Null geodesics, local CFT operators, and AdS/CFT for subregions

Abstract

We investigate the nature of the AdS/CFT duality between a subregion of the bulk and its boundary. In global AdS/CFT in the classical G N ¼ 0 limit, the duality reduces to a boundary value problem that can be solved by restricting to one-point functions of local operators in the conformal field theory (CFT). We show that the solution of this boundary value problem depends continuously on the CFT data. In contrast, the anti-de Sitter (AdS)-Rindler subregion cannot be continuously reconstructed from local CFT data restricted to the associated boundary region. Motivated by related results in the mathematics literature, we posit that a continuous bulk reconstruction is only possible when every null geodesic in a given bulk subregion has an endpoint on the associated boundary subregion. This suggests that a subregion duality for AdS-Rindler, if it exists, must involve nonlocal CFT operators in an essential way.

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

“…However, as we will demonstrate, at large momenta the imprint these tunneling modes leave at the boundary is exponentially small and as a consequence, a smearing function cannot be constructed. Our arguments closely follow those of [38,39], where first steps towards generalizing smearing functions to spaces other than pure AdS were made. It is important to note that the failure to construct a smearing function, which would connect local bulk data to local boundary data, does not necessarily prevent reconstruction of the bulk metric from nonlocal boundary data, such as JHEP01(2014)062 the 2-point boundary correlators considered in [40][41][42][43][44][45][46].…”

“…Therefore, one might wonder if the inability to construct smearing functions is simply due to the presence of singularities. This question has been raised before in the case of black hole solutions in AdS 14 [38,39]. Fortunately, in our case there are known ways to resolve the singularity, so we can directly test the conjecture that non-existence of smearing functions is related to singularities.…”

Bulk locality and quantum error correction in AdS/CFT

“…We will observe that this construction has several paradoxical features, which we will illuminate by recasting it on the CFT side in the language of quantum error correcting codes [9,10]. This language gives a new, more general perspective on the issue of bulk reconstruction, and we believe that it is the natural framework for understanding the idea of "subregion-subregion" duality [11][12][13][14]. In particular, the radial direction in the bulk is realized in the CFT as a measure of how well CFT representations of bulk quantum information are protected from local erasures.…”

“…To get a set of operators that really act within H C we can include projection operators onto H C on both sides of φ i ; these will be irrelevant except in studying high-point correlation functions, so we will not carry them around explicitly here. 13 Now consider a decomposition of the boundary Cauchy surface Σ into A and A. If our code subspace H C can protect against the erasure of A, then by our condition (3.21) it must be that we can find a representation of any operator on H C with support only in A.…”

“…The only fundamental limitation on the AdS-Rindler reconstruction comes from the backreaction considerations we discuss in section 5 below. 13 A subtlety here is that the states (4.1) are not mutually orthogonal; a state of the form φ m |Ω has a normalized overlap with the vacuum of order Ω|φ m |Ω √ Ω|φ 2m |Ω ∼ 2 −m . The scaling with m follows from counting Wick contractions at leading order in 1/N .…”