Double Holography: A Triple Duality Framework

• Double holography is a framework featuring three equivalent holographic descriptions—bulk, brane, and boundary—that offer a unified view of gravitational systems.
• It employs brane-world constructions with Neumann boundary conditions and induced gravity mechanisms to couple dynamical gravity with non-gravitating quantum field theories.
• The approach provides powerful tools to compute entanglement measures like the Page curve and quantum extremal surfaces, offering insights into the black hole information paradox.

Double holography, also known as doubly-holographic duality, refers to gravitational systems—typically brane-world constructions—where a single setup admits three distinct but equivalent descriptions, each mapping to a different effective field theory or holographic duality. This framework extends the conventional AdS/CFT paradigm by introducing an intermediate brane supporting dynamical gravity and matter, mediating between a higher-dimensional “bulk gravity” perspective and a lower-dimensional non-gravitational quantum field theory. Double holography underpins recent progress in the quantum information approach to semi-classical gravity, providing a unifying context for results on islands, entanglement wedge reconstruction, complementarity, nonlocality, and the Page curve describing the entropy of Hawking radiation.

1. Tripartite Structure: Definitions and Equivalent Descriptions

Double holography is characterized by three interconnected pictures (Ling et al., 2021, Omiya et al., 2021, Karch et al., 2022):

• Bulk (Higher-Dimensional) Perspective:

A $(d+1)$-dimensional asymptotically AdS spacetime with an end-of-the-world (ETW) brane (or Planck brane) supporting dynamical gravity via Neumann (mixed) boundary conditions, possibly including a DGP term for induced gravity (Ling et al., 2021).

• Brane (Intermediate) Picture:

A $d$-dimensional effective theory of dynamical gravity (Einstein-Hilbert action) coupled to conformal matter (CFT) localized on the brane, and further coupled through transparent boundaries to a non-gravitating “bath” CFT on the boundary of AdS (Jiang et al., 19 Dec 2024).

• Boundary (Defect/BCFT) Perspective:

A non-gravitating CFT, possibly with a (codimension-one) defect describing degrees of freedom at the brane–bath interface, with the boundary interpreted as a conformal defect in the ambient CFT (Ling et al., 2021, Liu et al., 2023).

The essence of double holography lies in the equivalence of these perspectives. The full dictionary relating fields and observables across these dualities is encoded via the near-brane expansion of bulk fields, relating brane-localized operators to subleading bulk coefficients (Neuenfeld, 2021).

2. Geometric Foundations and Gravity/Boundary Conditions

The geometric core of double holography is an arrangement where a $(d+1)$-dimensional bulk $N$ is sliced by an end-of-the-world brane $Q$, which intersects the conventional AdS boundary $M$ along a codimension-two locus $P$ (Ling et al., 2021, Ogawa et al., 2022). The bulk action generically takes the form: $$I_{\rm bulk} = \frac{1}{16\pi G^{(d+1)}} \bigg[\int_N (R + \tfrac{d(d-1)}{L^2}) + \text{GH terms} \bigg] + \text{brane-induced gravity/DGP terms}$$

The brane is subject to a Neumann boundary condition of the Israel type,

$$K_{ij} - K h_{ij} + \alpha h_{ij} = \lambda L (\tfrac{1}{2} R_h h_{ij} - R_{h,ij})$$

where $K_{ij}$ is extrinsic curvature, $\alpha$ is related to brane tension, and $\lambda$ encodes the ratio of bulk-to-brane Newton constants, controlling the strength of induced gravity (Liu et al., 2023).

The brane supports its own Einstein–Hilbert action with Newton constant $G_b^{(d)}$, and matter CFTs are coupled both to the brane and to the non-gravitating bath. A second, lower-dimensional locus $P$ frequently supports a conformal defect, allowing application of further holographic dualities (AdS/BCFT or similar) to obtain a non-gravitating CFT description (Liu et al., 2023, Karch et al., 2022).

3. Entanglement, Islands, and the Page Curve

Double holography provides a geometric realization of the quantum extremal surface (QES) and island rules, pivotal in recent resolutions to the black hole information paradox. In the brane picture, the entropy of a bath region is given by

$$S(\mathfrak R) = \min_\Sigma \left[ \frac{\mathrm{Area}(\Sigma)}{4G_d} + S_{\rm bulk}(\mathfrak R \cup I_\Sigma) \right]$$

where the minimization, or “island extremization,” selects the surface $\Sigma$ whose “island” $I_\Sigma$ is included in the entropy computation (Jiang et al., 19 Dec 2024). In the bulk, this is replaced by a standard minimization over HRT (Hubeny–Rangamani–Takayanagi) extremal surfaces in $(d+1)$ dimensions.

In systems with two-sided AdS black holes and non-gravitating baths, the Page curve emerges from a competition between “no-island” vertical membranes (Hartman–Maldacena phase) and “island” membranes (maximal-slope phase), reproducing semi-classical gravity predictions and matching chaotic many-body results via the entanglement membrane formalism (Jiang et al., 19 Dec 2024). This equivalence persists through the “minimize–then–compare” prescription, bridging semiclassical and coarse-grained circuit calculations.

