Equivalence of Emergent de Sitter Spaces from Conformal Field Theory

TL;DR

This work investigates whether two proposals for emergent de Sitter space from conformal field theory are equivalent beyond vacuum states. By analyzing the kinematic spaces of all nontrivial $3$D gravity solutions (global AdS$_3$, BTZ black string/black hole, and conical singularities), the authors show these spaces are globally hyperbolic subregions of $dS_2$ and that their dynamics can be governed by suitable boundary conditions at future infinity. They establish a precise match between the kinematic space construction and the auxiliary de Sitter approach in vacuum and thermal settings, notably for the BTZ case, and propose a refined prescription connecting entanglement entropy, modular Hamiltonians, and a boundary-to-bulk propagator on emergent $dS$. The results reinforce a deep link between boundary entanglement structure and emergent spacetime geometry, with implications for MERA-kinematic space connections and potential generalizations to higher dimensions and non-universal states.

Abstract

Recently, two groups have made distinct proposals for a de Sitter space that is emergent from conformal field theory (CFT). The first proposal is that, for two-dimensional holographic CFTs, the kinematic space of geodesics on a spacelike slice of the asymptotically anti-de Sitter bulk is two-dimensional de Sitter space (dS$_2$), with a metric that can be derived from the entanglement entropy of intervals in the CFT. In the second proposal, de Sitter dynamics emerges naturally from the first law of entanglement entropy for perturbations around the vacuum state of CFTs. We provide support for the equivalence of these two emergent spacetimes in the vacuum case and beyond. In particular, we study the kinematic spaces of nontrivial solutions of $3$d gravity, including the BTZ black string, BTZ black hole, and conical singularities. We argue that the resulting spaces are generically globally hyperbolic spacetimes that support dynamics given boundary conditions at future infinity. For the BTZ black string, corresponding to a thermal state of the CFT, we show that both prescriptions lead to an emergent hyperbolic patch of dS$_2$. We offer a general method for relating kinematic space and the auxiliary de Sitter space that is valid in the vacuum and thermal cases.

Figures (20)

• Figure 1: (a) A boundary-anchored geodesic on a constant time slice of AdS, with boundary interval in thick purple. The interval or equivalently the geodesic anchored to its endpoints is parametrized by the coordinates of its endpoints, $u$ and $v$, or by its midpoint angle $\theta$ and opening angle $\alpha$. (b) These geodesics enjoy a natural causal structure based on the containment relations of their boundary intervals (colored in thick purple and thicker green): geodesics are time-like separated if they have embedded boundary intervals (top left), null separated if they share a left or right endpoint (top right), and space-like separated if their boundary intervals are not embedded (bottom two).
• Figure 2: The kinematic space for pure AdS$_3$ is a 2-dimensional de Sitter space, represented (a) as a Penrose diagram, using coordinates defined in eqs. \ref{['B8']} and \ref{['B9']}, and (b) as a dS$_2$ hyperboloid embedded in flat space $R^{1,2}$, with constant $t$ lines (black) and constant $\theta$ lines (dashed). The dS$_2$ waist at $\alpha = \pi/2$ is highlighted in thick black to stress that kinematic space is the space of oriented geodesics: the entire expanding portion of de Sitter (above the waist) maps to all $H_2$ geodesics with one orientation, while the contracting region (under the waist) maps to the same geodesics but with opposite orientation. The geodesics cover the full constant time slice of AdS$_3$, represented in (c) as a Poincaré disk (cf. figure \ref{['PoincareDiskAdS']}).
• Figure 3: Kinematic space defines an emergent dS$_2$ on a CFT$_2$ by associating a point $p$ on de Sitter to a given CFT interval $[u,v]$, or equivalently, to the corresponding boundary-anchored $H_2$ geodesic $[u,v]$. (a) The CFT interval lies at the asymptotic future boundary of dS$_2$, and can be identified with a point $p$ in de Sitter at the tip of the lightcone extending into the bulk. (b) In embedding space, geodesics on $H_2$ are intersections of origin-centered planes with the $H_2$ hyperboloid (blue). Each such plane specifies a point in the dS hyperboloid via its outward pointing normal.
• Figure 4: The kinematic space for the Poincaré patch of AdS$_3$ is the planar patch of dS$_2$, depicted in red in (a) the Penrose diagram, using coordinates defined in eqs. \ref{['PlanarPenrose1']} and \ref{['PlanarPenrose2']}, and (b) the embedding diagram, with lines of constant time (solid) and constant $\theta$ coordinates (dashed). The geodesics cover the full constant time slice of AdS$_3$, represented in (c) as the half-plane (cf. figure \ref{['PoincareDiskAdS']}).
• Figure 5: The kinematic space for the 1-sided BTZ black string is the hyperbolic patch of de Sitter, depicted in green in (a) the Penrose diagram, using coordinates defined in eqs. \ref{['HyperbolicPenrose1']} and \ref{['HyperbolicPenrose2']}, and (b) the embedding diagram with constant time lines (solid) and constant $\theta$ lines (dashed). (c) The geodesics cover one half of the Poincaré disk, that is, one outside-horizon region (cf. figure \ref{['PoincareDiskBTZ']}).