de Sitter and Cosmological-Horizon Holography

De Sitter holography has several calculable interfaces but no established complete duality. Late-time wave functions organize data at spacelike infinity; static-patch algebras organize observables for one observer; horizon thermodynamics fixes temperature and entropy; lower-dimensional models reproduce selected spectra or correlators; and string constructions ask whether the background itself is controlled. These questions are complementary, not interchangeable. This chapter compares them under explicit choices of dimension, region, vacuum, observable class, and approximation, with all microscopic claims labeled proposal-level as of 2026-08-20.

Helpful background. Gauge Theory and String-Theory Preparation supplies constrained observables and analytic continuation; Quantum Gravity frames the duality problem; Quantum Information supplies algebra and entropy distinctions; and Causality and Global Structure supplies spacelike conformal boundaries and causal diamonds.

One geometry, several formulations

Global $(d+1)$-dimensional de Sitter space has

$$ds^2=-dt^2+L^2\cosh^2(t/L)d\Omega_d^2,$$

with spacelike past and future conformal boundaries. A geodesic observer instead uses a static patch bounded by a cosmological horizon,

$$ds^2=-\left(1-\frac{r^2}{L^2}\right)dt_s^2 +\frac{dr^2}{1-r^2/L^2}+r^2d\Omega_{d-1}^2.$$

The first description invites a wave functional of late-time fields; the second invites an observer algebra and KMS state at $T=1/(2\pi L)$, normalized to the geodesic observer at $r=0$. They do not ask for the same observable. Witten 2001, pp. 1–15 emphasizes the absence of a globally timelike positive energy and the difference between mathematical asymptotic quantities and measurements available to one observer.

This is also why an AdS dictionary cannot simply be continued by changing the sign of $L^2$. The D3 and D1–D5 examples rely on timelike AdS boundaries and controlled decoupling limits; NHEK relies on an extremal throat and explicit falloffs. De Sitter has neither structure globally. Analytic continuation can relate saddle calculations, but the contour, inner product, state space, and operational meaning must be supplied separately.

Read the chapter in this order

1. de Sitter Geometry and Holographic Obstacles compares global and static coordinates, derives the horizon temperature, and identifies the failed AdS reconstruction assumption.
2. dS/CFT Wave-Function Proposals distinguishes $\Psi[\phi_0]$ from a thermal partition function and tracks mass-dependent conformal weights and reality conditions.
3. Static-Patch Holography compares type-III, type-II$_1$, and finite-matrix descriptions and includes the 2026 negative-shock trace test.
4. Cosmological-Horizon Entropy and Microstates derives de Sitter and Schwarzschild–de Sitter thermodynamics while separating entropy from demonstrated degeneracy.
5. Matrix, SYK-Like, and Lower-Dimensional Models tests spectral, entropy, and correlator matches under one declared ensemble and dictionary.
6. Control Criteria for de Sitter String Constructions requires independent loop, $\alpha'$, scale-separation, localization, tadpole, stability, and decay checks.

The sequence moves from geometric facts to two proposed observable frameworks, then to entropy, toy-model evidence, and ultraviolet existence. A successful lower-dimensional model does not prove a controlled four-dimensional compactification; a controlled compactification would not by itself construct a holographic dual.

Results and limitations through 2026-08-20

The formulation axes are dimension, global or static region, observer, vacuum, asymptotic conditions, contour, algebra, entropy regulator, microscopic model, and compactification control.

Evidence or position	Controlled result	Limitation or adverse discriminator
dS/CFT proposal, Strominger 2001, §§5–8, pp. 7–14	Asymptotic symmetries and free scalar calculations yield Euclidean conformal structures at spacelike infinity	Complex weights and observer inaccessibility obstruct an unqualified conventional unitary-CFT interpretation
Static-patch observer algebra, Chandrasekaran, Longo, Penington, and Witten 2023, §§2–4, pp. 9–34	Semiclassical constraints with an observer yield a type-II$_1$ algebra and generalized-entropy relation	The factor is infinite-dimensional and does not supply a unique nonperturbative microscopic theory
Negative-shock adverse test, Chen, Stanford, Tang, and Yang 2026, §§2–4, pp. 8–31	Eikonal OTOCs including recoil and backreaction directly test a Euclidean trace proposal	Cyclicity and positivity conflict for the tested conjecture; the result does not exclude every modified observer algebra
Lower-dimensional models	Exact quantum-mechanical spectra and selected $\mathrm{dS}_3$ correlator or entropy matches provide controlled laboratories	Ensemble choice, incomplete observable maps, and absent higher-dimensional embeddings prevent a unique-dual conclusion
String-background candidates, Andriot and Ruehle 2024, §§2.3–4.3, pp. 14–33	Quantized classical searches and scaling analyses test radii, volume, and correction control in explicit compactifications	Enlarging total volume need not control every radius, localization, scale hierarchy, uplift, or decay channel

