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.
Highlights
One unforeseen consequence of this program was the identification of an auxiliary Lorentzian geometry from conformal field theory (CFT) entanglement data, distinct from the usual bulk AdS space
We offer a general method for relating kinematic space and the auxiliary de Sitter space that is valid in the vacuum and thermal cases
For the BTZ black string and the conical singularities, we show that the resulting kinematic spaces are, respectively, the hyperbolic patch and glued together “sub-de Sitter spaces” of de Sitter space, which are depicted in figures 5 and 13
Summary
As formulated in [15, 16], can be defined for any CFT in any state. our main interest is in two-dimensional holographic CFTs which have an asymptotically AdS3 bulk spacetime, where the kinematic space has a geometric interpretation as a space of boundary-anchored, oriented geodesics. Given any time-reflection symmetric asymptotically AdS3 spacetime, there is a time coordinate t such that all space-like extremal curves that anchor on boundary points with t = 0 are entirely confined to a space-like slice defined by the condition t = 0. By invoking results in integral geometry, [15] proposes a kinematic space prescription for deriving a metric on kinematic space entirely from the entanglement entropy Sent(u, v) of the boundary intervals:1 This spacetime is Lorentzian due to a natural causal structure inherited from the containment relation of boundary intervals: two geodesics contained within one another are time-like separated, otherwise they are space-like separated or, in the marginal case where they share a left or right endpoint, null (see figure 1). Their detailed specification depends on the CFT state under consideration
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