Holographic Entanglement and Poincare blocks in three dimensional flat space

TL;DR

The paper develops a covariant holographic framework for flat space in three dimensions, deriving a worldline-based, probe-limit prescription for holographic entanglement entropy and global BMS3 blocks, and then extends it with a full extrapolate dictionary to access non-probe regimes. It demonstrates that two- and three-point BMS3 correlators, as well as scalar and spinning Poincaré blocks, can be computed via extremal networks of geodesics and geodesic Feynman diagrams, with explicit results matching known BMS3 structures and AdS2/CFT1 results in the appropriate limits. The extrapolate dictionary is tested by connecting boundary correlators to bulk propagators integrated along null lines, reproducing the correct two-, three-, and four-point functions and blocks, and revealing a deep link between flat-space holography and AdS2-like geodesic networks. Overall, the work provides concrete tools and a coherent geometric dictionary to study holography in asymptotically flat 3D spacetimes with BMS3 symmetry, enabling computation of entanglement and correlation structures in this setting.

Abstract

We propose a covariant prescription to compute holographic entanglement entropy and Poincare blocks (Global BMS blocks) in the context of three-dimensional Einstein gravity in flat space. We first present a prescription based on worldline methods in the probe limit, inspired by recent analog calculations in AdS/CFT. Building on this construction, we propose a full extrapolate dictionary and use it to compute holographic correlators and blocks away from the probe limit.

Figures (6)

• Figure 1: a) Holographic entanglement entropy as understood in Jiang:2017ecm. The gray planes ($N_i$) correspond to the codimension one Cauchy Horizon of the flat space cosmological solution. The green lines are null, and we have dubbed them $\gamma_1$ and $\gamma_2$ throughtout this note. The black arrows stand for the modular flow vector in the bulk, which vanishes at the intersection between the two planes; $N_1\cap N_2$. The path is then formed by null lines tangent to modular flow (green) and the fixed points of replica symmetry (blue). b) Modular flow at the boundary. The axis are $\phi, u$. The red line is the boundary region ${\cal A}$ whose entanglement entropy is being studied.
• Figure 2: a) Calculation of Entanglement Entropy before extremizing. The red boundary points $x_i$ are connected to the blue points $y_i$, which live at the null lines $\gamma_i$. b) Geometrical picture after extremizing the total length. The bulk points live at the intersections $N_1\cap N_2\cap \gamma_i$.
• Figure 3: a) Set-up for the calculation of the holographic three point function before extremizing. The red boundary points $x_i$ are connected to the blue points $y_i$, which live at the null lines $\gamma_i$. The bulk points are then connected to a completely arbitrary vertex point $y_v$, drawn in purple. b) Set-up for the calculation of the holographic scalar global block before extremizing. The red boundary points $x_i$ are connected to the blue points $y_i$, which live at the null lines $\gamma_i$. The bulk points are then connected to two completely arbitrary vertex points $y_v$ and $y_{v'}$, drawn in purple. These two vertex are connected with a line where a particle with mass $\xi_p$ is propagating.
• Figure 4: Map from euclidean global AdS$_2$ (left) to the Poincaré patch (right). a) Transformation from the disk to the strip, implemented by the change of coordinates in equation \ref{['eq:CFT1map']}. b) Inverse Wick rotation from the euclidean strip to the lorentzian one, implemented by $\tau=i T$. c) Poincaré patching of the lorentzian strip, implemented by formula \ref{['eq:PoincareMap']}.
• Figure 5: a) Set-up for the calculation of the holographic spinning two-point function. The red points are located at the null boundary at $x_{i=1,2}$, equipped with boundary vectors $n^{\partial}_{i}=\partial_u$. The regulated blue points are located at $\gamma_{i}$, and they are equipped with blue vectors $n_{i}=\partial_r\vert_{y_{i}}$. The spinning particle propagates between the blue points, along the green straight line connecting $\gamma_1$ and $\gamma_2$. b) Set-up for the calculation of the holographic spinning three-point function. There are three boundary points drawn in red, and their regulated bulk versions are drawn in blue. At each blue point we have a blue vector $n_{i=1,2,3}=\partial_r\vert_{y_{i}}$ pointing along the null geodesics $\gamma_i$. The spining particles meet at a common purple point $y_v$, where we have a purple time-like vector $n_v$ for which the spinning total on-shell action must be extremized.