The controlled core consists of geometry, semiclassical thermodynamics, free or perturbative wave-function data, specific observer-algebra constructions, and exact calculations in declared toy models. Competing positions favor a global wave-function dual, a finite static-patch system, an algebraic observer theory, or a lower-dimensional microscopic model. No position has established a complete, unique, nonperturbative de Sitter dual. The unresolved limitation is simultaneous control of state space, observables, probability, horizons, and ultraviolet background.

Cross-checks across proposals

Every proposal must reproduce the same horizon temperature for the same observer and Killing-time normalization in the appropriate vacuum. A detector held away from $r=0$ measures the redshifted value $T_{\rm loc}=T/\sqrt{1-r^2/L^2}$; there is no finite universal detector temperature “at” the horizon. Agreement on temperature is weak evidence for a microscopic dual because it follows from geometry. Entropy is stronger only when the ensemble and additive convention match. A microscopic claim should identify which state space gives $A/(4G)$, reproduce its changes after inserting energy, and explain the Nariai limit.

Wave-function and static-patch claims require different conjugations. A coefficient in $\log\Psi$ becomes a probability contribution only through $|\Psi|^2$; a static-patch trace must satisfy cyclicity and positivity. A model that passes one check need not pass the other. Likewise, a type-II$_1$ trace is not proof of finite Hilbert-space dimension.

For string constructions, weak coupling and low curvature must be tested separately. A positive effective critical point remains a candidate until localized higher-dimensional equations, quantized tadpoles, the full mass spectrum, scale separation, and leading corrections are controlled. Holographic observables are then an additional layer, not evidence that can repair an uncontrolled background.

What could change these conclusions

The conclusions could change if a theorem constructs a complete positive-norm de Sitter state space and observer algebra, if an independent calculation reproduces or overturns the negative-shock result, or if a controlled microscopic model derives horizon entropy and correlators without fitted identifications. The conclusions also depend on whether a new nonperturbative wave-function contour, compactification family, no-go theorem, decay channel, or observational protocol changes one formulation’s domain.

Review the chapter

A proposed model matches $T=1/(2\pi L)$, one scalar two-point function, and the leading horizon entropy, but does not define an observer algebra and has no controlled higher-dimensional embedding. Determine the strongest justified claim and the next two decisive tests.

Solution

The model is a useful lower-dimensional interface because it matches three quantities in a declared regime, but the temperature is largely geometric and one entropy number may depend on normalization. Without an observer algebra, it has not defined all accessible observables or tested trace positivity; without an embedding, it has not described controlled higher-dimensional de Sitter gravity. The next tests should compute a real-time higher-point observable, including the negative-shock cyclicity and positivity discriminator when applicable, and derive a full state-space or spectral density relation that reproduces entropy variations. A separate compactification analysis would still be required before making a string-background claim.

References

• Andriot, David, and Fabian Ruehle. “On Classical de Sitter Solutions and Parametric Control.” Preprint (2024). arXiv:2403.07065.
• Chandrasekaran, Venkatesa, Roberto Longo, Geoff Penington, and Edward Witten. “An Algebra of Observables for de Sitter Space.” Journal of High Energy Physics 02 (2023) 082. arXiv:2206.10780.
• Chen, Yiming, Douglas Stanford, Haifeng Tang, and Zhenbin Yang. “Negative Shocks versus Static Patch Holography.” Preprint (2026). arXiv:2607.14042.
• Strominger, Andrew. “The dS/CFT Correspondence.” Journal of High Energy Physics 10 (2001) 034. arXiv:hep-th/0106113.
• Witten, Edward. “Quantum Gravity in de Sitter Space.” In Strings 2001. arXiv:hep-th/0106109.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• de Sitter and Cosmological-Horizon Holography