Radius, Couplings, and the Parameter Map

A holographic parameter map is a collection of independent approximation tests, not one large-$N$ slogan. In the canonical AdS$_5\times S^5$ example, $\lambda$ controls curvature in string units, $\lambda/N$ controls the perturbative string coupling, $N$ controls the effective bulk Planck hierarchy, and the compactification has no parametric Kaluza–Klein gap. This page derives every coefficient in one declared convention and then separates the conclusions that each ratio licenses.

Required background. Central charge, Newton coupling, and the Planck scale fixes the stress-tensor and reduced-Planck conventions used below. Anti-de Sitter geometry defines the curvature radius $L$. Helpful background. $\mathcal N=4$ SYM field content and couplings supplies the boundary theory card; gravitational EFT power counting explains why derivative, loop, and threshold expansions must be tested separately.

Freeze the convention before mapping parameters

Take the local gauge algebra to be $\mathfrak{su}(N)$, with Hermitian fundamental generators and Euclidean gauge term

$$\operatorname{Tr}(T^aT^b)=\frac12\delta^{ab}, \qquad S_E\supset \frac{1}{2g_{\mathrm{YM}}^2} \int d^4x\,\operatorname{Tr}F_{\mu\nu}F^{\mu\nu}.$$

In the standard D3-brane normalization, the gauge and type-IIB axiodilaton couplings are identified by

$$\tau_{\mathrm{YM}} \equiv \frac{\theta_{\mathrm{YM}}}{2\pi} +\frac{4\pi i}{g_{\mathrm{YM}}^2} =C_0+\frac{i}{g_s},$$

so that

$$g_{\mathrm{YM}}^2=4\pi g_s, \qquad \lambda\equiv g_{\mathrm{YM}}^2N.$$

Here $C_0$ is the asymptotic Ramond–Ramond axion, $N$ is both the $SU(N)$ number-of-colors parameter and the integer five-form flux, $g_s=e^{\Phi_\infty}$ is the asymptotic string coupling, $\alpha'=\ell_s^2$ defines the string length, and $L$ is the common radius of the AdS$_5$ and round $S^5$ factors. The Lie group itself has rank $N-1$. We use the asymptotically normalized Einstein metric

$$g^{(E)}_{MN}=\bigl(g_s e^{-\Phi}\bigr)^{1/2}g^{(s)}_{MN}.$$

It equals the string-frame metric on this constant-dilaton background, where $e^\Phi=g_s$. Thus the same $L$ enters the string-curvature test and the Einstein reduction below. A different constant frame convention is harmless only if its lengths and Newton constant are rescaled together Aharony et al. 2000, §1.3, eqs. (1.15)–(1.17), printed pp. 17–19, PDF.

For an unambiguous Planck comparison, define the reduced $D$-dimensional Planck length by

$$\ell_{P,D}^{\,D-2}\equiv 8\pi G_D.$$

Other authors absorb factors of $8\pi$ into their symbol $\ell_P$. The definition, rather than the name, should therefore accompany every numerical Planck-radius relation. The coupling identification and five-form normalization are summarized in Aharony et al. 2000, §3.1, eqs. (3.7)–(3.10), printed pp. 59–60, PDF.

Flux quantization fixes the string-scale curvature

In the conventional unrescaled, gauge-invariant Ramond–Ramond field normalization, write the sphere component as

$$\widetilde F_5\big|_{S^5}=\frac{4L^4}{g_s}\,\omega_5, \qquad \int_{S^5}\omega_5=\pi^3,$$

where $\omega_5$ is the volume form of the unit five-sphere. The integer condition is

$$\frac{1}{(2\pi)^4\alpha'^2}\int_{S^5}\widetilde F_5=N.$$

Here $H_3=F_3=0$, so the gauge-invariant flux and D3 Page charge coincide; backgrounds with Chern–Simons mixing require the appropriate Page-charge combination instead. The self-dual D3-brane solution then gives

$$L^4=4\pi g_sN\alpha'^2 =\lambda\alpha'^2.$$

Consequently,

$$\frac{L}{\ell_s}=\lambda^{1/4}, \qquad \epsilon_{\alpha'}\equiv\frac{\alpha'}{L^2}=\lambda^{-1/2}, \qquad g_s=\frac{\lambda}{4\pi N}.$$

The first small parameter measures the local derivative expansion in a background with curvature $O(L^{-2})$; the last quantity organizes perturbative string worldsheets. These are different tests. The radius relation follows directly from the D3 harmonic function in Maldacena 1998, §2, eqs. (2.2)–(2.3), printed p. 3, PDF.

