SCALE INVARIANT TOPOLOGICAL ONTOLOGY
Scale-Invariant
Reformulation of the RyuâTakayanagi Formula
Author: Rowan Brad Quni-Gudzinas
Affiliation: QNFO Contact:
rowan.quni@qnfo.org ORCID: 0009-0002-4317-5604
ISNI: 0000 0005 2645 6062 DOI:
10.5281/zenodo.17216372 Publication Date: 2025-09-28
Version: 1.0
Given
Let a \(d\)-dimensional boundary
conformal field theory (CFT) be holographically dual to a \((d+1)\)-dimensional asymptotically anti-de
Sitter (AdS) bulk spacetime of radius \(L\). Let \(A
\subset \partial \mathcal{M}\)be a spatial subregion of the
boundary with characteristic linear size \(R\), and let \(\gamma_A\)be the codimension-2 extremal
surface in the bulk that is homologous to \(A\)and satisfies \(\partial \gamma_A = \partial A\).
Assume the RyuâTakayanagi (RT) formula holds in its standard
form:
\[
SA = \frac{\mathrm{Area}(\gammaA)}{4G_N^{(d+1)}},
\]
where \(G_N^{(d+1)}\)is the \((d+1)\)-dimensional Newton constant.
Assume the AdS/CFT dictionary relates the bulk gravitational
parameters to boundary CFT data via:
\[
\frac{L^{d-1}}{GN^{(d+1)}} = kd \cdot a_d,
\]
where \(a_d\)is a dimensionless
central charge of the CFT (e.g., \(a_2 =
c/3\)in \(d=2\), \(a4 = a\)in \(d=4\)), and \(kd\)is a known dimensionless constant
depending only on \(d\).
Let \(\epsilon\)denote a UV cutoff
in the boundary theory, interpreted geometrically as a bulk cutoff at
radial coordinate \(z = \epsilon\).
Goal
Reformulate the RyuâTakayanagi formula in a
scale-invariant manner that eliminates dependence on
absolute scales (\(L\), \(\epsilon\), \(R\)) and isolates the universal,
dimensionless information-theoretic content of holographic entanglement
entropy.
Derivation
Step 1:
Dimensional Analysis of the Standard Formula
In natural units (\(\hbar = c =
1\)), the dimensions are:
\([S_A] = 1\)(dimensionless
entropy),
\([\mathrm{Area}(\gamma_A)] =
[\mathrm{length}]^{d-1}\),
\([G_N^{(d+1)}] =
[\mathrm{length}]^{d-1}\).
Thus, the RT formula is dimensionally consistent but **not
scale-invariant**, as it depends on the absolute AdS scale \(L\)through both \(\mathrm{Area}(\gammaA)\)and \(GN^{(d+1)}\).
Step
2: Express Gravitational Coupling via CFT Central Charge
From the AdS/CFT dictionary, the combination \(L^{d-1}/G_N^{(d+1)}\)is proportional to the
number of degrees of freedom in the boundary CFT. Define the
dimensionless central charge \(a_d\)by:
\[
ad := \frac{1}{kd} \cdot \frac{L^{d-1}}{G_N^{(d+1)}},
\]
so that:
\[
GN^{(d+1)} = \frac{L^{d-1}}{kd a_d}. \tag{1}
\]
This expresses the bulk gravitational coupling in terms of a
dimensionless boundary quantity.
Step 3:
Parametrize the Minimal Surface Area
For a region \(A\)with
characteristic size \(R\), the area of
the extremal surface \(\gamma_A\)in
pure AdS\(_{d+1}\)takes the form:
\[
\mathrm{Area}(\gammaA) = L^{d-1} \cdot \mathcal{F}d\left(
\frac{R}{\epsilon}, \frac{R}{L} \right), \tag{2}
\]
where \(\mathcal{F}_d\)is a
dimensionless function encoding both UV-divergent and finite
contributions. The dependence on \(R/L\)arises only in the finite part; the
divergent part depends solely on \(R/\epsilon\).
For example, in \(d=2\)with an
interval of length \(R\):
\[
\mathrm{Area}(\gamma_A) = 2L \log\left( \frac{R}{\epsilon} \right).
\]
Step
4: Substitute into the RyuâTakayanagi Formula
Substituting (1) and (2) into the RT formula:
\[
SA = \frac{L^{d-1} \cdot \mathcal{F}d(R/\epsilon, R/L)}{4 \cdot
(L^{d-1}/(kd ad))} = \frac{kd ad}{4} \cdot \mathcal{F}_d\left(
\frac{R}{\epsilon}, \frac{R}{L} \right). \tag{3}
\]
The factor \(L^{d-1}\)cancels,
leaving an expression that depends only on **dimensionless
ratios and the dimensionless central charge**
\(a_d\).
