Radius, Couplings, and the Parameter Map

A holographic parameter map contains several independent expansions. Large $N$ suppresses suitable bulk quantum loops; large ‘t Hooft coupling can suppress string-scale curvature corrections; compactification controls whether a lower-dimensional truncation is useful. None of these statements follows from another without model-specific flux quantization and normalization data.

Required background. Central charge, Newton coupling, and the Planck scale supplies the general large-$N$ scaling, and Anti-de Sitter geometry fixes the radius $L$. Helpful background. $\mathcal N=4$ SYM field content and couplings supplies the boundary example; gravitational EFT power counting explains the distinct derivative and loop expansions.

First application: the AdS5 × S5 parameter map

For type-IIB string theory on AdS$_5\times S^5$ dual to $SU(N)$ $\mathcal N=4$ super-Yang–Mills—the example introduced by Maldacena 1998, §§2–3 and sharpened at the correlator level by Gubser, Klebanov, and Polyakov 1998, pp. 109–112—choose

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

and normalize the self-dual five-form so that it carries $N$ units of flux. Flux quantization gives

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

Therefore

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

The first ratio controls the local stringy derivative expansion in a background whose curvature is $O(L^{-2})$. The second controls the string genus expansion. A weakly curved, weakly coupled string description requires both $\lambda\gg1$ and $N\gg\lambda$ in this convention. The classical supergravity limit takes these conditions together; “large $N$” alone is insufficient.

The ten-dimensional gravitational normalization is fixed by

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

Reducing on a round $S^5$ of volume $\pi^3L^5$ gives

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

Equivalently, the Weyl-anomaly normalization is

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

at leading large $N$ for $SU(N)$, with the exact free-field value differing by the decoupled $U(1)$-sized subtraction, $a=c=(N^2-1)/4$. This coefficient is an invariant checkpoint on the reduction and Newton-constant conventions, as reviewed in Aharony et al. 2000, §§3.1 and 4.1.

What each limit licenses

Boundary or top-down datum	Bulk ratio	What becomes controlled	What remains uncontrolled
$N\to\infty$ with suitable coupling scaling	$G_5/L^3\sim N^{-2}$	five-dimensional gravitational loops	string-scale derivatives
$\lambda\to\infty$	$\alpha'/L^2=\lambda^{-1/2}$	local $\alpha'$ corrections	genus corrections unless $g_s\ll1$
$g_s\to0$	genus weight $g_s^{2h-2}$	string loops	curvature in string units
compact spectrum	$m_{\mathrm{KK}}L$	truncation error below the KK scale	modes at or above the compactification scale

AdS$_5\times S^5$ has $m_{\mathrm{KK}}\sim L^{-1}$, so it does not have a parametric separation between the AdS and Kaluza–Klein scales. Five-dimensional gauged supergravity is consistent for selected fields, but a generic process at energy $E\sim L^{-1}$ cannot infer that the full KK tower is heavy.

The map is model specific. Other brane systems change powers, numerical coefficients, compact volumes, and the relation between central data and rank. Universal large-$N$ prose must never be used to manufacture a flux quantization formula.

Adversarial check: fixed small coupling is not a gravity limit

Take $N\to\infty$ at fixed $\lambda\ll1$. Then $g_s=\lambda/(4\pi N)\to0$ and bulk string loops are suppressed, but

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

The background is strongly curved in string units, so the two-derivative supergravity action receives unsuppressed $\alpha'$ corrections. The strongest surviving claim is a planar string description, not a weakly curved Einstein geometry. Reversing the order—first taking $\lambda\gg1$ and then choosing $N\gg\lambda$—can control both expansions, but any observable must still be below the relevant string and KK thresholds.

Controlled limits and handoff

The numerical coefficients displayed belong to AdS$_5\times S^5$ with its stated flux and reduction conventions; only the scaling logic generalizes automatically. The supergravity regime requires both $N\gg\lambda$ and $\lambda\gg1$, plus energies below string and Kaluza–Klein thresholds. Bulk Fields and Boundary Operators uses $L$ to make masses dimensionless, and later top-down chapters derive model-specific flux maps.

Exercise

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

Solution

Equating the two expressions gives $\pi L^3/(8G_5)=N^2/4$. Solving yields $G_5/L^3=\pi/(2N^2)$. This dimensionless ratio is insensitive to a simultaneous rescaling of dimensionful units and is a useful convention check.

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. arXiv. DOI.
• Gubser, Steven S., Igor R. Klebanov, and Alexander M. Polyakov. “Gauge Theory Correlators from Non-Critical String Theory.” Physics Letters B 428 (1998): 105–114. arXiv. DOI.
• Maldacena, Juan M. “The Large N Limit of Superconformal Field Theories and Supergravity.” Advances in Theoretical and Mathematical Physics 2 (1998): 231–252. arXiv. DOI.