplitude can be reinterpreted as evolving the insertion of the operator in the field theory in imaginary time an amount ǫ. This regularisation procedure has been considered in [12][13][14] for single trace operators. If one is interested in understanding the physics of these states in terms of some semiclassical bulk description, it is natural to ask, for a given ǫ, what the most populated state is.…”

“…There is an alternative way to regulate the problem that has been described in papers by Takayanagi et al [12][13][14]. One replaces the operator insertion O, by a regulated version that has been evolved in Euclidean time an amount ǫ…”

AdS/CFT correspondence in operator formalism

“…Furthermore, using the identification, we can construct the local field in AdS space from the operators in CFT. The bulk reconstruction of the local field has been intensively studied, for example, in [11]- [23] where a version of the GKPW relation, which implies that the boundary value of the bulk field is the CFT primary field, [5] was used. 6 In this paper, we start from the identification of the spectrum, therefore, the GKPW is not assumed.…”

“…23 Here, we have neglected the boundary term which corresponds to the renormalization. However, even including the boundary term, the relation (3.39) holds essentially as discussed in [34] in which the interactions in the bulk was also considered.…”