Central Charge, Newton Coupling, and the Planck Scale

The stress-tensor two-point coefficient $C_T$ is a measurable boundary normalization. In a holographic description where the Ward-normalized stress tensor sources a massless AdS graviton, it fixes that graviton’s effective kinetic coefficient and therefore its gravitational loop parameter. For two-derivative Einstein gravity this becomes an exact relation between $C_T$ and $L^{d-1}/G_{d+1}$. The result separates a robust Planck hierarchy from stronger claims: $C_T$ alone does not determine the string scale, the Kaluza–Klein scale, or the highest trustworthy bulk energy.

Required background. Large-N Factorization and Classical Bulk Scaling supplies the normalization-dependent connected-correlator hierarchy.

Helpful background. Current and Stress-Tensor CFT Data defines $C_T$. Heavy Thresholds, Species, and the Gravitational Cutoff explains why many light fields can lower the effective gravitational cutoff.

The Ward-normalized stress tensor

Work at separated Euclidean points in boundary dimension $d>2$. Fix the convention

$$\langle T_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle =\frac{C_T}{(x^2)^d}\mathcal{I}_{\mu\nu,\rho\sigma}(x),$$

with

$$I_{\mu\nu}(x) =\delta_{\mu\nu}-2\frac{x_\mu x_\nu}{x^2}, \qquad \mathcal{I}_{\mu\nu,\rho\sigma}(x) =\frac12\bigl( I_{\mu\rho}I_{\nu\sigma} +I_{\mu\sigma}I_{\nu\rho} \bigr) -\frac1d\delta_{\mu\nu}\delta_{\rho\sigma}.$$

The conformal stress tensor has scaling dimension $d$, so the powers of $x$ make $C_T$ dimensionless. Its normalization is not freely adjustable: the Ward identity fixes

$$P_\nu=\int_{S^{d-1}}\mathrm dS^\mu\,T_{\mu\nu}$$

to generate translations with the chosen coordinate normalization. An improvement can change contact terms, but after the conformal stress tensor and this charge normalization are fixed, multiplying $T_{\mu\nu}$ by an arbitrary constant would change the translation generator. Thus a quoted “central charge” must identify its tensor and Ward conventions. The tensor structure above is that of Osborn and Petkou 1994, § 2, eqs. (2.22)–(2.23).

From the metric source to the graviton kinetic term

Let $\gamma^{(0)}_{\mu\nu}$ be the boundary metric and define the connected Euclidean generating functional by

$$W[\gamma^{(0)}]=-\log Z_{\mathrm{CFT}}[\gamma^{(0)}].$$

With a covariant metric source, its first variation is

$$\delta W =\frac12\int \mathrm d^dx\,\sqrt{\gamma^{(0)}}\, \langle T^{\mu\nu}\rangle\,\delta\gamma^{(0)}_{\mu\nu}.$$

At leading order in a semiclassical bulk saddle,

$$W[\gamma^{(0)}]=S_{\mathrm{ren}}^{\mathrm{os}}[\gamma^{(0)}].$$

Consequently, the nonlocal quadratic response of the renormalized on-shell action to $\gamma^{(0)}_{\mu\nu}=\delta_{\mu\nu}+h^{(0)}_{\mu\nu}$ is the separated-point stress-tensor two-point function. Local terms are contact terms and can depend on the counterterm scheme; the coefficient of the nonlocal kernel cannot.

Use dimensionless Poincaré coordinates $Y^a=(z,u^\mu)$, where $z$ is radial, and write

$$g_{ab}=L^2\bigl(\bar g_{ab}+h_{ab}\bigr).$$

where $\bar g_{ab}$ is the unit-radius AdS metric and $h_{ab}$ is dimensionless. Define $G_{\mathrm{kin}}$ operationally by writing the transverse-traceless quadratic action as

$$S_{\mathrm{TT}}^{(2)} =\frac{L^{d-1}}{16\pi G_{\mathrm{kin}}}\, \mathcal Q_{\mathrm{TT}}[h],$$

where $\mathcal Q_{\mathrm{TT}}$ is dimensionless and has the standard Einstein normalization. In a two-derivative Einstein action, $G_{\mathrm{kin}}=G_{d+1}$. Since an action is dimensionless, $[G_{\mathrm{kin}}]=\text{length}^{d-1}$, and only $L^{d-1}/G_{\mathrm{kin}}$ can multiply this dimensionless quadratic form.