4. Quantum Information Measures and Entanglement Wedge Structures

Double holography allows direct calculation of advanced entanglement measures such as reflected entropy, via a two-step procedure: $$S^R(A:B) = \min_{E_{A:B}} \left[ \frac{\mathrm{Area}(E_{A:B})}{4G_b^{(d)}} + \frac{\mathrm{Area}(E(\Sigma^A : \Sigma^B))}{4G^{(d+1)}} \right]$$ where $E_{A:B}$ is a cross-section on the brane and $E(\Sigma^A : \Sigma^B)$ its lift to the bulk (Ling et al., 2021). This formula embodies both the classical geometry and quantum matter corrections (bulk entanglement) in a geometric language, enabling numerical and analytic interpolation between quantum and classical regimes.

Similar techniques apply to the entanglement and reflected entropy of defect subregions, with critical behavior controlled by the ratio of defect to bath central charges. Subregions experience entanglement phase transitions—collapse of the wedge and vanishing entropy in the subcritical regime, and stable “islands” with logarithmic divergent entropies in the supercritical regime (Liu et al., 2023).

5. Causal Structures, Nonlocality, and Complementarity

The interplay of causality across the three double holography perspectives demonstrates both compatibility and subtle violations:

• Bulk-BCFT compatibility: Causality in the AdS/BCFT picture is maintained, as established by a generalized Gao–Wald theorem—no bulk-induced shortcuts can violate boundary causality (Omiya et al., 2021).
• Intermediate picture nonlocality: The effective brane+bath theory exhibits IR-sensitive bilocality not present in the higher/lower-dimensional pictures, allowing “shortcuts” between brane and bath points that are only causal in the full bulk geometry (Omiya et al., 2021, Karch et al., 2022). This nonlocality arises essentially from wormhole-type saddles in the full gravitational path integral, and is required for the matching of causal structure.
• Entanglement wedge/outer wedge structure: The choice of homology constraint for RT surfaces depends on the picture: the boundary description allows RT surfaces ending on the brane, producing the Page transition, while the brane-only description does not, resulting in a growing entropy without saturation (Neuenfeld, 2021). Operator reconstruction in the brane theory is limited to the outer wedge; only a full quantum (boundary) description can reconstruct the entanglement wedge behind islands.

These features manifest a concrete mechanism for black hole complementarity, wherein two complementary but non-overlapping semiclassical descriptions coexist, each valid in distinct regimes of operator complexity.

6. Realizations in String Theory and Generalizations

Explicit string-theoretic constructions anchor double holography:

• In Type IIB, stacks of D3, D5, and NS5 branes engineer BCFTs and their defects, with near-horizon AdS geometries precisely realizing the threefold structure (Karch et al., 2022). The “proper” intermediate (defect brane gravity) picture is not a subregion of the full BCFT dual, but arises by removing the 4d node and dualizing only the defect, confirming the universality and locality of the construction.
• Generalizations to flat spacetime (“wedge holography”) allow the dual description between gravity on branes embedded in flat $(d+1)$-space and non-unitary CFTs on $S^{d-1}$, closely related to celestial holography (Ogawa et al., 2022).
• Time-dependent and entangled systems (e.g., thermal brane universes in AdS$_3$, TFD states) reveal wormhole phase transitions, mutual information structure, and protocols such as Hayden–Preskill decoding, all within the geometric language of double holography (Myers et al., 26 Mar 2024).

7. Limitations and Outlook

While double holography provides a powerful and unifying language for the interplay between gravity, quantum information, and holography:

• Homology conditions and entropy: The RT prescription depends on the chosen perspective. The island formula applies only when the connected surface is allowed (boundary view), but not if RT surfaces must be purely homologous to the bath (brane view) (Neuenfeld, 2021).
• Causal paradoxes and IR nonlocality: Nonlocality required in the intermediate theory is IR-dominated and unrelated to stringy UV effects, possibly arising from ensemble-averaged wormhole saddles (Omiya et al., 2021).
• Defect-bath phase transitions: The stability of islands is governed by the ratio of defect to bath central charges; insufficient defect central charge leads to a collapse of the entanglement wedge (Liu et al., 2023).
• Microscopic models: Toy models, such as qubit systems with scrambling, effectively reproduce the Page curve via an island minimization principle (Neuenfeld, 2021).
• Open directions: Generalizations to higher-curvature branes, JT gravity reductions, time-dependent or evaporating black holes, and flat-space/celestial holography remain areas of active research (Jiang et al., 19 Dec 2024, Ogawa et al., 2022).

In summary, double holography serves as both a conceptual and computational framework for connecting brane-world gravity, quantum information theory, and holographic QFTs, revealing deep insights into entanglement, nonlocality, and the structure of semi-classical spacetime (Ling et al., 2021, Jiang et al., 19 Dec 2024, Liu et al., 2023, Neuenfeld, 2021, Karch et al., 2022).