Hao Geng

A bstract: It has been established in both high and low spacetime dimensions that existing constructions of entanglement islands, at least in the study of black holes, are all in the context of massive gravity. In fact, later studies realized that the graviton mass is not an accident and it is necessary for the consistency of the holographic interpretation for entanglement islands. An important lesson we learned from these studies is that the graviton mass is a manifestation of the deep relationship between quantum entanglement and emergent geometry (EPR=ER). Nevertheless, various interesting questions and counter-arguments exist regarding whether this connection is of any relevance, either by modifying the definition of entanglement islands or denying the graviton mass as a physically relevant concept. In this paper, we will revisit the necessity of graviton mass for the consistency of entanglement islands and address the above questions in and out of the literature. Since the calculations in higher dimensions are already well understood, we will start with a demonstration of massive islands in a frequently used lower dimensional model — AdS ₂ Jackiw-Teitelboim (JT) gravity coupled with transparent conformal matter fields, for which the relationship between energy dissipation and the graviton mass manifests. We will then discuss the necessity of graviton mass for entanglement islands in general situations and clarify the physical meaning of graviton mass. In the end, we will comment on the attempts to construct entanglement islands in massless gravity and suggest the search for a new type of entanglement wedges in holography. The purpose of this paper is to summarize the recent progress, clarifying potential confusions and convey key messages to both experts and non-experts in the field.

A bstract: We provide a derivation of the Ryu-Takayanagi (RT) formula in 3D gravity for generic boundary subregions — including RT surface phase transitions — directly from the dual two-dimensional conformal field theory (CFT). Our approach relies on the universal statistics of the algebraic conformal data and the large- c behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFTs with and without Cardy boundaries. The RT formula is obtained directly from the CFT in the high-temperature regime. Two direct applications are: 1) A simple derivation of the multi-interval entanglement entropy for the vacuum state of a single CFT; 2) A CFT-based detection of the emergence of replica wormholes in the context of entanglement islands and black hole microstate counting. Our framework yields the first holographic random tensor network that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the eigenstate thermalization hypothesis (ETH) in the dual field theory in the large- c limit — a paradigm we refer to as It from ETH .

Some recent work has postulated the existence of an ‘observer’ for a consistent definition of subregion algebras in gravitational universes. The subregion algebras consist of operators dressed to this ‘observer’ and are typically Type II von Neumann algebras. Nevertheless, as opposed to standard physical systems, such an ‘observer’ was postulated to have a Hamiltonian H ^ obs linear in phase space variable. This linear form suggests that the complete dynamics of such an ‘observer’ should also be controlled by an external system or some underlying degrees of freedom within the system. In this paper, we show that this is exactly the case in the island model. In the island model, we have a gravitational asymptotically anti-de Sitter (AdS) spacetime coupled with a non-gravitational bath, and the diffeomorphism symmetries in the gravitational AdS are spontaneously broken due to the bath coupling. In this setup, the ‘observer’ is constructed using the Goldstone vector field associated with the spontaneously broken diffeomorphism symmetry, and the external system that also controls the dynamics of the ‘observer’ is the non-gravitational bath. The basic consistency of the entanglement wedge reconstruction requires operators in the entanglement island to be dressed to this ‘observer’. Thus, we establish the result that entanglement islands correspond to emergent Type II ∞ von Neumann algebras from the holographic dual perspective. This result relies on assuming the geometric modular flow conjecture. Our study also raises a question for earlier constructions of Type II 1 von Neumann algebras.

Motivated by the study of entanglement island in the Karch-Randall braneworld, it has been conjectured and proven in general that entanglement island is not consistent with long-range (massless) gravity. In this paper, we provide a careful check of this conclusion in a model of massless gravity that is constructed using the Karch-Randall braneworld. We show that there is indeed no entanglement island and hence not a nontrivial Page curve due to the diffeomorphism invariance if we are studying the correct question. We will also clarify the subtlety that is important to put this question in a correct manner. Moreover, we show that this conclusion is not affected by deforming the set-up with the Dvali-Gabadadze-Porrati (DGP) terms. Furthermore, we show that the consistency of holography in this model will provide nontrivial constraints to the DGP parameters. This study provides an example that causality and holography in anti-de Sitter space can be used to constrain low energy effective theories. We also clarify several subtleties in the braneworld gravity which were overlooked in the literature. We show that a direct implication of these clarifications is a resolution of the causality paradox in the Karch-Randall braneworld. At the end, we discuss the robustness of the above results against possible coarse-graining protocols to define a subregion in a gravitational theory. Contents 1 Introduction 1 2 No Entanglement Island in Massless Gravity 4 2.1 The Set-up 4 2.2 The Question and the Result without DGP Terms 5 2.3 Adding DGP Terms Doesn't Change the Result 8 2.4 A Question in the Boundary Description 9 3 Consistency of Wedge Holography and the Swampland Bounds 10 4 The Intermediate Picture and Coarse-Graining 11 4.1 The Precise Description of the Intermediate Picture 12 4.2 An Implication-Resolution of the Causality Paradox 15 4.3 The Derivation of the Modified Island Formula 19 4.4 A Potential Puzzle and Its Resolution 21 4.5 Robustness of the Result Under Coarse Graining 22 5 Conclusion 23

