Local bulk operators in AdS/CFT correspondence: A boundary view of horizons and locality

Abstract

We develop the representation of local bulk fields in AdS by non-local operators on the boundary, working in the semiclassical limit and using AdS 2 as our main example. In global coordinates we show that the boundary operator has support only at points which are spacelike separated from the bulk point. We construct boundary operators that represent local bulk operators inserted behind the horizon of the Poincaré patch and inside the Rindler horizon of a two dimensional black hole. We show that these operators respect bulk locality and comment on the generalization of our construction to higher dimensional AdS black holes.

What do non-relativistic CFTs tell us about Lifshitz spacetimes?

“…Moreover, the map (1.2) can be inverted in position space. As a result, local quantum fields in the bulk can be expressed in terms of boundary operators with the help of a so-called smearing function K [32][33][34]. Consequently, we can study CFTs to learn something about their gravitational duals [35][36][37].…”

“…In this section, we review the use of smearing functions as a way to reconstruct bulk physics from boundary dynamics [30][31][32][33][34][35][36]. Using the WKB formalism developed in appendix A, we will show that for Lifshitz spacetimes, and more generally for any flow involving Lifshitz, such reconstruction is not possible.…”

“…Another interesting question is what happens if we take a geometry with a smearing function, such as AdS [32][33][34], and add a small (planar) perturbation in the IR. It can be seen from (2.6) that e W must start with non-positive slope at the boundary for any background that is asymptotically AdS.…”

An infalling observer in AdS/CFT

“…Before we conclude this section, it is useful to compare our construction with that of [31]. The authors of that paper were working directly in position space, and attempted to write the bulk field as…”

Conformal complementarity maps

“…The main limitation of these considerations is the absence of an actual reconstruction of operators with approximate bulk locality, in the spirit of [23,[49][50][51][52][53][54][55]. In this sense, we have strived to characterize horizon complementarity in the absence of an actual 'horizon', using only deep UV data.…”