Constructing local bulk observables in interacting AdS/CFT

Abstract

Local operators in the bulk of AdS can be represented as smeared operators in the dual CFT. We show how to construct these bulk observables by requiring that the bulk operators commute at spacelike separation. This extends our previous work by taking interactions into account. Large-N factorization plays a key role in the construction. We show diagrammatically how this procedure is related to bulk Feynman

Bulk locality and quantum error correction in AdS/CFT

“…At higher orders in this perturbation theory we will need to confront the problem of defining local operators in a diffeomorphism invariant theory, but we postpone discussion of this until section 5. It is not obvious from the definition that the operators (2.2) have the expected commutators in the bulk; this has been checked perturbatively within low point correlation functions in [19], but must eventually break down in states with enough excitations to avoid a contradiction with the argument in our introduction. We will argue below that, within the subspace of states that are "perturbatively close" to the vacuum, it breaks down only at the level of non-perturbatively small corrections.…”

“…We can now give a version of this argument that includes the gravitational dressing; from figure 8, we see that we should modify the previous statement to "commutes with all local operators at the boundary except at one point". Were this to hold as an operator equation in the CFT, it would now not imply that the operator in the center must be trivial in the CFT, but it would imply that this operator can be nontrivial at 19 A subtlety here is that they send their geodesics from arbitrary boundary times, but take them to be orthogonal to the boundary in the temporal direction as well as the S d−1 directions. These operators agree with ours at t = 0, which is where we will study them, but as a matter of principle we have not made their choice because we want to restrict to Schrodinger-picture operators acting on a fixed time slice at the boundary.…”

“…In the case of empty AdS, where we take (2.1) to hold everywhere, explicit representations of the smearing function can be found in [7,18]. 3 1/N corrections can be systematically included [18,19], 3 One subtlety here is that for more general asymptotically AdS backgrounds, we are not aware of a rigorous argument for the existence of K, even in the distributional sense that we will see we need to allow in the following subsection. One obvious problem is that x could be behind a horizon, but even for geometries with no horizons the only precise argument for the existence of K (or more precisely the existence of the "spacelike Green's function" it is built from) requires spherical symmetry [18].…”

“…The study of these commutators was initiated in [18], and more recently elaborated in [44]. 19 A full analysis has not yet been completed, however, and one point that has not yet been addressed is essential for the consistency of the AdS/Rindler reconstruction at higher orders in 1/N : to all orders in 1/N perturbation theory around a fixed background, two dressed bulk operators with the property that all points on their dressing geodesics are mutually spacelike separated in that background must commute. 20 The reason this must be the case is illustrated in figure 8.…”

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.…”