Central Charge, Newton Coupling, and the Planck Scale

The stress-tensor two-point coefficient determines the normalization of the bulk graviton kinetic term. Consequently, its large-$N$ scaling fixes the dimensionless ratio $L^{d-1}/G_{d+1}$ for an $AdS_{d+1}$ description. It does not, without compactification and species data, determine every Planck, string, or ultraviolet cutoff.

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.

From the stress tensor to the graviton action

Fix the CFT convention

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

where the tensor structure ${\cal I}$ is fixed by conformal symmetry. Expanding the Einstein action about $AdS_{d+1}$ gives a quadratic graviton term of the form

$$S^{(2)}=\frac{L^{d-1}}{16\pi G_{d+1}} \int \mathrm d^{d+1}x\,\sqrt{\lvert\bar g\rvert}\;h{\cal K}h.$$

The GKPW prescription therefore yields

$$C_T=\kappa_d\frac{L^{d-1}}{G_{d+1}},$$

with a known convention-dependent coefficient $\kappa_d$. The scaling relation is robust only after the normalization of $T_{\mu\nu}$ and ${\cal I}$ has been specified.

This stress-tensor normalization follows by differentiating the renormalized bulk action with respect to the boundary metric. The anomaly calculation of Henningson and Skenderis 1998 provides a canonical $AdS_5/CFT_4$ example, while Osborn and Petkou 1994 fixes the general CFT tensor structures.

The effective Planck hierarchy

Define the $(d+1)$-dimensional Planck length by

$$G_{d+1}\sim \ell_{\mathrm P}^{d-1}.$$

Then

$$\frac{\ell_{\mathrm P}}{L} \sim C_T^{-1/(d-1)}.$$

If $C_T\sim N^2$, bulk quantum-gravity loops at the AdS scale are parametrically suppressed. In AdS5, for example, the five-dimensional relation gives $\ell_{\mathrm P}/L\sim N^{-2/3}$. A ten-dimensional Planck length follows only after including the compact $S^5$ volume and the ten-dimensional Newton constant.

Convention and species checks

Four-dimensional authors variously quote $a$, $c$, or $C_T$. These quantities are proportional only after conventions and the theory class are fixed. Substituting a symbol $c$ without the conversion coefficient can therefore produce a wrong Newton normalization.

Moreover, with $N_{\mathrm{sp}}$ light bulk species, a species estimate gives schematically

$$\Lambda_{\mathrm{sp}}^{d-1} \lesssim \frac{1}{N_{\mathrm{sp}}G_{d+1}}.$$

Holding $G_{d+1}$ fixed while increasing $N_{\mathrm{sp}}$ lowers the effective cutoff. Thus $C_T$ fixes the graviton kinetic normalization, not by itself the highest trustworthy energy.

Orders of limits, evidence ceiling, and handoff

The Planck hierarchy follows when $C_T\to\infty$ while the compactification data and number of light species are controlled. If $N_{\mathrm{sp}}$ grows proportionally to $C_T$, the species cutoff in AdS units need not grow at all. This is the adversarial convention-and-species check: two theories can share the same $C_T$ scaling yet have different conversion coefficients and different ultraviolet windows.

Matching the normalized stress-tensor two-point function establishes $L^{d-1}/G_{d+1}$ in the assumed bulk theory and the scaling of graviton-loop suppression. It does not establish that the assumed dictionary is correct or determine the string scale, Kaluza–Klein scale, compactification volume, or higher-derivative couplings. Those independent data are supplied in the spectrum, cutoff, and string-regime chapters.

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

• Henningson, Måns, and Kostas Skenderis. 1998. “The Holographic Weyl Anomaly,” Journal of High Energy Physics 07, 023.
• Maldacena, Juan M. 1998. “The Large N Limit of Superconformal Field Theories and Supergravity,” Advances in Theoretical and Mathematical Physics 2, 231–252.
• Osborn, Hugh, and Andreas Petkou. 1994. “Implications of Conformal Invariance in Field Theories for General Dimensions,” Annals of Physics 231, 311–362.