A proof of Generalized Connected Wedge Theorem

Abstract

In the context of asymptotic $2$-to-$2$ scattering process in AdS/CFT, the Connected Wedge Theorem identifies the existence of $O(1/G_N)$ mutual information between suitable boundary subregions, referred to as decision regions, as a necessary but not sufficient condition for bulk-only scattering processes, i.e., nonempty bulk scattering region $S_0$. Recently, Liu and Leutheusser proposed an enlarged bulk scattering region $S_E$ and conjectured that the non-emptiness of $S_E$ fully characterizes the existence of $O(1/G_N)$ mutual information between decision regions. Here, we provide a geometrical or general relativity proof for a slightly modified version of their conjecture.

Figures (9)

• Figure 1: Boundary set-up of $2$-to-$2$ scattering process. This figure is adjusted from LL2025superadditivity.
• Figure 2: Schematic illustrating $\mathcal{S}_0 \subseteq \mathcal{S}_E$ when both $\mathcal{E}(V_1 \cup V_2)$ and $\mathcal{E}(W_1 \cup W_2)$ are connected. Panel $(a)$ depicts $\partial J^-[r_1]$ and $\partial J^-[r_2]$ and their intersections with the Cauchy slice $\Sigma_1$. Panel $(b)$ shows $\Sigma_1$ with the causal surface of $V_1 \cup X_i \cup V_2$ (i.e., $J^-[r_i] \cap \Sigma_1$) in black and the HRRT surface of $X_i$ in red. Panel $(c)$ displays $\partial J^+[c_1]$ and $\partial J^+[c_2]$ and their intersections with $\Sigma_2$. Panel $(d)$ presents $\Sigma_2$ with the causal surface of $W_1 \cup Y_i \cup W_2$ (i.e., $J^+[c_i] \cap \Sigma_2$) in black and the HRRT surface of $Y_i$ in red. Note that panels (c) and (d) are rotated relative to panels (a) and (b).
• Figure 3: An explicit example of $\mathcal{E}(V_1\cup V_2)$ being connected while $\mathcal{E}(W_1\cup W_2)$ being disconnected. Here, the input points $c_1,c_2$ and output points $r_1,r_2$ as well as RT surfaces of $V_1, V_2$ and $W_1, W_2$ are all projected onto a single Cauchy slice (see Remark \ref{['rmk:project_global_hyperbolic']}). The four points $c_1,c_2,r_1,r_2$ are equally spaced along the circle. The shaded region indicates addition of matter in a non-spherically-symmetric way while maintaining the reflection symmetry across segment $r_1-r_2$ and segment $c_1-c_2$. This will make HRRT surfaces of $W_1, W_2$ (on $\Sigma_2$) and HRRT surfaces of $X_1, X_2$ (on $\Sigma_1$) to be the true minimum.
• Figure 4: Schematic for intersections among $\mathcal{N}_{X_i}$ and $\mathcal{N}_{Y_j}$ when $\mathcal{E}(V_1\cup V_2)$ and $\mathcal{E}(W_1 \cup W_2)$ are both connected. The past-pointing null sheets from $RT(Y_1)$ and $RT(Y_2)$ intersect at the ridge $R$, which intersects $\partial M$ at points $R_1$ and $R_2$. The future-pointing null sheets from $RT(X_1)$ and $RT(X_2)$ intersect at the ridge $T$, which intersects $\partial M$ at points $T_1$ and $T_2$. A case of $\mathcal{S}_E = \emptyset$ is shown in (a-b) while a case of $\mathcal{S}_E$ has positive measure is shown in (c-e) with (e) illustrating the geometry of $\mathcal{S}_E$. Shown in $(f)$ is an example that ridges $R$ and $T$ or their projections intersect more than once. Red curves are associated with $\mathcal{N}_{X_i}$ while green curves are associated with $\mathcal{N}_{Y_j}$.
• Figure 5: Schematic for $\mathcal{S}_E$ consisting of multiple points. That is, the ridge $R$ and the ridge $T$ intersect at more than one point. We get the same geometric structure as before when restricting to the prat of $R$ between the first and the last intersection point.

Theorems & Definitions (24)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Conjecture 2.4
• Example 2.5
• Theorem 2.6