dS/CFT Correspondence: Quantum Gravity Insights

• dS/CFT correspondence is a theoretical framework positing a holographic relation between de Sitter quantum gravity and a Euclidean conformal field theory at future infinity.
• It features distinct operator dictionaries where differentiation and extrapolation yield different scaling behaviors, highlighting key renormalization challenges.
• Analytic continuation from AdS to dS, alongside dynamic boundary conditions, underpins its role in modeling cosmic acceleration and aspects of inflationary cosmology.

The dS/CFT correspondence posits a holographic duality between a quantum theory of gravity in de Sitter (dS) spacetime and a conformal field theory (CFT) defined on its spacelike boundary at future infinity. Unlike the well-established AdS/CFT duality, which relates anti-de Sitter gravity to unitary, local boundary CFTs, the dS/CFT framework operates in a Lorentzian background with a cosmological horizon and seeks to capture quantum gravitational effects relevant for cosmic acceleration and inflationary cosmology. Its precise definition, operator dictionaries, and physical scope are the subject of extensive research and ongoing debate.

1. Operator Dictionaries and Inequivalence in dS/CFT

The mapping between bulk fields and boundary operators—the "operator dictionary"—is foundational in holographic dualities. In AdS/CFT, two prescriptions for this map are known to coincide up to scheme choices: (1) the "differentiate" dictionary, wherein boundary correlators are computed by differentiating the bulk partition function with respect to boundary sources (as in the Gubser-Klebanov-Polyakov-Witten procedure), and (2) the "extrapolate" dictionary, in which one extrapolates bulk correlators to the boundary and identifies their leading asymptotics.

For dS/CFT, these procedures are fundamentally inequivalent (1104.2621). In the de Sitter context:

• Differentiating the late-time wave function (the Bunch–Davies boundary condition) yields correlators of operators with a single conformal dimension, $\delta = \frac{d}{2} + \sqrt{\frac{d^2}{4} - m^2}$.
• Extrapolation of bulk correlators results in both $\delta$ and $d-\delta$ scaling behaviors; both modes reach the boundary, reflecting that both are allowed to fluctuate due to the absence of a fixed boundary condition at future infinity.

The physical origin of this dichotomy is that in de Sitter space the future boundary is not sharply specified by a boundary condition (as in AdS), but emerges dynamically from evolution out of an initial state (typically Euclidean or Bunch–Davies). As a result, the boundary theory accommodates contributions from both normalizable and non-normalizable modes, leading to operator dictionaries that cannot be equated through simple field redefinitions or renormalizations.

2. Analytic Continuation and Wave Functions

A central structural theme in dS/CFT is the use of analytic continuation from anti-de Sitter space to de Sitter space. At the level of the bulk wave function, a prescription is established whereby the IR component of the Euclidean AdS path integral, upon Wick rotation $L_{AdS} \to iL_{dS}$ and $z \to -iT$, yields the wave function for dS quantum gravity with Euclidean initial conditions (1104.2621). That is,

$$\Psi[\tilde{\phi}, T] = \exp \left\{ iS_\text{local} + c_\delta \, (-iT)^{2(\delta-d)} \int \frac{d^d k}{(2\pi)^d} \tilde{\phi}_{-k} \tilde{\phi}_k \left[ k^{2\delta-d} + \cdots \right] \right\}$$

The reality structure is deeply affected by this procedure: the cancellation of local terms in $|\Psi|^2$ means that expectation values in dS are not analytic continuations of their AdS counterparts. However, the underlying wave function itself does continue, and this analytic structure underpins much of the comparison between the two settings.

This analytic continuation also leads to the appearance of different scaling behaviors in late-time correlators (i.e., both $\delta$ and $d-\delta$) in the dS boundary theory, emphasizing the absence of a sharp boundary condition at future infinity.

