Equivalence of emergent de Sitter spaces from conformal field theory

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 space-like 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 3d 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.

An investigation of AdS2 backreaction and holography

“…Suppose we set T ++ (s) = T −− (s), as would follow from a reflecting boundary condition at the AdS 2 boundary. Equation (2.9) then gives that the deviation δΦ 2 of Φ 2 due to the energy momentum flux is equal to This formula precisely matches with the first law of entanglement thermodynamics of a 2d CFT (see for example [31,32]), that expresses the change in the entanglement entropy of the interval [x + , x − ] due to an injection of energy momentum. This relation gives an encouraging hint that it should be possible to reconstruct the complete bulk dynamics of the AP model from thermodynamic considerations.…”

Topological Complexity in AdS3/CFT2

“…We restrict our attention to pure AdS since the kinematic space has been worked out in great detail. However, there has been recent work on kinematic spaces corresponding to conical defects and black holes …”

Towards a Fisher-Information Description of Complexity in de Sitter Universe

“…This is consistent with the MERA/spacetime correspondence proposed in [11], where MERA is regarded as a quantum circuit and the dS metric is derived. A similar perspective can be found in [12][13][14][15][16], where MERA is viewed as a discretization of kinematic space-the space of bulk geodesics, instead of the time slice of the original bulk, and the kinematic space of an AdS space is of dS geometry. Second, the on-shell action of dS can be identified as an FIM, which is a description of the measure of information.…”

“…The kinematic space is defined by a set of boundary-anchored geodesics. The measure of a kinematic space is determined by [12,14] network. We have ignored disentangler because the spacetime volume is only interpreted as conditional mutual information.…”

“…One support of the duality (16) comes from a classical relation between central charge of the boundary theory and the gravitational constant of the gravity theory. From (14) and (15) we have 1 The reason we choose element gate like this is that the MERA network can be thought of as an iterative compression algorithm that maps the density matrix of an interval to a direct product state [13]. In the opposite direction, this tensor network maps a non-entangled state to an entangled state [34].…”