A UV-Finite Ryu-Takayanagi Relation from Relative Entropy in AdS$_3$/CFT$_2$

Technical Summary: A UV-Finite Ryu–Takayanagi Relation from Relative Entropy in AdS3/CFT2

Problem Statement
The Ryu–Takayanagi (RT) proposal posits a direct equivalence between the entanglement entropy of a spatial subregion in a holographic Conformal Field Theory (CFT) and the area of a minimal surface in the dual Anti-de Sitter (AdS) geometry. However, from the perspective of Operator-Algebraic Quantum Field Theory (AQFT), the von Neumann entanglement entropy is ill-defined for type III local algebras; it is ultraviolet (UV) divergent and regulator-dependent. Existing derivations of the RT relation often rely on this divergent quantity or its renormalized versions. The authors aim to establish a rigorous, UV-finite RT relation in the $AdS_3/CFT_2$ setting by replacing the von Neumann entropy with the relative entropy, which is mathematically well-defined and UV-finite.

Methodology
The paper constructs a correspondence between the boundary relative entropy and the variation of the bulk geodesic length through the following steps:

1. Boundary Setup (CFT): The authors consider a free massless scalar field on $M^{1+1}$, decomposing it into commuting chiral components. They focus on a diamond region $C_I$ on the boundary with a base interval $I$. The vacuum state $|\Omega\rangle$ is compared to a family of chiral Virasoro coherent states $|\Psi_f\rangle = U(f)|\Omega\rangle$, generated by unitary excitations $U(f)$ supported on $I$.
2. Relative Entropy Calculation: Using the Bisognano–Wichmann/Borchers theorem, the modular Hamiltonian $K_I$ for the vacuum is identified. The relative entropy $S^I_{\text{rel}}(\omega_0 \| \omega_f)$ between the vacuum and the excited state is computed. By employing Hollands' exact result for chiral relative entropy, this quantity is expressed as an integral involving the Schwarzian derivative $\{\rho(u), u\}$ of the boundary reparametrization $\rho$ generated by the excitation $f$.
3. Bulk Dual (Gravity): The coherent state $|\Psi_f\rangle$ is mapped to a dual bulk geometry using the Fefferman–Graham dictionary. The expectation value of the boundary stress tensor acts as the source for the bulk metric. In $AdS_3$, the most general asymptotically AdS solution consistent with Brown–Henneaux boundary conditions is the Bañados metric. The authors restrict to the right chiral sector ($\bar{L}=0$) and linearize the Bañados metric around pure $AdS_3$.
4. Geometric Variation: The authors calculate the first-order variation of the length of the RT geodesic ($\gamma_A$) in the perturbed Bañados geometry. A key technical observation is that the metric perturbation component $h_{uu}$ is independent of the radial coordinate $z$. This allows the variation of the geodesic length to be expressed directly in terms of the boundary source function $L(u)$.
5. Identification: The authors identify the boundary source $L(u)$ in the Bañados metric with the Schwarzian derivative of the reparametrization $\rho$ via the holographic dictionary and the Virasoro Ward identity. The rigidity of three-dimensional gravity (lack of local graviton degrees of freedom) ensures that this boundary identification uniquely fixes the bulk geometry.

Key Contributions and Results
The primary result is the derivation of an exact equality between the boundary relative entropy and the variation of the RT geodesic length, valid to linear order in the metric perturbation:

$$S^I_{\text{rel}}(\omega_0 \| \omega_f) = \frac{\delta \text{Length}(\gamma_A)}{4G_N} = \frac{c}{6} \delta A$$

where $\delta A$ is the dimensionless change in geodesic length, $c = 3R/(2G_N)$ is the Brown–Henneaux central charge, and $G_N$ is Newton's constant.

Specifically, the paper demonstrates:

• UV-Finiteness: The relation is established without reference to the divergent von Neumann entropy, utilizing the well-defined Araki–Uhlmann relative entropy.
• Operator-Algebraic Foundation: The derivation relies solely on the Bisognano–Wichmann/Borchers theorem and the holographic dictionary, providing an operator-algebraic counterpart to the RT relation.
• Exact Matching: The Schwarzian functional appearing in the boundary relative entropy (derived from Hollands' formula) is shown to be identical to the functional governing the variation of the bulk geodesic length, mediated by the Bañados source $L(u)$.

Significance and Claims
The authors claim to have established a "rigorous, UV-finite RT relation" that avoids the ambiguities of regularization inherent in entanglement entropy. By working with relative entropy, they provide a mathematically sound formulation of the holographic principle in $AdS_3/CFT_2$ that holds for coherent excitations of the vacuum.

The paper situates this result within the broader context of the Faulkner–Lewkowycz–Maldacena (FLM) and Jafferis–Lewkowycz–Maldacena–Suh (JLMS) relations. In their specific setup, the excitation corresponds to a pure gravitational back-reaction (a change in the metric with no bulk matter fields). Consequently, the bulk entanglement entropy and bulk modular energy variations vanish or cancel in a specific manner, reducing the general JLMS relation to the direct equality derived in the paper. The authors note that this treatment remains at the level of classical pure gravity; a quantum treatment involving the quantization of metric perturbations (gravitons) is left for future work.