The transverse-traceless boundary value $h^{(0)}_{\mu\nu}(x')$ at boundary point $x'$ is the source. Solving the linearized bulk equation gives a bulk-to-boundary kernel with scalar factor

$$\frac{z^d}{\bigl(z^2+\lvert u-x'\rvert^2\bigr)^d}$$

and inversion tensors that produce $\mathcal I_{\mu\nu,\rho\sigma}(u-x')$. On shell, the bulk integral reduces to a radial boundary term of the form $z^{1-d}h\,\partial_z h$. After the divergent local pieces are cancelled, differentiating the remaining nonlocal $h^{(0)}h^{(0)}$ term twice gives

$$A_d \equiv \frac{d+1}{d-1} \frac{\Gamma(d+1)}{\pi^{d/2}\Gamma(d/2)}, \qquad C_T=A_d\frac{L^{d-1}}{8\pi G_{\mathrm{kin}}}.$$

The radial boundary term, kernel normalization, and exact coefficient are displayed in Buchel et al. 2010, § 3.1, eqs. (3.11)–(3.15), pp. 6–7 for their bulk $D=d+1\ge5$ Gauss–Bonnet family; its Einstein limit gives the coefficient above, whose Einstein boundary calculation extends to $d>2$. Skenderis 2002, §§ 3–4 explains why the asymptotic solution, counterterms, and finite variation must be treated together.

The operational definition of $G_{\mathrm{kin}}$ is important beyond Einstein gravity. Higher-curvature terms can alter the transverse-traceless kinetic coefficient. Sen and Sinha 2014, § 1, eqs. (1.7) and (1.9)–(1.13), pp. 4–5, with the derivative extension on p. 6 treat the stated class of local curvature Lagrangians. In a nondegenerate massless-graviton sector, $C_T$ fixes its effective kinetic coefficient, but not a generic bare Newton parameter.

For later scale comparisons, write

$$G_L\equiv G_{\mathrm{kin}}(E\sim L^{-1})$$

for this AdS-scale effective coefficient. Threshold matching and running can make the coupling relevant at a much higher candidate cutoff differ from $G_L$.

The effective Planck hierarchy

Define the reduced effective Planck length associated with this kinetic term by

$$\ell_{\mathrm{kin}}^{d-1}=8\pi G_L.$$

Then

$$\frac{\ell_{\mathrm{kin}}}{L} = \left(\frac{A_d}{C_T}\right)^{1/(d-1)}.$$

Thus, at fixed $d$—or more generally when $C_T/A_d\gg1$—the AdS curvature radius separates from the effective Planck length. The dimensionless kinetic factor that enters gravitational power counting at energies of order $L^{-1}$ is

$$\frac{G_L}{L^{d-1}} =\frac{A_d}{8\pi C_T}.$$

If $C_T\sim N^2$ and a controlled bulk EFT with a fixed number of light fields exists, pure-graviton loops and matter-loop contributions involving only gravitational vertices are suppressed by $N^{-2}$ at $E\sim L^{-1}$, up to numerical factors. Loops governed by independent matter self-couplings need not be; their suppression requires additional connected-correlator data. This conclusion presupposes a controlled bulk EFT rather than deriving one from a large number alone.

The notation used for “central charge” varies by dimension. The conversion must precede the bulk inference.

$C_T$ for $d>2$. This is the coefficient of the Ward-normalized separated-point stress-tensor correlator and enters the kinetic formula directly.

$c$ in $d=4$. This is the coefficient of the Weyl-squared trace anomaly in the convention below. Convert it with $C_T=40c/\pi^4$ before using the kinetic formula.

$a$ in $d=4$. This is the Euler-density anomaly coefficient. It is independent in general and cannot replace $C_T$.

