Holography in flipped AdS/$\mathbb{Z}$: Another approach to dS holography

Technical Summary: Holography in flipped AdS/Z: Another approach to dS holography

Problem Statement
The conventional $dS/CFT$ correspondence, which posits a duality between quantum gravity in de Sitter (dS) space and a Euclidean conformal field theory (CFT) at the spacelike conformal boundaries $\mathcal{I}^\pm$, faces significant interpretational subtleties. Unlike the $AdS/CFT$ correspondence, the $dS/CFT$ dictionary is not uniquely defined; analytic continuation from $AdS$ often suggests an imaginary central charge ($c \sim i$), leading to imaginary entropies if standard thermal Cardy formulas are applied directly. Furthermore, the GKPW prescription and the extrapolate (BDHM) dictionary, which are equivalent in $AdS$, are generally inequivalent in $dS$. These issues suggest that the standard Euclidean CFT interpretation may not provide a complete or unambiguous microscopic description of de Sitter physics. The authors seek an alternative route to de Sitter holography by exploring whether a different bulk spacetime, related to $dS$ via analytic continuation, admits a well-defined holographic dual that can subsequently be mapped to $dS$.

Methodology
The paper employs analytic continuation as a constructive principle for holography. The core strategy involves three main steps:

1. Establishing the Bulk Relation: The authors identify a specific Lorentzian spacetime, termed "flipped AdS/Z" ($fAdS$), which arises from the same Euclidean parent geometry (the sphere $S_n$) as de Sitter space ($dS_n$) but through a distinct Wick rotation.

   • $dS_n$ is obtained via the continuation $\theta_1 \to i\tau$.
   • $fAdS_n$ is obtained via the continuation $\psi \to i\rho$.
     The metric of $fAdS_n$ is given by $ds^2/l^2 = -d\rho^2 - \sinh^2 \rho d\Omega_{n-2}^2 + \cosh^2 \rho dt^2$, where the time coordinate $t$ is compact ($t \sim t + 2\pi$) and $\rho$ serves as the radial evolution parameter. This makes $fAdS$ a multi-time spacetime with a single conformal boundary at $\rho \to \infty$.

2. Constructing the $fAdS/fCFT$ Correspondence:

   • Bulk QFT: The authors formulate Quantum Field Theory (QFT) in $fAdS$ by canonically quantizing the theory on constant-$\rho$ hypersurfaces. They define the Euclidean vacuum via a path integral over the sphere $S_n$ and construct $\rho$-ordered Feynman correlators. They demonstrate that the correlation functions in $fAdS$ and $dS$ are related by analytic continuation of the same Euclidean correlator on $S_n$.
   • Holographic Dictionary: They propose a holographic dual for $fAdS$, termed "flipped CFT" ($fCFT$), living on the boundary topology $T^{n-2,1} = S^{n-2} \times S^1$ with a signature-flipped metric ($dt^2 - d\Omega_{n-2}^2$). They derive the extrapolate dictionary for asymptotically $fAdS$ spacetimes, relating bulk field asymptotics to boundary operators.
   • Consistency Checks: The holographic two-point functions are computed in two independent ways: (i) by taking the asymptotic limit of the bulk propagator using the extrapolate dictionary, and (ii) by deriving vacuum correlators directly from the conformal symmetry $SO(n-1, 2)$ of the boundary theory. The agreement between these two methods validates the proposed dictionary.

3. Application to dS Entropy: The authors utilize the established $fAdS/fCFT$ duality to compute the entropy of de Sitter horizons. By analytically continuing the $fAdS$ geometry to $dS$, they apply the Cardy formula to the dual $fCFT_2$ (in $n=3$ dimensions). They treat the central charge and modular parameters as analytically continued quantities (e.g., imaginary central charge and imaginary temperature) appropriate for the Lorentzian torus boundary.

Key Contributions and Results

• Definition of Flipped AdS/Z and fCFT: The paper rigorously defines the $fAdS_n$ spacetime and its dual $fCFT_{n-1}$. It establishes that $fAdS$ possesses a conformal boundary with topology $T^{n-2,1}$ and that the dual theory is a multi-time QFT with signature-flipped metric.
• Derivation of the Holographic Dictionary: A complete extrapolate dictionary is constructed for free scalar fields in arbitrary-dimensional asymptotically $fAdS$ spacetimes. The authors explicitly compute the holographic two-point functions and verify them against independent conformal symmetry arguments, confirming the consistency of the $fAdS/fCFT$ correspondence.
• Reproduction of dS Entropy:
  • Pure $dS_3$: Using the Cardy formula for $fCFT_2$ with an analytically continued central charge $c = i \frac{3l}{2G}$ and temperature $T = \frac{1}{2\pi i}$, the authors successfully reproduce the Bekenstein-Hawking entropy of the $dS_3$ cosmological horizon ($S = \frac{\pi l}{2G}$).
  • Kerr-$dS_3$: The construction is extended to Kerr-$dS_3$. By identifying the Euclidean geometry as a conical defect quotient of $S^3$, the authors derive the modular parameters for the dual $fCFT_2$. The Cardy formula again reproduces the exact Bekenstein-Hawking entropy of the Kerr-$dS_3$ cosmological horizon.
• Analytic Continuation Framework: The paper demonstrates that $dS$ and $fAdS$ QFTs are related by analytic continuation of the same Euclidean path integral on $S_n$. This provides a controlled bulk map ($A_{bulk}$) that can potentially induce a boundary map ($A_{bdry}$) between $fCFT$ and a candidate dual for $dS$.

Significance and Claims
The paper claims that the $fAdS/fCFT$ correspondence provides a constructive starting point for an alternative formulation of de Sitter holography. By establishing a well-defined holographic dual for the "flipped" spacetime, the authors offer a new pathway to address the subtleties of the conventional $dS/CFT$ correspondence.

Specifically, the successful reproduction of the $dS_3$ and Kerr-$dS_3$ entropies using the $fCFT_2$ Cardy formula serves as a nontrivial check, suggesting that $fCFT$ captures the microscopic degrees of freedom associated with de Sitter cosmological horizons. The authors propose that the analytic continuation relating the bulk geometries ($fAdS \to dS$) may induce a corresponding continuation on the boundary ($fCFT \to QFT_{dS}$), potentially yielding a new holographic description of de Sitter physics.

The paper remains modest regarding the ultimate interpretation, acknowledging that the boundary dual for de Sitter space is generally non-unitary and that the microscopic interpretation of the Cardy formula in this non-unitary setting remains an open question. The work is presented as a foundational step toward a broader program of constructing de Sitter holography through induced analytic continuation, rather than a complete resolution of all issues in $dS$ quantum gravity.