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arXiv:2610.25969

Higher-Dimensional Test of the Holographic Dual of Entanglement Cohomology

R. ~C. ~Rashkov

Published Oct 19, 2026• 0 citations
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Abstract

We extend the holographic test of entanglement cohomology from $d=2$ to higher-dimensional conformal field theories. For a $d$-dimensional CFT with a spherical entangling region $A$, the vacuum modular Hamiltonian is known explicitly, and the boundary relative entropy of a primary-operator perturbation can be computed via conformal Ward identities. On the bulk side, the perturbation is dual to a scalar field in AdS$_{d+1}$, and the bulk canonical energy is computed using the JLMS relation. We carry out the computation explicitly for $d=3$ and $d=4$ and find that the boundary and bulk results agree at first order in the perturbation, with identical coefficients $κ_d Δ^2 / C_T$, where $κ_3 = π^2/12$ and $κ_4 = 3π^3/16$. The result extends the quantitative verification of the holographic dual of entanglement cohomology beyond the $d=2$ case treated in the companion paper and confirms that the modular Hamiltonian perturbation is the correct holographic image of the first entanglement cohomology class. We discuss the role of the conformal anomaly, the shape of the entanglement wedge, and an outlook on TsT-deformed backgrounds. \keywords{Entanglement entropy \and Cyclic cohomology \and von Neumann algebras \and Holography \and Modular Hamiltonian}

DOI
10.48550/arxiv.2610.25969

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We extend the holographic test of entanglement cohomology from d=2d=2d=2 to higher-dimensional conformal field theories. For a ddd-dimensional CFT with a spherical entangling region AAA, the vacuum modular Hamiltonian is known explicitly, and the boundary relative entropy of a primary-operator perturbation can be computed via conformal Ward identities. On the bulk side, the perturbation is dual to a scalar field in AdSd+1_{d+1}d+1​, and the bulk canonical energy is computed using the JLMS relation. We carry out the computation explicitly for d=3d=3d=3 and d=4d=4d=4 and find that the boundary and bulk results agree at first order in the perturbation, with identical coefficients κdΔ2/CTκ_d Δ^2 / C_Tκd​Δ2/CT​, where κ3=π2/12κ_3 = π^2/12κ3​=π2/12 and κ4=3π3/16κ_4 = 3π^3/16κ4​=3π3/16. The result extends the quantitative verification of the holographic dual of entanglement cohomology beyond the d=2d=2d=2 case treated in the companion paper and confirms that the modular Hamiltonian perturbation is the correct holographic image of the first entanglement cohomology class. We discuss the role of the conformal anomaly, the shape of the entanglement wedge, and an outlook on TsT-deformed backgrounds. \keywords{Entanglement entropy \and Cyclic cohomology \and von Neumann algebras \and Holography \and Modular Hamiltonian}
We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional bulk modular hamiltonian in an emergent AdS space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein’s equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.
The entanglement spectrum of a bipartite quantum system is given by the distribution of eigenvalues of the modular Hamiltonian. In this work, we compute the entanglement spectrum in the vacuum state for a subregion of a ddd-dimensional conformal field theory (CFT) admitting a holographic dual. In the case of a spherical (or planar) entangling surface, we recover known results in two dimensions, including the Cardy formula in the high energy regime. In higher dimensions d>2d>2d>2, we analytically determine a generalization of the Cardy formula valid at large energies and consistent with previous studies of CFT spectra in the literature. We also investigate numerically the spectrum at energy levels far above the modular ground state energy. We extend our analysis to the supersymmetric point of Einstein-Maxwell gravity, providing exact results when d=2,3d=2,3d=2,3, and a generalization of the Cardy formula at high energies in generic dimension ddd. We consider small shape deformations of a spherical entangling surface, for both the non-supersymmetric and the supersymmetric cases. In all cases we find that the high-energy scaling of the microcanonical entropy with the modular energy is unaffected by the shape deformation. This result suggests that the high-energy regime of the entanglement spectra carries universal information, independent of the shape of the entangling surface.
We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, KKK, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by 4GN4G_N4GN​. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS3_33​ we find that the KKK-dependent holographic entanglement entropy is governed by cm=3ℓ2GN.c_m=\frac{3 \ell}{2 G_N}.cm​=2GN​3ℓ​. For a thermal state at high conformal temperatures we find that, the entropy is governed by ceff=3ℓ2GNKℓ−K2ℓ2−42c_{\rm eff}=\frac{3 \ell}{2 G_N} \frac{K \ell- \sqrt{K^2 \ell^2-4}}{2}ceff​=2GN​3ℓ​2Kℓ−K2ℓ2−4​​, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal TTˉT \bar{T}TTˉ like operator - and find SEE=ceff3ln⁡ ⁣(2Rsin⁡ϕ0ε).S_{EE} =\frac{c_{\rm eff}}{3} \ln\!\left(\frac{2 R \sinϕ_0}ε\right) .SEE​=3ceff​​ln(ε2Rsinϕ0​​). This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of ceffc_{\mathrm{eff}}ceff​ while clarifying that this state is not the state dual to global AdS.
For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk fields. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the first-order result given by the entanglement first law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the first-order result in arXiv:1509.00113 .
A holographic correspondence has been recently developed between higher spin theories in three-dimensional anti-de Sitter space (AdS3) and two-dimensional Conformal Field Theories (CFTs) with extended symmetries. A class of such dualities involves SL(N, R) × SL(N, R) Chern-Simons gauge theories in the (2 + 1)-dimensional bulk space-time, and CFTs with WN symmetry algebras on the (1 + 1)-dimensional boundary. The topological character of the bulk theory forces one to reconsider standard geometric notions such as black hole horizons and entropy, as well as the usual holographic dictionary. Motivated by this challenge, in this note we present a proposal to compute entanglement entropy in the WN CFTs via holographic methods. In particular, we introduce a functional constructed from Wilson lines in the bulk Chern-Simons theory that captures the entanglement entropy in the CFTs dual to standard AdS3 gravity, corresponding to SL(2, R) × SL(2, R) gauge group, and admits an immediate generalization to the higher spin case. We explicitly evaluate this functional for several known solutions of the bulk theory, including charged black holes dual to thermal CFT states carrying higher spin charge, and show that it reproduces expected features of entanglement entropy, study whether it obeys strong subadditivity, and moreover show that it reduces to the thermal entropy in the appropriate limit.
In this paper, we study the entanglement entropy of a single interval on a cylinder in two-dimensional TT‾T\overline{T}TT-deformed conformal field theory (CFT). For such case, the (R\'enyi) entanglement entropy takes a universal form in a CFT. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of the conformal perturbation theory. We find that the correction to the entanglement entropy is nonvanishing in the finite temperature case, while it is vanishing in the finite size case. For the deformed holographic large ccc CFT, which is proposed to be dual to a AdS3{\mathrm{AdS}}_{3}AdS3​ gravity in a finite region, we find the agreement with the holographic entanglement entropy via the Ryu-Takayanagi formula. Moreover, we compute the leading order correction to the R\'enyi entropy, and discuss its holographic picture as well.