Step
5: Isolate the Universal (Scale-Invariant) Term
The function \(\mathcal{F}_d\)generally contains divergent
terms as \(\epsilon \to 0\), but also a
universal term \(\mathcal{F}_d^{\mathrm{univ}}\)that is
independent of \(\epsilon\)and depends
only on the geometry of \(A\)and the
ratio \(R/L\). Under a global scale
transformation:
\[
R \mapsto \lambda R, \quad L \mapsto \lambda L, \quad \epsilon \mapsto
\lambda \epsilon,
\]
the ratio \(R/L\)is invariant, and
thus:
\[
\mathcal{F}_d^{\mathrm{univ}}\left( \frac{R}{L} \right) \mapsto
\mathcal{F}_d^{\mathrm{univ}}\left( \frac{R}{L} \right).
\]
Hence, the universal part of \(S_A\)is
scale-invariant.
For spherical regions, this universal term is known explicitly:
In even \(d\):
\(SA^{\mathrm{univ}} = (-1)^{d/2} \cdot 4 ad
\log(R/\epsilon)\),
In odd \(d\):
\(SA^{\mathrm{univ}} = Fd(a_d)\), a
constant independent of \(R\).
In both cases, the coefficient is proportional to the \(a\)-anomaly (even \(d\)) or the \(F\)-quantity (odd \(d\)), both of which are
dimensionless and intrinsic to the
CFT.
Step
6: Construct a Scale-Invariant Entropy Functional
To extract the scale-invariant content without reference to \(\epsilon\)or \(L\), define the **universal
entanglement entropy** as:
\[
SA^{\mathrm{univ}} := \lim{\epsilon \to 0} \left[ S_A(R, \epsilon) -
S_A^{\mathrm{div}}(R, \epsilon) \right], \tag{4}
\]
where \(S_A^{\mathrm{div}}\)contains
all terms that diverge as \(\epsilon \to
0\). This subtraction is equivalent to the **holographic
renormalization** procedure in the bulk, which removes boundary
counterterms.
Equivalently, for two regions \(A\)and \(A_0\)of the same shape but different sizes,
the difference:
\[
\Delta S = SA(R) - S{A0}(R0)
\]
is scale-invariant when expressed in terms of the dimensionless ratio
\(R/R_0\), and its finite part is
universal.
Step
7: Verify Scale Invariance Under Weyl Transformations
Under a boundary Weyl transformation \(g_{\mu\nu} \mapsto e^{2\omega(x)}
g_{\mu\nu}\), the bulk metric in FeffermanâGraham coordinates
transforms as:
\[
ds^2 = \frac{L^2}{z^2} \left( dz^2 + e^{2\omega(x)} g_{\mu\nu}(x) dx^\mu
dx^\nu \right).
\]
The extremal surface \(\gamma_A\)shifts, and its area changes
by:
\[
\delta \mathrm{Area}(\gammaA) = \int{\partial A} \omega(x) \cdot
(\text{local geometric terms}) + (\text{anomalous term}).
\]
However, the universal part of the entropy
transforms covariantly and, for conformally invariant
states, remains invariant under global rescalings. This
is because the anomaly coefficients \(a_d\)are Weyl-invariant by
construction.
Thus, the combination:
\[
SA^{\mathrm{univ}} = \kappad \cdot a_d
\]
is invariant under \(R \mapsto \lambda
R\), \(L \mapsto \lambda L\),
confirming scale invariance.
Conclusion
Theorem (Scale-Invariant RyuâTakayanagi Formula).
The universal part of holographic entanglement entropy is
scale-invariant and given by:
\[
\boxed{SA^{\mathrm{univ}} = \kappad \cdot a_d}
\]
where:
\(a_d\)is the dimensionless central
charge of the \(d\)-dimensional
boundary CFT (e.g., \(a_2 = c/3\),
\(a_4 = a\)),
\(\kappa_d\)is a universal
numerical constant depending only on spacetime dimension \(d\),
The expression is invariant under global scale transformations \(x^\mu \mapsto \lambda x^\mu\).
For a spherical region of radius \(R\)in even dimensions, this yields:
\[
SA^{\mathrm{univ}} = (-1)^{d/2} \cdot 4 ad \log\left(
\frac{R}{\epsilon} \right),
\]
where the logarithmic dependence on the dimensionless ratio \(R/\epsilon\)reflects the conformal anomaly,
and the coefficient is independent of the AdS radius \(L\) and **Newtonâs constant
\(G_N^{(d+1)}\)**.
This reformulation demonstrates that the **physically
meaningful, observable content** of holographic entanglement
entropy resides entirely in **dimensionless, scale-invariant
data** characterizing the boundary CFT, consistent with the
principles of universal scale invariance and epistemic humility.
Q.E.D.