The generic derivative-counting unit is $\alpha'/L^2=\lambda^{-1/2}$, but maximal type-IIB supersymmetry removes the would-be local massless-field interactions at orders $\alpha'$ and $\alpha'^2$. The first such interaction in the ten-dimensional effective action is schematically $\alpha'^3R^4$. For observables whose finite-tension correction is governed by this local massless-field EFT, it contributes at order $\lambda^{-3/2}$. The classic thermal example is worked out in Gubser, Klebanov, and Tseytlin 1998, §1, eqs. (4)–(8), pp. 2–3, PDF; their $g_{\mathrm{YM}}^2=2\pi g_s$ convention means that $2g_{\mathrm{YM,GKT}}^2N=\lambda$ here. Protected quantities can receive fewer corrections, while extended-string observables can have $\lambda^{-1/2}$ worldsheet corrections, as the exact circular Wilson-loop asymptotics illustrate Erickson, Semenoff, and Zarembo 2000, §§3–4, PDF. Thus $\lambda^{-3/2}$ is neither a universal first correction nor a universal nonzero coefficient.

Newton constants and two Planck hierarchies

Normalize the Lorentzian ten-dimensional Einstein term by $S_{10}\supset(16\pi G_{10})^{-1}\int d^{10}x\sqrt{-g}\,R$. Standard type-IIB normalization gives Aharony et al. 2000, §1.3, eqs. (1.15)–(1.17), printed pp. 17–19, PDF

$$2\kappa_{10}^2=(2\pi)^7g_s^2\alpha'^4, \qquad 16\pi G_{10}=2\kappa_{10}^2,$$

and hence

$$G_{10}=8\pi^6g_s^2\alpha'^4, \qquad \frac{G_{10}}{L^8}=\frac{\pi^4}{2N^2}.$$

For the unwarped round sphere, $\operatorname{Vol}(S^5)=\pi^3L^5$. Dimensional reduction of the Einstein term therefore yields

$$G_5=\frac{G_{10}}{\pi^3L^5}, \qquad \frac{G_5}{L^3}=\frac{\pi}{2N^2}.$$

The reduced Planck definition now converts these Newton couplings into genuine length ratios:

$$\frac{\ell_{P,10}}{L} =\left(\frac{4\pi^5}{N^2}\right)^{1/8}, \qquad \frac{\ell_{P,5}}{L} =\left(\frac{4\pi^2}{N^2}\right)^{1/3}.$$

Both ratios vanish as $N\to\infty$, but with different powers because $G_{10}$ and $G_5$ have different mass dimensions. The five-dimensional ratio is the useful loop-counting scale for processes already captured by a five-dimensional truncation; the ten-dimensional ratio diagnoses quantum gravity before compactification.

Central data close the normalization chain

In four-dimensional CFT conventions with trace anomaly

$$\langle T^\mu{}_{\mu}\rangle =\frac{c}{16\pi^2}W^2 -\frac{a}{16\pi^2}E_4+ \text{scheme-dependent terms},$$

classical five-dimensional Einstein gravity gives

$$a=c=\frac{\pi L^3}{8G_5}=\frac{N^2}{4}.$$

This is the leading large-$N$ result. The exact protected anomaly coefficients of the interacting $SU(N)$ $\mathcal N=4$ theory are

$$a=c=\frac{N^2-1}{4}, \qquad C_T=\frac{40c}{\pi^4} =\frac{10(N^2-1)}{\pi^4}.$$

Thus the classical bulk calculation reproduces the $O(N^2)$ term; the $-1$ is an $O(N^0)$ quantum correction and cannot be demanded of tree-level supergravity. The complete Kaluza–Klein one-loop sum reproduces this subleading anomaly term Mansfield, Nolland, and Ueno 2003, pp. 207–210, PDF. The holographic anomaly calculation appears in Henningson and Skenderis 1998, §3.2, eqs. (20)–(23), pp. 7–8, PDF, the exact supersymmetric anomaly relations in Anselmi et al. 1998, §§2–4, PDF, and the $C_T$ conversion in Osborn and Petkou 1994, §8, PDF.