$c$ in $d=2$. This is the Virasoro central charge and uses the separate Brown–Henneaux relation.

In odd boundary dimension there is no analogous local Weyl-anomaly coefficient on a smooth closed manifold, so $C_T$ is the unambiguous label to retain.

AdS₅/CFT₄ normalization checkpoint

In four dimensions, let $c$ denote the coefficient of the Weyl-squared term in the trace anomaly,

$$\langle T^\mu{}_{\mu}\rangle =\frac{c}{16\pi^2}W_{\mu\nu\rho\sigma}^2 -\frac{a}{16\pi^2}E_4 +\text{local scheme-dependent terms}.$$

For this convention, Osborn and Petkou 1994, § 8 gives $C_T=40c/\pi^4$; Henningson and Skenderis 1998, §§ 2–3 derive the corresponding anomaly from renormalized five-dimensional gravity. Inserting $d=4$ into $A_d$ then yields

$$C_T=\frac{5}{\pi^3}\frac{L^3}{G_{\mathrm{kin}}} =\frac{40c}{\pi^4}.$$

In four-dimensional $\mathcal N=4$ $SU(N)$ super-Yang–Mills theory, free-field counting and stress-tensor nonrenormalization give the exact protected result

$$C_T=\frac{10(N^2-1)}{\pi^4}, \qquad c=\frac{N^2-1}{4};$$

see Arutyunov and Frolov 1999, § 3, eq. (3.30) and the following Ward-identity equation, p. 8. In the classical two-derivative bulk limit $G_{\mathrm{kin}}=G_5$, and only the leading large-$N$ part is retained:

$$C_T=\frac{10N^2}{\pi^4}+O(N^0), \qquad \frac{L^3}{G_5}=\frac{2N^2}{\pi}+O(N^0),$$

and, with $\ell_{\mathrm{kin}}^3=8\pi G_5$ in this limit,

$$\frac{\ell_{\mathrm{kin}}}{L} = \left(\frac{4\pi^2}{N^2}\right)^{1/3} \bigl[1+O(N^{-2})\bigr] \sim N^{-2/3}.$$

The exact $-1$ in $N^2-1$ is an order-$N^0$ effect, the same large-$N$ order at which bulk quantum corrections enter; it should not be silently folded into a tree-level Einstein coefficient. The checkpoint also guards against replacing $C_T$ by $c$ without the conversion factor. Inferring a ten-dimensional Planck length additionally requires the compact $S^5$ volume and the ten-dimensional Newton constant, while the string scale requires the ‘t Hooft coupling.

Species multiplicity and the gravitational window

A small single-field coupling need not control a sum over many fields. Let $N_{\mathrm{eff}}(E)$ denote the number of species active in a declared gravitational observable at bulk energy $E$, including mass thresholds and spin- and channel-dependent weights. Define $\Lambda_{\mathrm{sp}}$ as the candidate scale at which this collective correction becomes order one, and let $G_{\mathrm{sp}}\equiv G_{\mathrm{eff}}(E\sim\Lambda_{\mathrm{sp}})$. The estimate is implicit because $N_{\mathrm{eff}}$ and the matched effective coupling must be evaluated self-consistently there; species well above that scale are not part of its active count.

The threshold and species analysis in gravity EFT explains the distinction between loop, black-hole, and entropy criteria. The four-dimensional black-hole argument of Dvali and Redi 2008, §§ II–III, pp. 045027-2–045027-6 and the diverse-dimensional EFT interpretation of van de Heisteeg et al. 2024, §§ 2–3 share the following parametric form while differing in hypotheses and numerical factors:

$$\Lambda_{\mathrm{sp}}^{d-1} \sim \frac{1}{N_{\mathrm{eff}}(\Lambda_{\mathrm{sp}})G_{\mathrm{sp}}}.$$

It is convenient to define

$$\kappa_d = \frac{d+1}{d-1} \frac{\Gamma(d+1)} {8\pi^{d/2+1}\Gamma(d/2)}.$$

so that $C_T=\kappa_d L^{d-1}/G_L$. Combining the two relations gives

$$\bigl(\Lambda_{\mathrm{sp}}L\bigr)^{d-1} \sim \frac{C_T} {\kappa_d N_{\mathrm{eff}}(\Lambda_{\mathrm{sp}})} \frac{G_L}{G_{\mathrm{sp}}}.$$

