dS/CFT Duality: From AdS Analytic Continuation

Updated 6 April 2026

dS/CFT duality is a framework that relates quantum gravity in de Sitter space to nonunitary conformal field theories at its boundaries via analytic continuation from AdS.
It employs higher-spin gravity and static patch cut-off models to derive thermodynamic entropy, late-time correlators, and spectral data that match Bekenstein-Hawking results.
Key constructs include pseudoentropy, complex central charges, and entanglement structures akin to ER=EPR, highlighting its mathematical subtlety and physical richness.

The dS/CFT duality is a conjectured framework relating quantum gravity in de Sitter (dS) spacetimes to conformal field theories (CFTs) defined at the spacelike future (or past) infinity of dS. Rooted in the paradigm established by AdS/CFT, dS/CFT replaces the anti-de Sitter boundary with dS conformal infinity $\mathscr{I}^+$ (or $\mathscr{I}^-$ ), and typically involves nonunitary, sometimes exotic CFTs. While it lacks a fully rigorous, universally-applicable microscopic formulation, in higher-spin gravity and certain low-dimensional examples it achieves a degree of precision comparable to that of AdS/CFT, enabling calculations of perturbative observables, entropy, and late-time correlators. Notably, the analytic continuation of the AdS/CFT dictionary underlies most explicit results. The dual CFTs generally lack positive-definite norm and reflection positivity and admit complex or even imaginary central charge, endowing dS/CFT with both mathematical subtlety and physical richness.

1. Analytic Continuation: From AdS to de Sitter

The core mechanism for constructing explicit dS/CFT dualities is analytic continuation. In the embedding-space formalism, Euclidean AdS $_{d+1}$ and Lorentzian dS $_{d+1}$ are both realized as hyperboloids in $\mathbb{R}^{d+2}$ with signature $(-,+,\dots,+)$ . By Wick rotation, one maps the AdS line element

$- X_0^2 + X_1^2 + \cdots + X_{d+1}^2 = -L^2$

to the dS counterpart

$- X_0^2 + X_1^2 + \cdots + X_{d+1}^2 = +H^2$

with $H = L$ , using $X_0 \to X_1$ and $\mathscr{I}^-$ 0 as the analytic continuation. The static patch of dS $\mathscr{I}^-$ 1 arises as the analytic continuation of the interior region of AdS $\mathscr{I}^-$ 2 up to a finite radial shell. In particular, one can construct the dS $\mathscr{I}^-$ 3 static patch by continuing from the shell region $\mathscr{I}^-$ 4 of Euclidean AdS $\mathscr{I}^-$ 5, mapping $\mathscr{I}^-$ 6 to $\mathscr{I}^-$ 7 and interchanging the temporal and spatial embedding directions; the resulting metric is

$\mathscr{I}^-$ 8

This connection is exploited in higher-spin realizations and governs all local operator correlator matching via double Wick rotation (Karch et al., 2013).

2. Higher-spin Realizations and the Static Patch/Cut-off CFT Duality

The most explicit microscopic constructions of dS/CFT involve Vasiliev's higher-spin gravity. These models utilize dualities between AdS $\mathscr{I}^-$ 9 higher-spin gravity and free or critical $_{d+1}$ 0 vector models, then apply an analytic continuation $_{d+1}$ 1, $_{d+1}$ 2 to reach dS gravity coupled to Sp( $_{d+1}$ 3) vector models with anticommuting scalars (Anninos et al., 2011). In the static patch context, the correspondence is localized:

Higher-spin gravity on the dS $_{d+1}$ 4 static patch with radial cutoff $_{d+1}$ 5 is dual to a cut-off CFT on $_{d+1}$ 6 with UV cutoff $_{d+1}$ 7.
Each bulk higher-spin field $_{d+1}$ 8 induces a source at the cutoff which couples to a spin- $_{d+1}$ 9 conserved current in the CFT.
The dimension of the CFT Hilbert space grows as $_{d+1}$ 0 with $_{d+1}$ 1, reproducing the Bekenstein-Hawking entropy upon $_{d+1}$ 2 (Karch et al., 2013).

Bulk quasinormal frequencies are mapped to poles of CFT two-point functions at frequencies $_{d+1}$ 3, reflecting the spectrum of scalar perturbations in the static patch. This enables a direct microscopic derivation of thermodynamic entropy and spectral data from the dual CFT perspective.

3. Structure of the dS/CFT Dictionary and Observables

The dS/CFT dictionary primarily identifies the Bunch-Davies wavefunction of the bulk, $_{d+1}$ 4, with the generating functional of the boundary CFT, $_{d+1}$ 5: $_{d+1}$ 6 The precise definition of $_{d+1}$ 7 (boundary source) and the dual operator $_{d+1}$ 8 depends on the spectrum of the bulk field (principal versus complementary series). For principal series scalars, the late-time data are encoded by coherent states, and the resulting CFT correlators correctly satisfy conformal Ward identities, including complex scaling dimensions. The two-point and three-point functions at late times have the conformal form $_{d+1}$ 9, but with complex central charge and non-unitary reflection properties (Dey et al., 2024).

Cut-off dependent terms and divergences encode bulk Hamiltonian constraints. Notably, the two standard AdS/CFT dictionaries ("differentiate" and "extrapolate") are inequivalent in dS/CFT: "differentiate" isolates a single fall-off mode, while "extrapolate" generically yields a mixture of both fall-offs, corresponding to two CFT operators with complex conjugate dimensions (Harlow et al., 2011, Kalvakota et al., 2024).

