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Register a new userCFT descriptions of bulk local states in the AdS black holes
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We present a new method for reconstructing CFT duals of states excited by the bulk local operators in the three dimensional AdS black holes in the AdS/CFT context. As an important procedure for this, we introduce a map between the bulk points in AdS and those on the boundary where CFT lives. This gives a systematic and universal way to express bulk local states even inside black hole interiors. Our construction allows us to probe the interior structures of black holes purely from the CFT calculations. We analyze bulk local states in the single-sided black holes as well as the double-sided black holes.
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Thermal states in some quantum field theories (QFTs) correspond to black holes in asymptotically AdS spacetime in the AdS/CFT correspondence. We propose a direct procedure to construct holographic images of the black hole in the bulk from a given response function of the QFT on the boundary. The response function with respect to an external source corresponds to the asymptotic data of the bulk field generated by the source on the AdS boundary. According to the wave optics, we can obtain the images from the bulk field propagating in the bulk spacetime. For a thermal state on two-dimensional sphere dual to Schwarzschild-AdS$_4$ black hole, we demonstrate that the holographic images gravitationally lensed by the black hole can be constructed from the response function. In particular, the Einstein rings on the image can be clearly observed and their radius depends on the total energy of the QFT thermal state. These results are consistent with the size of the photon sphere of the black hole calculated in geometrical optics. This implies that, if there exists a dual gravitational picture for a given quantum system, we would be able to probe existence of the dual black hole by the Einstein rings constructed from observables of the quantum system.
The Schwarzschild, Schwarzschild-AdS, and Schwarzschild-de Sitter solutions all admit freely acting discrete involutions which commute with the continuous symmetries of the spacetimes. Intuitively, these involutions correspond to the antipodal map of the corresponding spacetimes. In analogy with the ordinary de Sitter example, this allows us to construct new vacua by performing a Mottola-Allen transform on the modes associated with the Hartle-Hawking, or Euclidean, vacuum. These vacua are the `alpha-vacua for these black holes. The causal structure of a typical black hole may ameliorate certain difficulties which are encountered in the case of de Sitter alpha-vacua. For Schwarzschild-AdS black holes, a Bogoliubov transformation which mixes operators of the two boundary CFTs provides a construction of the dual CFT alpha-states. Finally, we analyze the thermal properties of these vacua.
We develop the representation of free spinor fields in the bulk of Lorentzian anti-de Sitter space in terms of smeared operators in the dual conformal field theory. To do this we expand the bulk field in a complete set of normalizable modes, work out the extrapolate dictionary for spinor fields, and show that the bulk field can be reconstructed from its near-boundary behavior. In some cases chirality and reality conditions can be imposed in the bulk. We study the action of the CFT modular Hamiltonian on bulk fermions to show that they transform with the expected spinor Lie derivative, and we calculate bulk--boundary two-point functions starting from CFT correlators.
It is a long-standing conjecture that any CFT with a large central charge and a large gap $Delta_{text{gap}}$ in the spectrum of higher-spin single-trace operators must be dual to a local effective field theory in AdS. We prove a sharp form of this conjecture by deriving numerical bounds on bulk Wilson coefficients in terms of $Delta_{text{gap}}$ using the conformal bootstrap. Our bounds exhibit the scaling in $Delta_{text{gap}}$ expected from dimensional analysis in the bulk. Our main tools are dispersive sum rules that provide a dictionary between CFT dispersion relations and S-matrix dispersion relations in appropriate limits. This dictionary allows us to apply recently-developed flat-space methods to construct positive CFT functionals. We show how AdS$_{4}$ naturally resolves the infrared divergences present in 4D flat-space bounds. Our results imply the validity of twice-subtracted dispersion relations for any S-matrix arising from the flat-space limit of AdS/CFT.
These lectures review recent developments in our understanding of the emergence of local bulk physics in AdS/CFT. The primary topics are sufficient conditions for a conformal field theory to have a semiclassical dual, bulk reconstruction, the quantum error correction interpretation of the correspondence, tensor network models of holography, and the quantum Ryu-Takayanagi formula.
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