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A bstract Thermal partition functions for gravitational systems have traditionally been studied using Euclidean path integrals. But in Euclidean signature the gravitational action suffers from the conformal factor problem, which renders the action unbounded below. This makes it difficult to take the Euclidean formulation as fundamental. However, despite their familiar association with periodic imaginary time, thermal gravitational partition functions can also be described by real-time path integrals over contours defined by real Lorentzian metrics. The one caveat is that we should allow certain codimension-2 singularities analogous to the familiar Euclidean conical singularities. With this understanding, we show that the usual Euclidean-signature black holes (or their complex rotating analogues) define saddle points for the real-time path integrals that compute our partition functions. Furthermore, when the black holes have positive specific heat, we provide evidence that a codimension-2 subcontour of our real Lorentz-signature contour of integration can be deformed so as to show that these black holes saddles contribute with non-zero weight to the semiclassical limit, and that the same is then true of the remaining two integrals.

A bstract We use a microcanonical path integral closely related to that introduced by Brown and York in 1992 to add new entries to the AdS/CFT dictionary concerning the interiors of small black holes. Stationary points of such path integrals are also stationary points of more standard canonical-type path integrals with fixed boundary metric, but the condition for dominance is now maximizing Hubeny-Rangamani-Takayanagi entropy at fixed energy. As a result, such path integrals can bring to the fore saddles that fail to dominate in more familiar contexts. We use this feature to argue that the standard Kruskal-like two-sided extension of small AdS black holes with energy E 0 is dual to a microcanonical version of the thermofield double state for AdS black holes that maximize the microcanonical bulk entropy at this energy. We also comment on entanglement in such states and on quantum effects that become large when the energy-width of the microcanonical ensemble is sufficiently small.

A bstract In the 1980’s, work by Coleman and by Giddings and Strominger linked the physics of spacetime wormholes to ‘baby universes’ and an ensemble of theories. We revisit such ideas, using features associated with a negative cosmological constant and asymptotically AdS boundaries to strengthen the results, introduce a change in perspective, and connect with recent replica wormhole discussions of the Page curve. A key new feature is an emphasis on the role of null states. We explore this structure in detail in simple topological models of the bulk that allow us to compute the full spectrum of associated boundary theories. The dimension of the asymptotically AdS Hilbert space turns out to become a random variable Z , whose value can be less than the naive number k of independent states in the theory. For k > Z , consistency arises from an exact degeneracy in the inner product defined by the gravitational path integral, so that many a priori independent states differ only by a null state. We argue that a similar property must hold in any consistent gravitational path integral. We also comment on other aspects of extrapolations to more complicated models, and on possible implications for the black hole information problem in the individual members of the above ensemble.

A bstract The CPT map allows two states of a quantum field theory to be sewn together over CPT-conjugate partial Cauchy surfaces R 1 , R 2 to make a state on a new spacetime. We study the holographic dual of this operation in the case where the original states are CPT-conjugate within R 1 , R 2 to leading order in the bulk Newton constant G , and where the bulk duals are dominated by classical bulk geometries g 1 , g 2 . For states of fixed area on the R 1 , R 2 HRT-surfaces, we argue that the bulk geometry g 1 # g 2 dual to the newly sewn state is given by deleting the entanglement wedges of R 1 , R 2 from g 1 , g 2 , gluing the remaining complementary entanglement wedges of R ¯ 1 , R ¯ 2 together across the HRT surface, and solving the equations of motion to the past and future. The argument uses the bulk path integral and assumes it to be dominated by a certain natural saddle. For states where the HRT area is not fixed, the same bulk cut-and-paste is dual to a modified sewing that produces a generalization of the canonical purification state ρ discussed recently by Dutta and Faulkner. Either form of the construction can be used to build CFT states dual to bulk geometries associated with multipartite reflected entropy.

A bstract We reformulate recent insights into black hole information in a manner emphasizing operationally-defined notions of entropy, Lorentz-signature descriptions, and asymptotically flat spacetimes. With the help of replica wormholes, we find that experiments of asymptotic observers are consistent with black holes as unitary quantum systems, with density of states given by the Bekenstein-Hawking formula. However, this comes at the cost of superselection sectors associated with the state of baby universes. Spacetimes studied by Polchinski and Strominger in 1994 provide a simple illustration of the associated concepts and techniques, and we argue them to be a natural late-time extrapolation of replica wormholes. The work aims to be self-contained and, in particular, to be accessible to readers who have not yet mastered earlier formulations of the ideas above.

