The Black Hole Interior in AdS/CFT and the Information Paradox

We show that, within the AdS/CFT correspondence, recent formulations of the information paradox can be reduced to a question about the existence of certain kinds of operators in the CFT. We describe a remarkably simple construction of these operators on a given state of the CFT. Our construction leads to a smooth horizon, addresses the strong subadditivity paradox, while preserving locality within effective field theory, and reconciles the existence of the interior with the growth of states with energy in the CFT. We also extend our construction to non-equilibrium states.Comment: 5 pages; v2: clarified discussion of conserved charges. minor change in notatio

State-Dependent Bulk-Boundary Maps and Black Hole Complementarity

We provide a simple and explicit construction of local bulk operators that describe the interior of a black hole in the AdS/CFT correspondence. The existence of these operators is predicated on the assumption that the mapping of CFT operators to local bulk operators depends on the state of the CFT. We show that our construction leads to an exactly local effective field theory in the bulk. Barring the fact that their charge and energy can be measured at infinity, we show that the commutator of local operators inside and outside the black hole vanishes exactly, when evaluated within correlation functions of the CFT. Our construction leads to a natural resolution of the strong subadditivity paradox of Mathur and Almheiri et al. Furthermore, we show how, using these operators, it is possible to reconcile small corrections to effective field theory correlators with the unitarity of black hole evaporation. We address and resolve all other arguments, advanced in arxiv:1304.6483 and arxiv:1307.4706, in favour of structure at the black hole horizon. We extend our construction to states that are near equilibrium, and thereby also address the "frozen vacuum" objections of arxiv:1308.3697. Finally, we explore an intriguing link between our construction of interior operators and Tomita-Takesaki theory.Comment: (v1) 92 pages; mathematica file included with source. (v2) 97 pages; added discussion of non-Abelian symmetries; fixed typo

Comments on the Necessity and Implications of State-Dependence in the Black Hole Interior

We revisit the "state-dependence" of the map that we proposed recently between bulk operators in the interior of a large AdS black hole and operators in the boundary CFT. By refining recent versions of the information paradox, we show that this feature is necessary for the CFT to successfully describe local physics behind the horizon --- not only for single-sided black holes but even in the eternal black hole. We show that state-dependence is invisible to an infalling observer who cannot differentiate these operators from those of ordinary quantum effective field theory. Therefore the infalling observer does not observe any violations of quantum mechanics. We successfully resolve a large class of potential ambiguities in our construction. We analyze states where the CFT is entangled with another system and show that the ER=EPR conjecture emerges from our construction in a natural and precise form. We comment on the possible semi-classical origins of state-dependence.Comment: 136 pages, 16 figure

Quantum teleportation through time-shifted AdS wormholes

Based on the work of Gao-Jafferis-Wall and Maldacena-Stanford-Yang, we observe that the time-shifted thermofield states of two entangled CFTs can be made traversable by an appropriate coupling of the two CFTs, or alternatively by the application of a modified quantum teleportation protocol. This provides evidence for the smoothness of the horizon for a large class of entangled states related to the thermofield by time-translations. The smoothness of these states has some relevance for the firewall paradox and the proposal that some observables in quantum gravity may be state-dependent. We notice that quantum teleportation through these entangled states could be used in a laboratory setup to implement a time-machine, which allows the observer to travel far in the future.Comment: 18 page

Exact correlation functions in SU(2) N=2 superconformal QCD

We report an exact solution of 2- and 3-point functions of chiral primary fields in SU(2) N=2 super-Yang-Mills theory coupled to four hypermultiplets. It is shown that these correlation functions are non-trivial functions of the gauge coupling, obeying differential equations which take the form of the semi-infinite Toda chain. We solve these equations recursively in terms of the Zamolodchikov metric that can be determined exactly from supersymmetric localization on the four-sphere. Our results are verified independently in perturbation theory with a Feynman diagram computation up to 2-loops. This is a short version of a companion paper that contains detailed technical remarks, additional material and aspects of an extension to SU(N) gauge group.Comment: 5 page

AdS/CFT and the cosmological constant problem

Within the context of the AdS/CFT correspondence we attempt to formulate the cosmological constant problem in the dual conformal field theory. The fine-tuning of the bulk cosmological constant is related to an apparent fine-tuning in the effective description of the CFT in terms of its light operators: while the correlators of single particle operators satisfy a large N expansion, the expansion does not appear to be natural. Individual terms contributing to correlators have parametrically larger value than the one dictated by large N counting. The final 1/N suppression of correlators is achieved via delicate cancellations between such terms. We speculate on the existence of underlying principles which might make the bulk theory (secretly) natural.Comment: 38 page

Nonsupersymmetric Flux Vacua and Perturbed N=2 Systems

We geometrically engineer N=2 theories perturbed by a superpotential by adding 3-form flux with support at infinity to local Calabi-Yau geometries in type IIB. This allows us to apply the formalism of Ooguri, Ookouchi, and Park [arXiv:0704.3613] to demonstrate that, by tuning the flux at infinity, we can stabilize the dynamical complex structure moduli in a metastable, supersymmetry-breaking configuration. Moreover, we argue that this setup can arise naturally as a limit of a larger Calabi-Yau which separates into two weakly interacting regions; the flux in one region leaks into the other, where it appears to be supported at infinity and induces the desired superpotential. In our endeavor to confirm this picture in cases with many 3-cycles, we also compute the CIV-DV prepotential for arbitrary number of cuts up to fifth order in the glueball fields.Comment: 70 pages (47 pages + 4 appendices), 10 figure

Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories

We present some new exact results for general four-dimensional superconformal field theories. We derive differential equations governing the coupling constant dependence of chiral primary correlators. For N=2 theories we show that the Zamolodchikov metric on the moduli space and the operator mixing of chiral primaries are quasi-topological quantities and constrained by holomorphy. The equations that we find are the four-dimensional analogue of the tt* equations in two-dimensions, discovered by the method of "topological anti-topological fusion" by Cecotti and Vafa. Our analysis relies on conformal perturbation theory and the superconformal Ward identities and does not use a topological twist.Comment: 43 pages (29 pages + 4 appendices), 2 figures, harvmac, v2: fixed typos, improved some derivations, numerical coefficients in tt* equations correcte

A toy model of black hole complementarity

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.Comment: 24 page