Holographic Dual of Synge's Identity in AdS/CFT

Technical Summary: Holographic Dual of Synge's Identity in AdS/CFT

Problem Statement
The paper addresses a specific question within the framework of holographic duality: if a finite bulk observable and a conformal field theory (CFT) observable are already known to be exactly related, can a differential identity satisfied by the geometric (bulk) member be directly transported to a corresponding equation on the CFT side? Unlike standard holographic renormalization, which often relies on boundary regulators, heavy-operator limits, or geodesic-saddle approximations to generate Ward identities, this work seeks to derive CFT equations directly from exact, finite observable pairs without additional approximations. The specific geometric identity under investigation is Synge's identity (the Hamilton–Jacobi equation for geodesic flow) applied to finite bulk geodesics in arbitrary dimensions.

Methodology
The authors utilize recently established "exact finite pairs" between bulk and boundary observables:

1. Scalar Two-Point Function: An exact relation between a scalar two-point function in the Weyl frame of an open solid torus and a finite geodesic in the associated AdS interior [11].
2. Disjoint Entanglement Entropy: An exact relation between the finite disjoint entanglement entropy ($S_{\text{disj}}$) between disjoint balls in arbitrary-dimensional CFT and the volume of the entanglement wedge cross-section (EWCS) [10].

The derivation proceeds in three stages:

• Symmetric Representative: The authors first work in a symmetric open-solid-torus geometry where the bulk geodesic is radial. In this frame, the relationship between CFT endpoints ($P, Q$) and bulk endpoints ($p, q$) is explicit, with the bulk geodesic length $\ell$ equating to the Weyl-frame hyperbolic separation $L(P, Q) = \log(R_2/R_1)$.
• Application of Synge's Identity: Synge's identity, $g_{\mu\nu}\partial_\mu \ell \partial_\nu \ell = 1$, is applied to the bulk geodesic. This yields specific endpoint derivative relations (Hamilton–Jacobi branches) for the bulk length $\ell$ with respect to the bulk radial coordinates.
• Transport to CFT: Using the exact observable relations (which express CFT quantities as functions of $\ell$), the bulk derivative relations are pulled back to the CFT parameters ($R_1, R_2$). This generates differential equations for the CFT observables.
• Conformal Invariance: Finally, the derived equations are rewritten in terms of the conformally invariant inversive product $\varrho$, generalizing the results from the symmetric representative to arbitrary disjoint spherical configurations.

Key Contributions and Results

1. Scalar Correlator Sector:

   • Applying Synge's identity to the exact finite pair for the scalar two-point function $G_W$ yields two endpoint equations.
   • The sum of these equations recovers the planar dilatation Ward identity in the planar frame, demonstrating that the common-scale invariance of the correlator is the CFT dual of the bulk geodesic equation.
   • The difference of the equations yields a finite relative-scale evolution equation, retaining information about the separation scale that is lost in the standard Ward identity.
   • In the conformally invariant frame (variable $\varrho$), these combine into a first-order differential equation: $(\varrho^2 - 1) \frac{d}{d\varrho} \log G_W(\varrho) = 2\Delta$. Integrating this recovers the known exact finite correlator.

2. Entanglement Entropy Sector:

   • For the disjoint entanglement entropy $S_{\text{disj}}$, Synge's identity induces endpoint equations that govern the variation of entropy with respect to the boundary scales.
   • The relative-scale equation is governed by a transverse hyperbolic area term ($\sinh^{D-2}(\ell/2)$).
   • In two dimensions ($D=2$), this equation reduces exactly to the previously identified renormalization-group (RG) equation for entanglement entropy [9].
   • In higher dimensions ($D>2$), the equation represents a finite relative-scale evolution rather than a linear RG equation, reflecting the growth of the transverse area.
   • The conformally invariant form is given by $\frac{dS_{\text{disj}}}{d\varrho} = -2\pi E_D \Omega_{D-2} \frac{2^{(D-1)/2}}{(\varrho-1)^{D/2}\sqrt{\varrho+1}}$. Integrating this with the physical boundary condition ($S_{\text{disj}} \to 0$ as $\varrho \to \infty$) reproduces the exact hypergeometric entropy formula derived in Ref. [10].

3. Cavity Limit:

   • The paper demonstrates that in the limit where the disjoint regions become adjacent (the "cavity" limit, $\epsilon \to 0$), the derived equations correctly reproduce the standard area-law scaling for $D>2$ and the logarithmic law for $D=2$, consistent with the Ryu–Takayanagi relation.

Significance and Claims
The paper claims that the primary significance of this work lies in the directness and simplicity of the derivation. Because the starting point consists of exact, finite observable pairs, the translation of Synge's geometric identity to CFT equations does not require:

• Additional boundary regulators.
• Heavy-operator limits.
• Geodesic-saddle approximations.

The authors emphasize that the resulting CFT equations (the endpoint and scale equations) are the genuine holographic counterparts of Synge's identity. The known closed-form solutions for correlators and entanglement entropy are recovered here not as starting points, but as integrated solutions of these invariant differential equations. This establishes a practical method where, once an exact observable-level map is established, the differential relations obeyed by the paired observables can be studied directly, bypassing asymptotic limiting prescriptions. The work clarifies the structural link between bulk geodesic dynamics and boundary scale invariance/evolution in a finite, non-perturbative setting.