3. Renormalization and Composite Operators

When accounting for interactions or composite bulk operators, further subtleties emerge. In AdS, the equivalence of the operator dictionaries depends on the renormalization of bulk composite operators: the terms arising from higher-point bulk couplings must be matched to renormalized boundary insertions by subtracting non-universal contact terms, yielding an operator map that is consistent under the RG flow (1104.2621).

In contrast, in dS, the process of squaring the wave function for Born rule probabilities removes the peaked local (non-universal) contributions, and both falloffs contribute in the correlators. There is, consequently, no analogous mechanism in dS/CFT to achieve a unique renormalized operator correspondence, making the construction of a boundary CFT with conventional properties more intricate.

4. Interacting Fields, Fixed Backgrounds, and Dynamical Gravity

For interacting bulk fields on fixed (A)dS backgrounds, the analysis shows that in AdS, interactions become irrelevant near the boundary, preserving the leading boundary scaling upon proper renormalization of composite operators. In dS, boundary fluctuations remain significant, and both falloff modes contribute to the spectrum of correlations derived from bulk interactions (1104.2621).

Extending the discussion to cases with dynamical bulk gravity, the full partition function can be formally split into UV and IR components, leading to a functional Schrödinger-type equation for the UV wave function—even in the presence of dynamical metrics. Analytic continuation from AdS to dS remains valid perturbatively in the gravitational coupling, though nonperturbative phenomena (such as Stokes phenomena or the discretization of dual field theory parameters) may require further care (1104.2621).

5. Scaling Dimensions and Ward Identities

The dimension $\delta = \frac{d}{2} + \sqrt{\frac{d^2}{4} - m^2}$ plays a crucial role as the conformal weight of the boundary dual to a massive bulk field—determined by late-time behavior of the bulk wave function. The source/operator mapping at future infinity is governed by this dimension in the "differentiate" dictionary. For the "extrapolate" prescription, both $\delta$ and $d-\delta$ dimensions appear, and the structure of correlators is more intricate.

Despite the lack of a unitary boundary theory and the absence of a fixed norm, the nonlocal terms in correlation functions computed via analytic continuation can be shown to satisfy the conformal Ward identities of a CFT (Dey et al., 2 Jul 2024).

6. Implications and Conceptual Challenges

The dS/CFT correspondence frames de Sitter quantum gravity with a Euclidean CFT "living" on ${\cal I}^+$, but unlike AdS/CFT, the dual theory in dS is not unitary and generally fails to be described by a standard local QFT. The lack of reflection positivity and the presence of multiple scaling behaviors in boundary correlators underscore this distinction.

Various proposals, including higher-spin dualities (e.g., Vasiliev theory matching with Sp(N) anticommuting scalar models) (1108.5735), holographic RG flows with exotic fixed points (Das et al., 2013), and new kinetic or nonlocal structures in the boundary theory, exemplify the novel features that emerge in pursuing this duality. In three dimensions, Virasoro symmetry and entropy matching via Cardy’s formula in certain toy models support the existence of a two-dimensional CFT structure, though the resulting boundary theory may have negative central charge or admit an imaginary structure (Buyl et al., 2013, Hikida et al., 2022).

7. Summary Table: Operator Dictionaries in AdS/CFT vs. dS/CFT

Feature	AdS/CFT	dS/CFT
Operator Dictionary	Differentiate ≡ Extrapolate	Differentiate $\neq$ Extrapolate
Leading Scaling	Single scaling dim. ($\Delta$)	Both $\delta$ and $d-\delta$ appear
Boundary Condition	Fixed (Dirichlet/Neumann) at conformal $\infty$	Future boundary emerges dynamically; no fixed scalar data
Renormalization	Composite operator renormalization enables dictionary equivalence	No analogous mechanism; local terms cancel in expectation values
Analytic Continuation	At level of partition/wave function	Only wave function continues; expectation values do not

This structure highlights both the technical machinery and physical subtleties characterizing dS/CFT correspondence. The ongoing research continues to clarify these aspects, with implications for quantum cosmology, holography beyond AdS, and the nature of time and entropy in expanding universes.