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arXiv:2606.30723

Ryu-Takayanagi area from Virasoro modular data

Jennifer Lin

Published Jun 29, 2026• 0 citations
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Abstract

We show that in holographic 2d CFTs, the entanglement entropies across several choices of global state and subregion can be written in a way that at once has a microscopic interpretation and matches the leading large$-c$ organization of the Ryu-Takayanagi formula. This representation is obtained by applying crossing symmetry to the replica manifolds. From the boundary point of view, each rewritten entropy looks like an algebraic entanglement entropy for the Virasoro algebra restricted to the region, with center labels obtained by coarse-graining heavy primaries of the BCFT on the regulated region into bins labeled by Liouville momenta. At large $c$, a resulting sum over bins is dominated by a saddle, and the $O(c)$ part of the entropy comes from the Cardy density of heavy primaries in the dominant bin. We identify this $O(c)$ part of the entropy with the Ryu-Takayanagi area. Physically, this suggests a concrete statistical origin for the Ryu-Takayanagi area as coming from coarse-grained Virasoro intertwiners across the entangling cut. The result also provides quantitative criteria for the amount of coarse-graining allowed for the consistency of the interpretation.

DOI
10.48550/arxiv.2606.30723

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We show that in holographic 2d CFTs, the entanglement entropies across several choices of global state and subregion can be written in a way that at once has a microscopic interpretation and matches the leading large−c-c−c organization of the Ryu-Takayanagi formula. This representation is obtained by applying crossing symmetry to the replica manifolds. From the boundary point of view, each rewritten entropy looks like an algebraic entanglement entropy for the Virasoro algebra restricted to the region, with center labels obtained by coarse-graining heavy primaries of the BCFT on the regulated region into bins labeled by Liouville momenta. At large ccc, a resulting sum over bins is dominated by a saddle, and the O(c)O(c)O(c) part of the entropy comes from the Cardy density of heavy primaries in the dominant bin. We identify this O(c)O(c)O(c) part of the entropy with the Ryu-Takayanagi area. Physically, this suggests a concrete statistical origin for the Ryu-Takayanagi area as coming from coarse-grained Virasoro intertwiners across the entangling cut. The result also provides quantitative criteria for the amount of coarse-graining allowed for the consistency of the interpretation.
A bstract We provide a derivation of the Ryu-Takayanagi (RT) formula in 3D gravity for generic boundary subregions — including RT surface phase transitions — directly from the dual two-dimensional conformal field theory (CFT). Our approach relies on the universal statistics of the algebraic conformal data and the large- c behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFTs with and without Cardy boundaries. The RT formula is obtained directly from the CFT in the high-temperature regime. Two direct applications are: 1) A simple derivation of the multi-interval entanglement entropy for the vacuum state of a single CFT; 2) A CFT-based detection of the emergence of replica wormholes in the context of entanglement islands and black hole microstate counting. Our framework yields the first holographic random tensor network that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the eigenstate thermalization hypothesis (ETH) in the dual field theory in the large- c limit — a paradigm we refer to as It from ETH .
Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.
We consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of ‘entropy operators’ in general systems with a large number of degrees of freedom.
The Ryu-Takayanagi formula equates the area of a minimal surface with the von Neumann entropy of a boundary subregion, and leaves two things about that identification open. The first is how sharply a geometry fixes an entropy. Fannes-Audenaert answers with a Hilbert-space dimension, which diverges as the cutoff is removed however close the two states are. We replace it with the capacity of entanglement, the variance of the modular energy, whose square root grows like the square root of the entangling area where the dimensional factor grows like the regulated volume. The bound is dimension-free and saturated, and it makes the ambiguity of the entropy subextensive for any perturbation whose capacity is small compared with SvN2S_{vN}^2SvN2​ times the trace norm. The second is what the area means for a single state, since compression and dilution rates are defined only for many copies while a geometry describes one. When a single replica saddle dominates near α=1α= 1α=1, every smooth Rényi entropy at fixed α>1α> 1α>1 agrees with SvNS_{vN}SvN​ to O(SvN)\textit{O}(\sqrt{S_{vN}})O(SvN​​), as do the smooth min- and max-entropies. The minimal surface therefore fixes every one-shot entropy of the state at once, with large central charge playing the role of large copy number in the asymptotic equipartition property. As a consequence we bound how far outside the holographic entropy cone a holographic state can appear to fall, leaving estimation and certification open.
In this note, I revisit the problem of computing the entanglement entropy of a single interval in the ground state of a 2d CFT. I write the leading-order result in three different ways: once by doing the replica trick with the nnn-replicated cylinder partition function computed in the ``closed string channel"; once by computing the nnn-replicated cylinder partition function in the ``open string channel", where the entanglement entropy can be related to the density of states in a boundary CFT defined on the interval; and for holographic CFTs, once as the log of a Plancherel measure for a certain noncompact quantum group. I comment on the implications for what the Ryu-Takayanagi area term might be counting in holographic CFTs.
A bstract We use the framework of 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.