The following card collects the map without hiding which entries are exact, model-specific, or asymptotic.

AdS₅ × S⁵ parameter map in the declared SU(N) and type-IIB conventions

Relation	Physical meaning	Coefficient source	Status
gYM2 = 4πgs	Boundary gauge coupling versus asymptotic dilaton	D3-brane action and trace convention	Exact convention-dependent identification
L4 = 4πgsNα′2 = λ α′2	AdS and sphere radius in string units	Five-form flux quantization in the D3 solution	Exact within this top-down background and normalization
G10/L8 = π4/(2N2)	Ten-dimensional quantum-gravity strength at the AdS scale	Type-IIB Einstein normalization plus the radius relation	Algebraic identity in the stated two-derivative Einstein normalization
G5/L3 = π/(2N2)	Five-dimensional gravitational loop scale	Reduction over a round S5 of volume π3L5	Tree-level two-derivative reduction identity
a = c = N2/4	Weyl anomaly inferred from classical Einstein gravity	Renormalized five-dimensional on-shell action	Leading large-N result; exact field theory has (N2 − 1)/4
mKKL = O(1)	Sphere harmonics lie at the AdS scale	Equal radii and the S5 harmonic spectrum	Model-specific obstruction to a parametric KK gap

What each limit actually licenses

Independent approximation tests; passing one row does not pass the others

Limit or hierarchy	Small parameter	What becomes controlled	What does not follow
N → ∞ at fixed λ	G5/L3 ∼ N−2; at fixed external process, each extra closed-string handle is relatively gs2 ∼ N−2	Effective bulk loops and the fixed-λ genus expansion	Small curvature in string units or a KK gap
λ → ∞	α′/L2 = λ−1/2	The local string derivative expansion; the first massless type-IIB EFT interaction contributes at O(λ−3/2) when it governs the observable	Weak string coupling unless λ/N → 0
λ/N → 0	gs = λ/(4πN)	Perturbation theory in closed-string topology; genus h + 1 is relatively gs2 to genus h for a fixed process	Small α′ corrections
E/mKK → 0 with mKKL ≫ 1	E/mKK	A lower-dimensional EFT obtained by integrating out generic KK modes	A consistent truncation; that is a nonlinear closure property, not a mass hierarchy

For AdS$_5\times S^5$, scalar and tensor harmonics form towers whose lightest nonzero or omitted modes have $m_{\mathrm{KK}}L=O(1)$; higher harmonics grow up the tower Kim, Romans, and van Nieuwenhuizen 1985, Tables III–VI, pp. 397–399. There is therefore no generic low-energy argument that discards all KK modes while retaining physics at the AdS scale. Nevertheless, selected fields close under the nonlinear equations and define consistent five-dimensional truncations: every solution of the truncated theory uplifts, even though the omitted modes are not parametrically heavy Cvetič et al. 2000, §§1–2, PDF. Consistency does not make that truncation adequate for an arbitrary observable.

The familiar weakly coupled type-IIB supergravity corner combines

$$\lambda\gg1, \qquad g_s=\frac{\lambda}{4\pi N}\ll1,$$

or parametrically

$$1\ll\lambda\ll N.$$

For the vacuum and fixed AdS-scale processes with no additional large local invariant, this controls ten-dimensional classical supergravity, including its KK tower: local curvatures and invariant momenta must still remain small in string and Planck units. A few-field five-dimensional description additionally needs either a genuine hierarchy for the process in question or a consistent truncation containing every sourced field. The inequality is also duality-frame specific: when the displayed $g_s$ is large, type-IIB $SL(2,\mathbb Z)$ can sometimes provide a weakly coupled frame. It does not turn the original perturbative genus series into a controlled one, and every transformed charge and coupling must be remapped. In the boundary theory, the same duality can also change the global form and genuine-line lattice; it need not act within one theory specified only as global $SU(N)$.

