Coarse-Grained Fixed-Point Tensor Networks and Holographic Reflected Entropy in 3D Gravity

TL;DR

The paper develops a boundary-centric, CFT-based derivation of the holographic relation between reflected entropy and entanglement wedge cross sections by building fixed-point BCFT tensor networks from triangulated Euclidean state-preparation manifolds. Through coarse-graining universal heavy CFT data at large central charge, the authors connect boundary path integrals to emergent hyperbolic geometry via Liouville theory with ZZ boundary conditions, enabling geometric interpretations of RE as geodesic lengths. By constructing canonical purifications with BCFT cutting-and-gluing and performing ensemble averaging of OPE data in replica partition functions, they recover the entanglement wedge cross sections (EW, EW$_3$) as the dominant geometric objects, thus establishing $RE = \frac{2\,EW}{4G_N}$ for bipartite and multipartite cases. The results provide a precise CFT mechanism for geometry emergence and offer a framework to explore phase transitions and subleading $1/c$ corrections, quantum corrections, and extensions beyond vacuum AdS$_3$/CFT$_2$. The work thus strengthens the bridge between intrinsic CFT data, tensor-network pictures, and emergent bulk geometry, with potential implications for more general holographic dualities and purification-based entanglement measures.

Abstract

We use the framework of $\textit{fixed-point BCFT tensor networks}$ to present a microscopic CFT derivation of the correspondence between reflected entropy (RE) and entanglement wedge cross section (EW) in AdS$_3$/CFT$_2$, for both bipartite and multipartite settings. These fixed-point tensor networks, obtained by triangulating Euclidean CFT path integrals, allow us to explicitly construct the canonical purification via cutting-and-gluing CFT path integrals. Employing modular flow in the large-$c$ limit, we demonstrate that these intrinsic CFT manipulations reproduce bulk geometric prescriptions, without assuming the AdS/CFT dictionary. The emergence of bulk geometry is traced to coarse-graining over heavy states in the large-$c$ limit. Universal coarse-grained BCFT data for compact 2D CFTs, through the relation to Liouville theory with ZZ boundary conditions, yields hyperbolic geometry on the Cauchy slice. The corresponding averaged replica partition functions reproduce all candidate EWs, arising from different averaging patterns, with the dominant one providing the correct RE and EW. In this way, many heuristic tensor-network intuitions in toy models are made precise and established directly from intrinsic CFT data.

Figures (29)

• Figure 1: An illustration of $\text{RE}=\text{EW}$: The entanglement wedge cross section $\text{EW}(A:B)$ is represented by the cyan line in the left panel. As will be reviewed in Sec \ref{['purificationreview']}, from the AdS bulk point of view the canonical purification is obtained by cutting the manifold open along the pink RT surfaces and gluing it to a CPT-conjugate copy. The entanglement entropy of the resulting pure state, which defines the reflected entropy $\text{RE}(A:B)$, is dual to the RT surface highlighted in cyan. Clearly, it is two times the size of $\text{EW}(A:B)$.
• Figure 2: The tripartite entanglement wedge cross section $\text{EW}_3(A:B:C)$ is defined by minimizing $l_{1}+l_{2}+l_{3}$, where the geodesics $l_i$'s are anchored on the RT surface of $ABC$ and form a closed cycle.
• Figure 3: The general BCFT tensor network representation of a CFT quantum state prepared by a Euclidean path integral is obtained by introducing tiny holes as regulators on the state-preparation manifold and performing the OPE block decomposition; the resulting network is defined on the dual graph, indicated by the green lines.
• Figure 4: Left: Euclidean path integral for computing $\textrm{Tr}\, \rho^4_{AB}$ in the boundary theory and its dual bulk saddle. Each wedge (bounded by solid lines) represents a path integral that prepares $\rho_{AB}$. Right: slicing open the bulk path integral for $\textrm{Tr}\, \rho_{AB}^4$ produces the analogue of the Hartle–Hawking state $\ket{\rho_{AB}^2}$ in the bulk, with the dual CFT state defined on the red dots.
• Figure 5: We depict the vacuum AdS$_3$ solution together with a "three-boundary black hole" solution in hyperbolic slicing. The manifolds are cut open along the $\tau=0$ Cauchy slice, shown in red. On this slice, the cyan curves mark the $1D$ asymptotic boundaries of the Cauchy slice, which are described by ZZ boundary conditions on the Liouville field. The grey surface corresponds to the CFT state-preparation manifold.