4. Entropy, Entanglement, and Pseudoentropy in dS/CFT

In dS/CFT, the concept of entanglement entropy requires significant modification. The holographic calculation following the Ryu–Takayanagi prescription for extremal surfaces typically yields a complex quantity in dS, known as pseudoentropy. Perturbative expansions around symmetric entangling regions (e.g., round spheres) show that the leading shape dependence of pseudoentropy is governed by the stress tensor central charge $\mathbb{R}^{d+2}$ 0 of the non-unitary dual CFT. The round sphere is a local extremum, and deformations yield quadratic corrections proportional to $\mathbb{R}^{d+2}$ 1 (Anastasiou et al., 1 Dec 2025). For quadratic-curvature corrections in the bulk, the pseudoentropy retains the same universal dependence, modified only by rescaling parameters.

For two-dimensional $\mathbb{R}^{d+2}$ 2-deformed CFTs at finite dS cutoff, the trace-flow equations and entanglement entropy can be obtained by analytic continuation from AdS results. In this setting, the nonunitarity and nonlocality of the boundary theory manifest in the violation of strong subadditivity of pseudoentropy, a signature of the intrinsically nonlocal character of dS holography (Chang et al., 2024).

5. Quantum State Structure, Entanglement, and ER=EPR

Global dS spacetime contains two asymptotic boundaries, and the dS/CFT dictionary naturally prescribes the global wavefunction as a thermofield double state entangling two CFT Hilbert spaces (Cotler et al., 2023). Tracing over one factor yields a thermal density matrix describing a single static patch. In this sense, the global geometry is reconstructed from entanglement, realizing a precise dS analog of ER=EPR. The Hamiltonian of the static patch is mapped to the dilation operator on the corresponding CFT. The quasinormal mode decomposition of bulk fields corresponds directly to CFT operator insertions, and the thermal structure matches the Gibbons-Hawking temperature of the static patch horizon.

6. Mathematical Formulation and Open Questions

At the formal level, the dS/CFT duality involves identifying the Wheeler–DeWitt wavefunctional at $\mathbb{R}^{d+2}$ 3 as a sum of two functionals $\mathbb{R}^{d+2}$ 4 weighted by eikonal-like universal phase factors,

$\mathbb{R}^{d+2}$ 5

with $\mathbb{R}^{d+2}$ 6 encoding local counterterms. Unlike AdS/CFT, no explicit Lagrangian realization for $\mathbb{R}^{d+2}$ 7 is known in general, nor is unitarity required or present (Kalvakota et al., 2024). The absence of a concrete local CFT dual, the lack of geometric subregion-subalgebra duality, and the impossibility of defining entanglement wedges or modular flow in the standard sense are prominent obstacles.

Open questions include:

Is there a nonunitary CFT, perhaps with complexified sources, furnishing $\mathbb{R}^{d+2}$ 8?
Can a regime be identified in which one branch dominates?
What is the precise manner in which dS counterterms correspond to $\mathbb{R}^{d+2}$ 9-type deformations?
Can subregion duality or emergent bulk geometry be algebraically constructed in dS via a novel modular theory?

Recent work suggests that in higher-spin dS/CFT, priority is given to global HS symmetry over boundary spacetime locality; spin-local boundary actions, formulated in twistor space, lead to well-behaved Hartle–Hawking wavefunctions, even in the absence of conventional CFT locality (David et al., 2020).

7. Extensions: Lower Dimensions, Emergent Strings, and Algebraic Constructions

In three dimensions, dS/CFT corresponds to dualities between dS $(-,+,\dots,+)$ 0 gravity (or higher-spin Chern–Simons theory) and WZW models at "critical level" $(-,+,\dots,+)$ 1, with central charge $(-,+,\dots,+)$ 2. This approach reproduces the classical partition functions, two-point functions, and entanglement entropy—including topological entanglement entropy that coincides with semiclassical dS entropy (Hikida et al., 2022, Chen et al., 2022). In 1+1 dimensions, bottom-up derivations from quartic fermion models recover AdS $(-,+,\dots,+)$ 3/CFT $(-,+,\dots,+)$ 4 and its analytic continuation to dS $(-,+,\dots,+)$ 5/CFT $(-,+,\dots,+)$ 6, embedding emergent string theory at the "Strominger point" and matching central charges, operator dimensions, modular invariants, and GSO projections (Haddad, 29 Mar 2026).

Further, models based on Yang-Baxter operators for $(-,+,\dots,+)$ 7 representations construct dual pairs of chiral (bulk) and Euclidean (boundary) CFTs, realizing bulk–boundary operator maps and symmetries precisely and providing a rigorous algebraic model for dS/CFT (Hollands et al., 2016).

The dS/CFT correspondence thus stands as a mathematically and physically rich generalization of holographic duality, with explicit higher-spin, Chern–Simons, and exactly solvable instances. It is characterized by analytic continuation from AdS/CFT, a nonunitary and sometimes nonlocal boundary theory, the necessity of novel entropy and entanglement constructs ("pseudoentropy"), and deep connections between entanglement structure and the emergence of bulk spacetime. Numerous foundational and technical challenges remain, including the construction of a microscopic boundary theory, the formulation of modular and subregion dualities, and the systematic understanding of quantum gravitational information in de Sitter space (Karch et al., 2013, Cotler et al., 2023, Kalvakota et al., 2024, Anastasiou et al., 1 Dec 2025, Haddad, 29 Mar 2026, Hikida et al., 2022).