Adversarial limits

Large $N$ at fixed small $\lambda$. Let $N\to\infty$ with $\lambda\ll1$ fixed. Then $g_s\to0$ and $G_5/L^3\to0$, but

$$\frac{L}{\ell_s}=\lambda^{1/4}\ll1.$$

Bulk loops are suppressed while the background remains strongly curved in string units, so infinitely many $\alpha'$ corrections are unsuppressed. The surviving statement is the planar large-$N$ organization of the gauge theory—and, conditional on the exact duality, a genus-zero string description—not a weakly curved Einstein geometry or a calculable supergravity limit.

Large $\lambda$ at fixed $N$. Curvature in string units becomes small, but $g_s=\lambda/(4\pi N)$ eventually becomes large in the displayed frame. Classical geometry by itself has not controlled string topology.

Large $N$ and large $\lambda$ with no KK hierarchy. Even inside $1\ll\lambda\ll N$, the ten-dimensional derivative and loop expansions can be excellent while $m_{\mathrm{KK}}L$ remains order one. “Classical supergravity” and “a generic five-dimensional low-energy EFT with only a few fields” are therefore different claims.

A reusable parameter-map procedure

For another top-down background, do not copy the powers or numerical factors above. Rebuild the map in this order:

1. Fix the boundary action, generator trace, theta-angle periodicity, global form, and definition of every large parameter.
2. State the higher-dimensional action and Newton normalization, the string or M-theory length, and the metric frame.
3. Impose flux or brane-charge quantization to derive radii in microscopic units.
4. Reduce the gravitational kinetic term. For a direct unwarped product this uses the internal volume; warped backgrounds require the appropriate weighted effective volume.
5. Match an independently normalized observable such as $C_T$, $a$, $c$, a current coefficient, or a protected spectrum. Do not count an input used to calibrate $G_{d+1}$ as a new prediction.
6. List derivative, quantum-loop, string-topology, KK-threshold, backreaction, and observable-specific controls in separate rows, including an explicit failed limit.

This procedure separates universal statements such as $L^{d-1}/G_{d+1}\sim C_T$ from model-specific statements such as $L^4=4\pi g_sN\alpha'^2$.

Common pitfalls

Treating $g_s^{2h-2}$ as a universal correction factor. That is the vacuum topology weight for a closed genus-$h$ worldsheet. External-string vertices add their own powers. For a fixed normalized process, the safe comparison is that adding one handle gives a relative factor $g_s^2$.

Calling $(N^2-1)/4$ a free-field value. It can be computed at weak coupling, but supersymmetric anomaly relations protect it along the $\mathcal N=4$ conformal manifold. The mismatch with $N^2/4$ diagnoses the order of the bulk approximation, not coupling dependence of the exact anomaly.

Using consistent truncation as a synonym for heavy omitted modes. A consistent truncation is closed under the nonlinear equations. Scale separation instead requires $E/m_{\mathrm{KK}}\ll1$, and an AdS-scale separation requires $m_{\mathrm{KK}}L\gg1$.

Exercises

Recover the five-dimensional Newton coupling

Use $a=\pi L^3/(8G_5)$ and the leading result $a=N^2/4$ to recover $G_5/L^3$.

Solution

Equating the two expressions gives $\pi L^3/(8G_5)=N^2/4$. Solving,

$$\frac{G_5}{L^3}=\frac{\pi}{2N^2}.$$

This checks the dimensional reduction without referring separately to $g_s$ or $\alpha'$.

Find a simultaneous classical-string window

Let $\lambda=N^p$ as $N\to\infty$. For which $p$ do curvature corrections and the perturbative string genus expansion both become parametrically small in the displayed type-IIB frame?

Solution

Curvature control requires

$$\frac{\alpha'}{L^2}=N^{-p/2}\to0,$$

so $p>0$. String-loop control requires

$$g_s=\frac{N^{p-1}}{4\pi}\to0,$$

so $p<1$. Both hold precisely for $0<p<1$. At $p=0$ curvature does not improve parametrically; at $p=1$ the string coupling approaches a constant; for $p>1$ it grows.

Convert Newton constants to reduced Planck lengths

Starting from $G_{10}/L^8=\pi^4/(2N^2)$ and $G_5/L^3=\pi/(2N^2)$, derive the two reduced Planck-radius ratios.