The simpler power $\Lambda_{\mathrm{sp}}L\sim[C_T/(\kappa_dN_{\mathrm{eff}})]^{1/(d-1)}$ follows when threshold matching and running do not make $G_{\mathrm{sp}}/G_L$ parametrically large or small. All versions remain estimates with dimension-, channel-, spin-, threshold-, and criterion-dependent factors. At fixed normalized $C_T$ and fixed $G_{\mathrm{sp}}/G_L$, increasing the active species count lowers the candidate cutoff even though the AdS-scale graviton kinetic coefficient is unchanged.

What the hierarchy licenses

The strongest conclusion follows from a short inference chain. First fix the Ward normalization of $T_{\mu\nu}$. Then match its $C_T$ to the quadratic massless-graviton action. Finally, check the spectrum and every cumulative loop sum in the intended energy window. Skipping a link changes what has actually been established.

Massless-graviton kinetic term. Normalized $C_T$ supplies $G_L/L^{d-1}$. The additional assumption is a holographic dictionary in which the Ward-normalized stress tensor couples to that graviton.

Graviton loops at energy $L^{-1}$. Normalized $C_T$ supplies parametric $1/C_T$ suppression. Dimensionless vertices and the rest of the bulk EFT expansion still require control.

Species-sensitive cutoff. Normalized $C_T$ supplies the AdS-scale coupling $G_L$. One must additionally provide the active spectrum, thresholds, spin weights, observable, breakdown criterion, and matching from $G_L$ to $G_{\mathrm{sp}}$.

String, Kaluza–Klein, and higher-derivative scales. Normalized $C_T$ does not determine them directly. They require spectral gaps, compactification data, and microscopic couplings.

At fixed $d$, $C_T\to\infty$ by itself gives $\ell_{\mathrm{kin}}/L\to0$ within the assumed dictionary. A controlled EFT window above $L^{-1}$ additionally requires compactification and higher-derivative control and

$$\frac{C_T}{N_{\mathrm{eff}}(\Lambda_{\mathrm{sp}})} \frac{G_L}{G_{\mathrm{sp}}} \longrightarrow\infty.$$

When $G_{\mathrm{sp}}/G_L$ has no parametric growth, this reduces to $N_{\mathrm{eff}}/C_T\to0$. If instead $N_{\mathrm{eff}}\sim C_T$, the Planck length still becomes small while the species scale can remain $O(L^{-1})$. Once $d$, the Ward convention, and the operational definition of $G_{\mathrm{kin}}$ are fixed, $A_d$ is universal. The two adversarial tests are distinct: different quantities denoted $c$ require convention conversion, while different spectra or threshold matching can give theories with the same normalized $C_T$ different ultraviolet windows.

The matching assumes that the Ward-normalized boundary stress tensor sources the relevant massless graviton. In a higher-curvature theory, $G_{\mathrm{kin}}$ is the renormalized coefficient of the transverse-traceless mode around the chosen AdS vacuum, not necessarily a bare action parameter. The result does not establish a unique local bulk dual, nor does it determine compactification data or the derivative expansion.

The Radius, Couplings, and the Parameter Map keeps the AdS, Planck, string, and compactification scales separate. The D3-Branes and AdS5/CFT4 regime analysis shows how additional microscopic data relate them in a top-down example. The next article moves from the graviton normalization to interaction strengths for other normalized single-trace operators.

Common pitfalls

Rescaling the stress tensor as though it were an arbitrary operator. The translation Ward identity fixes its physical normalization. Convert between published tensor and anomaly conventions instead of rescaling $T_{\mu\nu}$ silently.

Calling every large quantity a central charge. In four dimensions, $a$, $c$, and $C_T$ are different data even when a special theory makes some of them equal. State the anomaly convention and perform the conversion before using a Newton relation.

