Hot multiboundary wormholes from bipartite entanglement

TL;DR

This work analyzes the CFT states dual to hot multiboundary AdS$_3$ wormholes, showing that in the high-temperature regime the states factorize into sewn copies of the thermofield double, yielding localized bipartite entanglement away from vertices where multiple sheets meet. The pair-of-pants geometry serves as the fundamental building block, with a detailed CFT path-integral and bulk-HRT analysis confirming that the boundary entanglement structure is captured by a tensor-network sewn from $|TFD\rangle$ blocks, up to exponential corrections. Finite-temperature corrections introduce a code subspace encoding near-horizon degrees of freedom and reveal phase transitions in the entanglement structure, while the causal shadow area remains fixed, signaling a residual tripartite component. The results extend naturally to general $n$-boundary and planar configurations, suggesting a broad tensor-network/combinatorial framework for understanding bulk connectivity in AdS/CFT and guiding future constructions of \Sigma-prime" wormholes and higher-dimensional generalizations.

Abstract

We analyze the 1+1 CFT states dual to hot (time-symmetric) 2+1 multiboundary AdS wormholes. These are black hole geometries with high local temperature, $n \ge 1$ asymptotically-AdS$_3$ regions, and arbitrary internal topology. The dual state at $t=0$ is defined on $n$ circles. We show these to be well-described by sewing together tensor networks corresponding to thermofield double states. As a result, the entanglement is spatially localized and bipartite: away from particular boundary points ("vertices") any small connected region $A$ of the boundary CFT is entangled only with another small connected region $B$, where $B$ may lie on a different circle or may be a different part of the same circle. We focus on the pair-of-pants case, from which more general cases may be constructed. We also discuss finite-temperature corrections, where we note that the states involve a code subspace in each circle.

Figures (17)

• Figure 1: A simple tensor network displaying the localized purely-bipartite entanglement characteristic of holographic $|TFD\rangle$ states at large $N$ on scales longer than the thermal scale. Each node represents a region in the CFT of scale longer than the thermal scale. We focus mainly on CFT states on $S^1 \times {\mathbb R}$ where one takes a high-temperature limit in order to fit many such long-distance regions onto the circle, though one may equally-well consider the planar case. The solid links are the entangling tensors implied by \ref{['eq:TFD']}. The dashed lines guide the eye by linking neighbouring regions in each of the two CFTs.
• Figure 2: Two topologically-distinct ways in which three copies of the tensor network in figure \ref{['fig:TFDnet']} can be sewn together (left figures) into a single tensor network (right figures) defining a state on 3 copies of the system. The dashed lines (red in color version) internal to the left diagrams guide the eye toward recognising the 3 constituent copies of the network in figure \ref{['fig:TFDnet']}. Links that meet across adjoining pairs of dashed lines are contracted, establishing entanglement between the remaining boundaries (marked 1, 2, and 3). In the bottom-left figure, two parts of the outermost tensor network are contracted with each other, resulting in two well-separated regions of boundary 1 becoming entangled with each other as shown in the bottom-right figure. As discussed below, all 3-boundary time-symmetric vacuum wormholes with pair-of-pants topology (orientable with no handles) and large horizons correspond at the moment of time-symmetry to one of the cases shown, or to the degenerate case that interpolates between them, when described in the "round" conformal frame in which the energy density is taken to be constant along each of the 3 boundaries. Although we show only a simplified cartoon of the full tensor network, we argue below that sewing the actual $|TFD\rangle$ tensor networks together in this way describes the corresponding CFT states with exponential accuracy away from the two 'vertices' in each diagram where 3 $|TFD\rangle$'s meet.
• Figure 3: The quotient of the hyperbolic plane $H^2$ by $\Gamma$. The pair of labeled geodesics are identified by $g$, so the region between them forms a fundamental domain for the quotient. The minimal closed geodesic $H$ is the horizon for the resulting BTZ geometry.
• Figure 4: The surface $\Sigma$ as a quotient of the Poincaré disc for $n=3$. The pairs of labeled geodesics (blue and red in colour version) are identified by the action of $\Gamma$. The region of the Poincaré disc bounded by these geodesics provides a fundamental domain for the quotient. $B_3$, $B_2$ and $B_1 = B_{1+} \cup B_{1-}$ become the desired three circular boundaries. There are corresponding minimal closed geodesics $H_3$, $H_2$ and $H_1 = H_{1+}\cup H_{1-}$. The lengths $L_a$ of these geodesics fully characterize the geometry of $\Sigma$.
• Figure 5: The region $\Sigma_{+}$ bounded by the geodesics $G_{ab}$, half each of $B_2,B_3$, and $B_{1+}$ shown in the Poincaré disc (left) and the BTZ frame strip (right). The BTZ presentation is chosen to place the half-horizon $H_{3+}$ along the BTZ horizon. The geodesics $G_{13}$, $G_{23}$ are respectively the lines $x=- \frac{L_3}{4}$, $x=\frac{L_3}{4}$. In contrast, $G_{12}$ lies in the upper half of the strip; its endpoints have $x=x_1,x_2$ with $\rho=+\infty$. Half each of $B_1, B_2$ is mapped respectively to the line segments $x \in [-\frac{L_3}{4}, x_1]$, $x \in [x_2,\frac{L_3}{4}]$ at $\rho=\infty$ respectively, whilst half of $B_3$ is mapped to $\rho=-\infty$. The corresponding $\Sigma_-$ is the symmetric region below $G_{23}$ in the Poinaré disk (left) and has an identical representation in the BTZ strip.