Higher-Dimensional Test of the Holographic Dual of Entanglement Cohomology

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 $κ_d Δ² / C_T$, where $κ_3 = π<sup>2/12$ and $κ_4 = 3π<sup>3/16$. 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}