Abstract
We consider the algebra of simple operators defined in a time band in a CFT with a holographic dual. When the band is smaller than the light crossing time of AdS, an entire causal diamond in the center of AdS is separated from the band by a horizon. We show that this algebra obeys a version of the Reeh-Schlieder theorem: the action of the algebra on the CFT vacuum can approximate any low energy state in the CFT arbitrarily well, but no operator within the algebra can exactly annihilate the vacuum. We show how to relate local excitations in the complement of the central diamond to simple operators in the band. Local excitations within the diamond are invisible to the algebra of simple operators in the band by causality, but can be related to complicated operators called "precursors". We use the Reeh-Schlieder theorem to write down a simple and explicit formula for these precursors on the boundary. We comment on the implications of our results for black hole complementarity and the emergence of bulk locality from the boundary.
Highlights
Means that we can approximate φ(x1) as accurately as we wish, but true equality can only be obtained by taking the limit of an infinite sequence of polynomials
We argue that the CFT dual of this division is that operators in the CFT can be naturally divided into two classes: simple operators —which consist of single trace operators and polynomials of an O (1) number of single trace operators — and complex operators, where the number of single-trace components starts scaling as a function of N
While the region of AdS near the boundary is reconstructed by simple operators, we argue that the region near the center of AdS is reconstructed by complicated ones
Summary
We consider a large N CFT with a holographic dual, defined on Sd−1 × [time]. We take the CFT in the ground state |0. We consider a time band B in the CFT that is defined to be the set of points Sd−1 ×. As was first explained in [23] and elaborated in [28], we must understand KT as a distribution that is integrated only against correlators in the CFT Leaving aside this subtlety, the bottom line is that we expect local operators in D to be related to simple CFT operators in the time band. The bottom line is that we expect local operators in D to be related to simple CFT operators in the time band Note that no such direct construction is possible for operators inside the diamond D within effective field theory.
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