A bstract Bulk quantum fields are often said to contribute to the generalized entropy A 4 G N + S bulk only at O (1). Nonetheless, in the context of evaporating black holes, O (1 /GN ) gradients in S bulk can arise due to large boosts, introducing a quantum extremal surface far from any classical extremal surface. We examine the effect of such bulk quantum effects on quantum extremal surfaces (QESs) and the resulting entanglement wedge in a simple two-boundary 2 d bulk system defined by Jackiw-Teitelboim gravity coupled to a 1+1 CFT. Turning on a coupling between one boundary and a further external auxiliary system which functions as a heat sink allows a two-sided otherwise-eternal black hole to evaporate on one side. We find the generalized entropy of the QES to behave as expected from general considerations of unitarity, and in particular that ingoing information disappears from the entanglement wedge after a scambling time β 2 π log ΔS + O 1 in accord with expectations for holographic implementations of the Hayden-Preskill protocol. We also find an interesting QES phase transition at what one might call the Page time for our process.

A bstract Recent work showed holographic error correcting codes to have simple universal features at O (1 /G ). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order ( O (1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G , ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.

A bstract We use the Einstein-Hilbert gravitational path integral to investigate gravita- tional entanglement at leading order O (1 /G ). We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies Sn are independent of n at this order. The n -dependence of Sn in more general states then arises from sums over the RT/HRT- area, which are generally dominated by different values of this area for each n . This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of alge- braic centers which arise in the quantum error-correcting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.

A bstract We use Euclidean path integrals to explore the set of bulk asymptotically AdS spacetimes with good CFT duals. We consider simple bottom-up models of bulk physics defined by Einstein-Hilbert gravity coupled to thin domain walls and restrict to solutions with spherical symmetry. The cosmological constant is allowed to change across the domain wall, modeling more complicated Einstein-scalar systems where the scalar potential has multiple minima. In particular, the cosmological constant can become positive in the interior. However, in the above context, we show that inflating bubbles are never produced by smooth Euclidean saddles to asymptotically AdS path integrals. The obstacle is a direct parallel to the well-known obstruction to creating inflating universes by tunneling from flat space. In contrast, we do find good saddles that create so-called “bag-of-gold” geometries which, in addition to their single asymptotic region, also have an additional large semi-classical region located behind both past and future event horizons. Furthermore, without fine-tuning model parameters, using multiple domain walls we find Euclidean geometries that create arbitrarily large bags-of-gold inside a black hole of fixed horizon size, and thus at fixed Bekenstein-Hawking entropy. Indeed, with our symmetries and in our class of models, such solutions provide the unique semi-classical saddle for appropriately designed (microcanonical) path integrals. This strengthens a classic tension between such spacetimes and the CFT density of states, similar to that in the black hole information problem.

A bstract Recent results suggest that new corrections to holographic entanglement entropy should arise near phase transitions of the associated Ryu-Takayanagi (RT) surface. We study such corrections by decomposing the bulk state into fixed-area states and conjecturing that a certain ‘diagonal approximation’ will hold. In terms of the bulk Newton constant G , this yields a correction of order O ( G − 1 / 2 ) near such transitions, which is in particular larger than generic corrections from the entanglement of bulk quantum fields. However, the correction becomes exponentially suppressed away from the transition. The net effect is to make the entanglement a smooth function of all parameters, turning the RT ‘phase transition’ into a crossover already at this level of analysis. We illustrate this effect with explicit calculations (again assuming our diagonal approximation) for boundary regions given by a pair of disconnected intervals on the boundary of the AdS 3 vacuum and for a single interval on the boundary of the BTZ black hole. In a natural large-volume limit where our diagonal approximation clearly holds, this second example verifies that our results agree with general predictions made by Murthy and Srednicki in the context of chaotic many-body systems. As a further check on our conjectured diagonal approximation, we show that it also reproduces the O ( G − 1 / 2 ) correction found Penington et al. for an analogous quantum RT transition. Our explicit computations also illustrate the cutoff-dependence of fluctuations in RT-areas.