A bstract: It has been argued that the existence of a geodesic shortcut in the Karch-Randall (KR) braneworld rules out the possibility for a low-energy intermediate theory, even when gravity is turned off. We study this problem in an explicit example with two symmetrically placed KR branes. We find that there can be a consistent quantization of bulk matter fields with a holographic intermediate description. In our model, there is no causality violation. We use this example to study potential implications of the bulk geodesic shortcut. We find that there are possible enhancements of the correlation or entanglement between the two branes that is directly correlated with the extra-dimensional geometry. This could affect the low-energy regime when there are interactions between low and high energy modes. Our model makes it clear that any possible causality violation can only arise from the UV. Independently of the quantization, an intermediate description should be valid as an EFT.

A bstract: Entanglement islands are subregions in a gravitational universe whose information is fully encoded in a disconnected non-gravitational system away from it. In the context of the black hole information paradox, entanglement islands state that the information about the black hole interior is encoded in the early-time Hawking radiation. Nevertheless, it was unclear how this seemingly nonlocal information encoding scheme emerges from a manifestly local theory. In this paper, we provide an answer to this question by uncovering the mechanism behind this information encoding scheme. As we will see, the early understanding that graviton is massive in island models plays an essential role in this mechanism. As an example, we will discuss how this mechanism works in detail in the Karch-Randall braneworld. This study also suggests the potential importance of this mechanism to the ER=EPR conjecture.

This paper summarizes recent discussions of the Page curve and the information paradox, and responds to the reasoning and examples from arXiv:2506.04311. We review arguments demonstrating that in quantum gravity the algebra of observables at infinity is complete, both in AdS and in asymptotically flat space. This completeness implies that the bulk Hilbert space in quantum gravity does not factorize along the radial direction, undermining a key common assumption in Hawking's argument for information loss and in initial derivations of the Page curve. As a consequence, in a standard theory of gravity, information does not ``emerge'' from a black hole in the manner suggested by the Page curve; rather, it is already encoded in asymptotic observables. Relatedly, the full black hole interior, and not just an ``island'', can be reconstructed from exterior data. Page curves and islands can be obtained by removing the Hamiltonian from the exterior algebra. This may be implemented operationally by restricting access to part of the asymptotic region (a detector with a ``blind spot'') or, in the special case of null infinity in asymptotically flat spacetimes, by formally discarding the Hamiltonian from the set of observables despite its physical accessibility. Such Page curves describe only the redistribution of information between measured and unmeasured degrees of freedom, rather than fundamental information recovery. Finally, Page curves and islands also arise when a black hole is coupled to a nongravitational bath, a setup that yields a nonstandard theory of gravity. We show how, even in this setting, the unusual localization of information in gravity provides a concrete physical mechanism for information transfer from the gravitational system into the bath.

It has been conjectured and proven that entanglement island is not consistent with long-range (massless) gravity in a large class of spacetimes, including typical asymptotically anti-de Sitter spacetimes, in high spacetime dimensions. The conjecture and its proof are motivated by the observation that existing constructions of entanglement islands in high dimensions are all in gravitational theories where the graviton is massive for which the standard gravitational Gauss’ law doesn’t apply. In this letter, we show that this observation persists to lower dimensional cases. We achieve this goal by providing a unified description of the gravitational Gauss’ law violation in island models that can work in any dimensions. This unified description teaches us new lessons on entanglement islands and subregion physics in quantum gravity. We focus on the case of the (1+1)-dimensional Jackiw-Teitelboim (JT) gravity for the purpose of demonstration.