CFT sewing as the dual of AdS cut-and-paste

TL;DR

This work analyzes the holographic dual of CPT sewing of two CPT-conjugate CFT states over a region $R$ in the leading $G$ limit, focusing on time-symmetric configurations. For fixed-area HRT surfaces, the bulk dual emerges from a bulk cut-and-paste: remove the entanglement wedges of $ar{R}$ from $g_1$ and $g_2$, glue the remaining wedges across $oldsymbol{ar{R}}$, and evolve the glued initial data to obtain $g(A)=g_1(A)\\#_R g_2(A)$; the path integral saddle implements this gluing. When the area is not fixed, a generalized sewing yields a bulk dual described by the canonical purification $ ext{√} ho_{ar{R}}$ with a dominant area $A_{ ext{dom}}$, linking the sewn state to a bulk geometry governed by that area. The results extend prior work on Dutta–Faulkner’s canonical purification and relate to multipartite reflected entropy, providing a concrete CFT dual to bulk cut-and-paste geometries and offering a path to include higher-derivative corrections.

Abstract

The CPT map allows two states of a quantum field theory to be sewn together over CPT-conjugate partial Cauchy surfaces $R_1,R_2$ to make a state on a new spacetime. We study the holographic dual of this operation in the case where the original states are CPT-conjugate within $R_1,R_2$ to leading order in the bulk Newton constant $G$, and where the bulk duals are dominated by classical bulk geometries $g_1,g_2$. For states of fixed area on the $R_1,R_2$ HRT-surfaces, we argue that the bulk geometry $g_1 \# g_2$ dual to the newly sewn state is given by deleting the entanglement wedges of $R_1,R_2$ from $g_1,g_2$, gluing the remaining complementary entanglement wedges of ${\bar R}_1, {\bar R}_2$ together across the HRT surface, and solving the equations of motion to the past and future. The argument uses the bulk path integral and assumes it to be dominated by a certain natural saddle. For states where the HRT area is not fixed, the same bulk cut-and-paste is dual to a modified sewing that produces a generalization of the canonical purification state $\sqrtρ$ discussed recently by Dutta and Faulkner. Either form of the construction can be used to build CFT states dual to bulk geometries associated with multipartite reflected entropy.

Figures (3)

• Figure 1: Various representations of $\alpha \#_R \beta$. Top Row: Tensor network states $\alpha, \beta$ both have boundaries containing regions called $R$ and associated RT surfaces $\gamma_R$ (dashed arcs). At left, the outputs in $R$ are contracted using the bilinear form $B$. This defines $\alpha \#_R \beta$. If $B$ is constructed from the adjoint map and a local anti-linear CPT operation, the contraction over $R$ becomes local as shown at right. Center Row: The isometries $V_1,V_2$ defined by the entanglement wedges have been used to rewrite the contraction defining $\alpha \#_R \beta$ as acting on truncated tensor networks from which the entanglement wedge of $R$ has been excised. The truncated networks define new states $\alpha_{trunc}, \beta_{trunc}$. At left, the contraction is implemented by the bilinear form $\tilde{B} = B \circ (V_1 \otimes V_2)$. If $V_1, V_2$ are CPT conjugate under a local CPT operation, this contraction becomes local on $\gamma_R$ as at right. Bottom Row: The analogous cut-and-paste construction for time-symmetric slices of bulk spacetimes with CPT-conjugate regions $R$. Slices of the original spacetimes are shown at left, while the result of the cut-and-paste is shown at right.
• Figure 2: Constructing a bulk saddle for $Z = |\psi_1 \#_R \psi_2|^2$. Top Row: When the regions $R=R_1$ and $R= R_2$ are CPT-conjugate, path integrals (semi-circles, shown separately at left) for pure CFT states $\psi_1$ on $R_1{\bar{R}}_1$ and $\psi_2$ on $R_2{\bar{R}}_2$ can be sewn together to give a path integral representation of $\psi_1 \#_R \psi_2$ on ${\bar{R}_1}{\bar{R}}_2$ (right). In the time-symmetric case, the path integrals may be taken to be Euclidean. Center Row: The corresponding path integral for the norm $Z = |\psi_1 \#_R \psi_2|^2$ of the sewn state is shown at left. Another representation of the same path integral is shown at right, differing only by a diffeomorphism that compresses to semi-circles those path-integral-parts that were (nearly) entire circles at left. Bottom Row: The dominant bulk saddles for path integrals computing the individual norms $Z_1 = |\psi_1|^2$, $Z_2 = |\psi_2|^2$ can be cut open through the entanglement wedges of $R_1, R_2$ as shown at left. Since the two states are CPT conjugate on $R$, the data on the upper cuts is also CPT-conjugate, as is the data on the lower cuts. As a result, we can join the cut saddles to build a saddle for $Z = |\psi_1 \#_R \psi_2|^2$ as shown at right. Dashed lines indicate $t=0$ in the wedges dual to ${\bar{R}}_1$, ${\bar{R}}_2$. The $t=0$ surface in the glued saddle (right) is obtained by gluing together these parts of $t=0$ from the original saddles (left).
• Figure 3: A cut-and-paste on a single bulk geometry. Top Row: At left, two boundary regions $R_1,R_2$ and their RT surfaces $\gamma_{R_1}, \gamma_{R_2}$ are marked on a time-symmetric slice of a bulk geometry $g_\psi$. The wedges of $R_1,R_2$ are assumed to be CPT-conjugate. At right, a new intermediate boundary region $R_{int}$ satisfying ${\bar{R}}_2 \supset R_{int} \supset R_1$ has been marked along with its RT surface $\gamma_{R_{int}}$. Center Row: On the boundary of $g_\psi$ we have defined $R_{sew} = R_{int} \cup R_2$ (left). At right is the geometry $g_{\sqrt{\rho_{int}}}$ dual to the canonical purification $\sqrt{\rho_{R_{int}}}$ of $\rho_{R_{int}}$. The upper half of its boundary is $R_{int}$, while the lower half defines ${\bar{R}}_{int}$. Here we take $R_{sew} = {\bar{R}}_{int} \cup R_1$. Assuming conditions (ii) and (iii) from the main text, the entanglement wedge of $R_{sew}$ in $g_\psi$ (left) is the union of wedges for $R_{int}$ and $R_2$ (shaded regions), while in $g_{\sqrt{\rho_{int}}}$ (right) the entanglement wedge of $R_{sew}$ is the union of wedges for ${\bar{R}}_{int}$ and $R_1$ (also shaded). The shaded regions in the two geometries are CPT-conjugate. Bottom: Excising the entanglement wedge of $R_{sew}$ from both $g_\psi$ and $g_{\sqrt{\rho_{int}}}$ and gluing the remaining (unshaded) parts along $\gamma_{R_{sew}} = \gamma_{R_{int}} \cup \gamma_{R_{1,2}}$ gives $g_\psi \#_{R_{sew}} g_{\sqrt{\rho_{R_{int}}}}$. The gluing along $\gamma_{R_{int}}$ simply reassembles the full wedge dual to ${\bar{R}}_1 \cap {\bar{R}}_2$ in $g_\psi$. Gluing along the remaining piece of $\gamma_{R_{sew}}$ identifies $\gamma_{R_1}$ with $\gamma_{R_2}$.