$T \bar T$ and EE, with implications for (A)dS subregion encodings

Abstract

We initiate a study of subregion dualities, entropy, and redundant encoding of bulk points in holographic theories deformed by $T \bar T$ and its generalizations. This includes both cut off versions of Anti de Sitter spacetime, as well as the generalization to bulk de Sitter spacetime, for which we introduce two additional examples capturing different patches of the bulk and incorporating the second branch of the square root dressed energy formula. We provide new calculations of entanglement entropy (EE) for more general divisions of the system than the symmetric ones previously available. We find precise agreement between the gravity side and deformed-CFT side results to all orders in the deformation parameter at large central charge. An analysis of the fate of strong subadditivity for relatively boosted regions indicates nonlocality reminiscent of string theory. We introduce the structure of operator algebras in these systems. The causal and entanglement wedges generalize to appropriate deformed theories but exhibit qualitatively new behaviors, e.g. the causal wedge may exceed the entanglement wedge. This leads to subtleties which we express in terms of the Hamiltonian and modular Hamiltonian evolution. Finally, we exhibit redundant encoding of bulk points, including the cosmological case.

Figures (8)

• Figure 1: Patches we will work with depicted in purple within the $\text{AdS}$ and $\text{dS}$ Penrose diagrams. The top left is $\text{dS}/\text{dS}$, with a fixed $w=w_c$ slice indicated by the dashed line. The purple and orange together cover the full dS/dS patch of $dS_3$, while the purple indicates the region that remains after introducing the cutoff at a fixed $w_c$ in the coordinate system (\ref{['mets']}). The top right is $\text{dS}/{\text{cylinder}}$, with again a fixed radial position $r=r_c$ indicated by the dashed line. The bottom left similarly depicts cut off $\text{AdS}/\text{dS}$ and the bottom right (a slice of) cut off $\text{AdS}/{\text{Poincar\'e}}$.
• Figure 2: Setup for the spatial SSA (left) and boosted SSA (right). The causal domains of the intervals are also shown. In both cases, the union and intersection are spatial intervals with lengths $L_1$ and $L_2$ respectively.
• Figure 3: A schematic illustration of the relationships between the regions described in the text, on a timeslice which we can take for simplicity to be at the moment of time symmetry in the gravity-side geometry. We show both the $Z_2$ symmetric case where the interval $R$ and its complement $\bar{R}$ are of the same size (left panel) and the case where they are unequal (right panel). The Ryu-Takayanagi surface is indicated in blue, and it separates the bulk effective field theory subregion $r$ from its complement $\bar{r}$. The causal wedge of the smaller subregion is indicated in light green in the asymmetric case, indicating that it exceeds the entanglement wedge (indicated by blue hatchmarks). This leads to the intersection between the CW of the smaller subregion and the EW of the complementary region discussed in the text. The purple arrows indicate the action of the bulk modular Hamiltonian Faulkner:2017vdd on probe bulk fields near the RT surface near the boundary, projected to the pictured spatial slice. In the left picture, similarly to asymptotically AdS spacetime, the RT surface intersects the boundary orthogonally, while generically in the deformed theories this is not the case, and the modular evolution in the bulk has a component that is orthogonal to the boundary.
• Figure 4: The regions $D[R]$ and $D[\bar{R}]$ for an asymmetric division of the system in the case of a boundary $dS_2$ spacetime, whose Penrose diagram we depict here. The region $E[R]$ for $R$ the smaller subregion is indicated by the dark blue hatchmarks. Operators at the point $p$ can be causally influenced from either region and are not in EW[R] or EW$[{\bar{R}}]$; this is familiar in asymptotic AdS/CFT. The point $q$ illustrates a point in $D[R]$ which is not in $E[R]$. Operators at that point do not commute with those in the algebra ${\cal A}_{\bar{R}_s}$ corresponding to the complementary subsystem to $R$. We note that this point $q$ on the boundary is also not part of the EW of either $R$ or $\bar{R}$, since it can be reached causally from both.
• Figure 5: CW vesus EW in cutoff AdS. When we move the boundary from $z=0$ to $z=z_c$, CW becomes larger than EW. The modular Hamiltonian evolution near the boundary is given by the inward pointing red arrow for $R$, and points outwardds for $\bar{R}$.