Identifying the Planck hierarchy with the full ultraviolet window. A small $\ell_{\mathrm{kin}}/L$ controls the AdS-scale graviton kinetic expansion. Species sums, running and threshold matching, higher-derivative corrections, compactification modes, and strings require independent information.

Counting fields without their thresholds. The species estimate uses the weighted fields active in the declared observable at the candidate scale. A heavy field can be absent from that active count while still leaving matched local interactions below threshold.

Exercises

Convention-conversion stress test

Assuming two-derivative Einstein gravity, so $G_{\mathrm{kin}}=G_5$, insert $d=4$ into the general formula and infer $L^3/G_5$ from the anomaly coefficient $c$. Then repeat the calculation incorrectly by substituting $c$ for $C_T$. By what factor does the incorrect answer differ from the correct one?

Solution — convention conversion

For $d=4$, $(d+1)/(d-1)=5/3$, $\Gamma(5)=24$, and $\Gamma(2)=1$. Therefore

$$C_T=\frac53\frac{24}{\pi^2}\frac{L^3}{8\pi G_5} =\frac{5}{\pi^3}\frac{L^3}{G_5}.$$

Using $C_T=40c/\pi^4$ gives

$$\left.\frac{L^3}{G_5}\right|_{\mathrm{correct}} =\frac{\pi^3}{5}C_T =\frac{8c}{\pi}.$$

If one instead sets $C_T=c$, one obtains $L^3/G_5=\pi^3c/5$. The ratio of the incorrect result to the correct one is $\pi^4/40$. The error is a convention factor, not a physical disagreement.

Growing-species stress test

Suppose $C_T\sim N^2$, $N_{\mathrm{eff}}(\Lambda_{\mathrm{sp}})\sim N^\alpha$, and $G_{\mathrm{sp}}/G_L\sim N^0$. How does the species scale behave in AdS units? For which $\alpha$ does it fail to become parametrically large, even though $\ell_{\mathrm{kin}}/L\to0$?

Solution — growing species sector

The Planck hierarchy is $\ell_{\mathrm{kin}}/L\sim N^{-2/(d-1)}$ and therefore improves for every $\alpha$. By contrast, the species estimate gives

$$\Lambda_{\mathrm{sp}}L \sim N^{(2-\alpha)/(d-1)}$$

up to an $N$-independent coefficient. It grows for $\alpha<2$, stays order one for $\alpha=2$, and decreases for $\alpha>2$. Thus a large $C_T$ guarantees the effective Planck hierarchy under the assumed dictionary, but not a parametrically high species scale.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Arutyunov, Gleb, and Sergey Frolov. 1999. “Three-Point Green Function of the Stress-Energy Tensor in the AdS/CFT Correspondence,” Physical Review D 60, 026004. Open PDF.
• Buchel, Alex, Jorge Escobedo, Robert C. Myers, Miguel F. Paulos, Aninda Sinha, and Michael Smolkin. 2010. “Holographic GB Gravity in Arbitrary Dimensions,” Journal of High Energy Physics 03, 111. Open PDF.
• Dvali, Gia, and Michele Redi. 2008. “Black Hole Bound on the Number of Species and Quantum Gravity at LHC,” Physical Review D 77, 045027. Open PDF.
• Henningson, Måns, and Kostas Skenderis. 1998. “The Holographic Weyl Anomaly,” Journal of High Energy Physics 07, 023. Open PDF.
• Osborn, Hugh, and Anastasios C. Petkou. 1994. “Implications of Conformal Invariance in Field Theories for General Dimensions,” Annals of Physics 231, 311–362. Open PDF.
• Sen, Kallol, and Aninda Sinha. 2014. “Holographic Stress Tensor at Finite Coupling,” Journal of High Energy Physics 07, 098. Open PDF.
• Skenderis, Kostas. 2002. “Lecture Notes on Holographic Renormalization,” Classical and Quantum Gravity 19, 5849–5876. Open PDF.
• van de Heisteeg, Lars, Cumrun Vafa, Max Wiesner, and David H. Wu. 2024. “Species Scale in Diverse Dimensions,” Journal of High Energy Physics 05, 112. Open PDF.