Static Patch Holography in de Sitter Space

• dS static patch holography is a framework that defines finite-dimensional quantum duals on the cosmological horizon, capturing the observer’s accessible bulk physics and horizon entropy.
• It employs methods such as minisuperspace quantization, finite Hilbert space truncation, and T𝑇̄+Λ₂ deformations to reproduce and refine the Bekenstein–Hawking entropy with explicit microstate counting.
• The approach clarifies the role of operator algebras, entanglement entropy via extremal surfaces, and nonlocal bulk-to-boundary mappings, advancing insights into de Sitter thermodynamics and quantum gravity.

De Sitter (dS) static patch holography is the paper of holographic correspondences in de Sitter spacetimes restricted to the static patch—the causal diamond associated with a single comoving observer. Unlike global dS/CFT, which posits a dual at spacelike conformal infinity, static patch holography seeks a dual quantum description localized on, or in the vicinity of, the cosmological horizon—the observer's holographic screen. This framework aims to define quantum mechanical, finite-dimensional duals that account for the horizon entropy, encode observer-accessible bulk physics, and clarify the microstate structure underlying semi-classical de Sitter thermodynamics and entanglement.

1. Static Patch Geometry and Causal Structure

The static patch of dS$_{d+1}$ is defined by the line element: $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2\,d\Omega_{d-1}^2, \quad f(r) = 1 - \frac{r^2}{L^2}, \quad 0 \leq r < L,$$ where $L$ is the dS radius and $r = L$ is the cosmological horizon. The observer at $r = 0$ experiences a horizon temperature $T = 1/(2\pi L)$ and horizon area $A = \Omega_{d-1}L^{d-1}$. The observer's accessible region is bounded by their causal diamond, covered by a thermal density operator.

The global dS spacetime comprises two antipodal static patches. The causal boundary coincides with the horizon for each observer, motivating proposals for dual quantum theories defined on one or both horizons. In various approaches, these horizons play the role of holographic screens encoding the reduced density matrix for the observer's patch (Franken et al., 2023).

2. Microscopic Duals: Quantum Mechanics and Hilbert Space Structure

Static patch holography postulates that the finite horizon entropy $S_{\rm dS} = A/(4G_N)$ counts the microstates of an intrinsically quantum system associated to the static patch. Several explicit models have been constructed:

• Quantum Mechanical Duals: Minisuperspace reduction followed by canonical quantization of dS-Schwarzschild gravity leads to a dual quantum mechanics, with the partition function $Z_{\rm QM}(g_{tt}, R; c) = \mathrm{Tr}[e^{-icH}]$, where $c$ is conjugate to the black hole mass and plays the role of emergent time. The dual admits a centrally extended $\mathfrak{e}(1,1)$ symmetry and reproduces the Bekenstein–Hawking entropy via an averaging procedure over the quasi-local metric (Blacker et al., 2023).
• Finite Hilbert Space: Holographic models systematically truncate the Hilbert space to be finite-dimensional, matching the entropy scaling. For higher-spin/Sp($2N$) models, the observer's patch maps to a subset of modes associated with antipodal identification, yielding a finite number of degrees of freedom (Neiman, 2018, Karch et al., 2013).
• $T\bar T + \Lambda_2$ Flows: In $dS_3$, the dual field theory on the cylinder undergoes an irrelevant deformation by $T\bar T + \Lambda_2$, implementing a cutoff at the cosmological horizon. The real spectrum of the deformed theory precisely matches the refined entropy count, including logarithmic corrections (Batra et al., 2 Mar 2024, Chang et al., 20 Nov 2025).

The above frameworks consistently realize $\dim\mathcal{H} \sim \exp(S_{\rm dS})$ for the static patch, with late-time complexity growth saturating only on timescales $\sim\exp(\mathcal{O}(S_{\rm dS}))$ (Mohan et al., 13 Aug 2025).

3. Operator Algebras, Symmetries, and Quantum Structure

The quantum systems dual to the static patch exhibit operator algebras and symmetry structures that interpolate between the global dS isometries and residual symmetries of the patch:

• Higher-Spin/Free Vector Models: For higher-spin holography, the static patch symmetry group $SO(1,1)\times O(d-1)$ results from observer-dependent breaking of the global $SO(1, d+1)$ or corresponding higher-spin extensions. Quantum commutator algebra, Hamiltonian structure, and operator expansions are directly constructed from boundary CFT correlators via Wigner–Weyl and Penrose transforms (Neiman, 2018).
• Central Extensions: In gravitational minisuperspace models, the Hamiltonian, dilation, and translation operators close into a centrally extended $\mathfrak{e}(1,1)$ algebra, crucial for encoding the gravitational phase and entropy (Blacker et al., 2023).
• Thermal Structure and Entanglement: The Bunch–Davies vacuum reduced to the static patch yields a thermal density matrix at $T = 1/(2\pi L)$, connecting the observer's physics to the (entangled) degrees of freedom on the horizon. In antipodally-identified models, the patch's state is a thermofield double between the two horizons (Neiman, 2017, Franken et al., 2023).

Bulk field reconstruction in the patch is implemented by matching mode expansions between bulk and boundary (or horizon) modes, and in higher-spin theories, this extends to the unfolded/twistor formalism (Neiman, 2018).

