Abstract
We revisit the holographic construction of (approximately) local bulk operators inside an eternal AdS black hole in terms of operators in the boundary CFTs. If the bulk operator carries charge, the construction must involve a qualitatively new object: a Wilson line that stretches between the two boundaries of the eternal black hole. This operator - more precisely, its zero mode - cannot be expressed in terms of the boundary currents and only exists in entangled states dual to two-sided geometries, which suggests that it is a state-dependent operator. We determine the action of the Wilson line on the relevant subspaces of the total Hilbert space, and show that it behaves as a local operator from the point of view of either CFT. For the case of three bulk dimensions, we give explicit expressions for the charged bulk field and the Wilson line. Furthermore, we show that when acting on the thermofield double state, the Wilson line may be extracted from a limit of certain standard CFT operator expressions. We also comment on the relationship between the Wilson line and previously discussed mirror operators in the eternal black hole.
Highlights
We showed that an essential ingredient of the bulk field φis the boundary-to-boundary Wilson line, a pure-gauge configuration that only exists on manifolds with two boundaries and is charged under Q =
We have shown that in order to correctly reproduce bulk perturbation theory in presence of charged operators in the background of an eternal black hole, a new gaugeinvariant operator needs to be included in the holographic dictionary, namely a boundary-toboundary Wilson line
This operator appears to only exist around entangled states of the two CFTs that are dual to connected two-sided geometries, which suggests it is a state-dependent operator
Summary
One of the most remarkable aspects of the AdS/CFT correspondence is that it gives us a definition of quantum gravity in anti-de Sitter space-time [1]. The gauge-invariant bulk operator that we will study throughout this paper is a charged scalar field φ( y), connected via a Wilson line to a point xR the right boundary. The action of the Wilson line on Ψtfd can be entirely determined from its commutators (around Ψtfd) with the low-lying CFT operators and its action on the thermofield double state The former can be inferred from bulk perturbation theory, whereas the latter can be obtained from a path integral argument. In 2.3, after carefully discussing the choice of gauge, we work out the Dirac brackets of the Wilson line with the bulk gauge field and scalar operators Upon quantization, these will yield the commutators of our newly-found operator with the usual low-lying CFT operators around states dual to smooth two-sided geometries
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