Solution

Using $\ell_{P,D}^{D-2}=8\pi G_D$ gives

$$\left(\frac{\ell_{P,10}}{L}\right)^8 =8\pi\frac{G_{10}}{L^8} =\frac{4\pi^5}{N^2},$$

and

$$\left(\frac{\ell_{P,5}}{L}\right)^3 =8\pi\frac{G_5}{L^3} =\frac{4\pi^2}{N^2}.$$

Taking the eighth and third roots produces the stated formulas. The unequal exponents reflect dimensional analysis, not different large-$N$ physics.

Separate KK decoupling from consistent truncation

Write $m_{\mathrm{KK}}L=\mu$ and $EL=x$. What controls integrating out a generic KK mode? What changes when $\mu=O(1)$ but the retained fields form a consistent truncation?

Solution

The threshold ratio is

$$\frac{E}{m_{\mathrm{KK}}}=\frac{x}{\mu}.$$

For processes at the AdS scale, $x=O(1)$, generic KK decoupling needs $\mu\gg1$. In AdS$_5\times S^5$, $\mu=O(1)$, so that mass-hierarchy argument fails. A consistent truncation can still make a selected nonlinear sector classically closed at the two-derivative supergravity level: solutions within it uplift without sourcing omitted modes. It does not imply that the omitted modes are heavy or irrelevant to observables outside that sector.

Where the map is used next

Bulk Fields and Boundary Operators uses $L$ to turn bulk masses into boundary dimensions. D3-Branes and AdS5/CFT4 derives the brane decoupling regimes; Flux Quantization, Compact Factors, and Kaluza–Klein Towers generalizes the charge and compactification steps; and Consistent Truncations and Lower-Dimensional Effective Actions separates nonlinear closure from scale separation.

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

• Aharony, Ofer, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, and Yaron Oz. “Large $N$ Field Theories, String Theory and Gravity.” Physics Reports 323 (2000): 183–386. DOI. Open PDF.
• Anselmi, Damiano, Daniel Z. Freedman, Marcus T. Grisaru, and Andreas A. Johansen. “Nonperturbative Formulas for Central Functions of Supersymmetric Gauge Theories.” Nuclear Physics B 526 (1998): 543–571. DOI. Open PDF.
• Cvetič, Mirjam, Hong Lü, Christopher N. Pope, Azadeh Sadrzadeh, and Tuan A. Tran. “Consistent $SO(6)$ Reduction of Type IIB Supergravity on $S^5$.” Nuclear Physics B 586 (2000): 275–286. DOI. Open PDF.
• Erickson, John K., Gordon W. Semenoff, and Konstantin Zarembo. “Wilson Loops in $\mathcal N=4$ Supersymmetric Yang–Mills Theory.” Nuclear Physics B 582 (2000): 155–175. DOI. Open PDF.
• Gubser, Steven S., Igor R. Klebanov, and Arkady A. Tseytlin. “Coupling Constant Dependence in the Thermodynamics of $\mathcal N=4$ Supersymmetric Yang–Mills Theory.” Nuclear Physics B 534 (1998): 202–222. DOI. Open PDF.
• Henningson, Måns, and Kostas Skenderis. “The Holographic Weyl Anomaly.” Journal of High Energy Physics 07 (1998): 023. DOI. Open PDF.
• Kim, H. J., Larry J. Romans, and Peter van Nieuwenhuizen. “Mass Spectrum of Chiral Ten-Dimensional $N=2$ Supergravity on $S^5$.” Physical Review D 32 (1985): 389–399. DOI.
• Maldacena, Juan M. “The Large $N$ Limit of Superconformal Field Theories and Supergravity.” Advances in Theoretical and Mathematical Physics 2 (1998): 231–252. DOI. Open PDF.
• Mansfield, Paul, David Nolland, and Tatsuya Ueno. “Order $1/N^2$ Test of the Maldacena Conjecture II: The Full Bulk One-Loop Contribution to the Boundary Weyl Anomaly.” Physics Letters B 565 (2003): 207–210. DOI. Open PDF.
• Osborn, Hugh, and Anastasios C. Petkou. “Implications of Conformal Invariance in Field Theories for General Dimensions.” Annals of Physics 231 (1994): 311–362. DOI. Open PDF.