4. Entanglement Entropy, Minimal Surfaces, and the Bilayer Proposal

The entanglement structure of static patch holography is captured by extremal (Ryu–Takayanagi-like) surfaces extending from the screens/horizons:

• Bulk Minimal Surfaces: In both DS/dS and codimension-2 defect approaches (e.g., Liouville description on $\mathbb{Z}_q$-fixed spheres), extremal surfaces anchored to horizons yield, via area/entropy relations, the Gibbons–Hawking entropy. One-parameter families of extremal surfaces exist, interpolating between “horizon-hugging” (across the horizon) and “global-slice” (inter-patch) types, all with identical area in the pure gravity case (Geng et al., 2019, Franken et al., 2023, Arias et al., 2019).
• Bilayer Prescription: Entanglement basis comes from two coupled horizon screens—each encoding its associated patch—such that mutual entanglement builds the bridge (“barrel”) region between them. Quantum extremal surface transitions (phase transitions in the entanglement wedge) indicate the nonlocal, dynamically encoded geometry of the full spacetime (Franken et al., 2023).
• Quantum Corrections and Matter: In the presence of matter satisfying the null energy condition, the entropy associated with tracing out one CFT/inter-patch region can exceed the Gibbons–Hawking value, signaling potential swampland bounds for dS holography with matter (Geng et al., 2019).

These results establish that the static patch entanglement wedge structure is intimately tied to the encoding capacity of the horizon screens and that its fine structure is regulated by quantum and matter contributions.

5. Exact Models: Liouville Theory and $T\bar T + \Lambda_2$ Deformations

Several solvable models elucidate static patch holography:

• Liouville/Replica Construction: In $dS_4$, a $\mathbb{Z}_q$ orbifold introduces codimension-2 defects whose effective dynamics are governed by Liouville field theory on the singular 2-spheres. The modular/replica entropy derived from the Liouville action yields the correct Gibbons–Hawking entropy, and in the $q\to\infty$ limit, reduces to dS$_3$/CFT$_2$ holography with central charge matching the Cardy formula (Arias et al., 2019).
• Composite $T\bar T + \Lambda_2$ Flows: In $dS_3$, the composite irrelevant deformation interpolates between T$\bar T$ (for spatial/spacelike cutoffs outside the horizon) and T$\bar T+\Lambda_2$ (for timelike/horizon cutoffs inside). This flow tracks the movement of a boundary from future infinity into the static patch, ensuring continuity of both energy and entanglement entropy across the cosmological horizon and providing a unifying RG framework connecting the two holographic regimes (Chang et al., 20 Nov 2025, Batra et al., 2 Mar 2024).

The structure of the spectrum and entropy in these models is robust under matter perturbations, with precise microstate counting agreeing with bulk expectations and quantum corrections.

6. Bulk Locality, Holographic Screens, and Entanglement Wedge Connectivity

Static patch holography imposes nontrivial constraints on the bulk-to-boundary dictionary and entanglement wedge reconstruction:

• Connected Wedge Theorem: In $dS_3$, the connected wedge theorem relates bulk scattering between points inside the patch to mutual information between screen subregions of order $O(1/G_N)$, requiring the entanglement wedge to be connected for operationally meaningful holographic encoding (Franken et al., 11 Oct 2024).
• Nonlocal Operator Mapping: Unlike AdS/CFT, where boundary local operators map to bulk local wavepackets, in static patch holography a bulk local excitation corresponds to a nonlocal (smeared) operator on the holographic screen, reflecting the causal structure induced by lightcones from null infinity (Franken et al., 11 Oct 2024).
• Wedge Holography and Multibrane Constructions: Generalized wedge holography places the static patch as a region between two end-of-the-world branes. Via double holographic constructions, the dual is a lower-dimensional quantum gravity theory or a defect CFT, with mutual communication between universes possible by brane gluing (Yadav, 31 Mar 2024).

These constructions ensure causal consistency, capture the encoding of observer-accessible information, and provide mechanisms for the emergence of communicating multiverse sectors.

7. Extensions: Complexity and Quantum Chaos, Gauge Fields, and Lower-Dimensional Models

• Holographic Complexity: The late-time growth of holographic complexity in the static patch, defined as the regulated volume of extremal timelike surfaces anchored to the observer's worldline or stretched horizon, is linear in proper time and proportional to the horizon entropy. This scaling supports a chaotic, finite-dimensional quantum system as the dual, with saturation timescales exponential in $S_{\rm dS}$ (Mohan et al., 13 Aug 2025).
• Bulk Gauge Fields and Open Quantum Systems: Static patch holography for bulk electromagnetism systematically encodes Hawking radiation, radiation reaction, and the open-system influence phase in the observer’s static patch through Schwinger–Keldysh techniques. These results extend to higher-spin fields and clarify the boundary conditions and renormalization procedures necessary for observer-centric open holographic duality (Loganayagam et al., 28 Feb 2025).
• Lower-Dimensional Realizations: The “centaur” geometry—a solution interpolating between AdS$_2$ in the UV and dS$_2$ static patch in the IR—provides a 2D toy model showing how patch-restricted observables and entropy can be reproduced holographically as a thermal state in dual conformal quantum mechanics (Anninos et al., 2017).

This synthesis demonstrates the wide applicability and concrete realizability of static patch holography across diverse setups, including deformed CFTs, defect models, and open-system approaches.

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In total, dS static patch holography offers a robust framework for the quantum gravitational microphysics of an observer's causal patch. Models constructed in this paradigm exhibit entropy-encoding consistent with the Gibbons–Hawking law, accommodate observer-accessible correlation functions, and introduce finite-dimensional chaotic dynamics suggestive of a closed quantum system with a sharply bounded Hilbert space. The connections between patch, global, and cosmological boundary holography are underpinned by explicit flows ($T\bar T + \Lambda_2$), extremal surface structure, higher-spin duals, and careful treatment of bulk gauge and matter sectors (Chang et al., 20 Nov 2025, Franken et al., 2023, Batra et al., 2 Mar 2024, Mohan et al., 13 Aug 2025, Neiman, 2018, Neiman, 2017, Arias et al., 2019).