A first AdS5 times S5 dictionary

Large rank suppresses nonplanar color contractions. Why should it also make a gravitational description classical, and why is a second limit needed to see a smooth geometry? We will derive the conversion factors needed to answer both questions.

Use double-line color counting and the first AdS/CFT dictionary. The latter defines anti-de Sitter space, a boundary source, and its response; those definitions apply here without another source-response derivation. Our boundary theory is four-dimensional $\mathcal N=4$ $SU(N)$ Yang–Mills in the conformal vacuum, with zero scalar expectation values. Here $N\geq2$ is the fundamental representation’s dimension; the Lie-group rank is $N-1$.

The proposed dual is type IIB string theory, a ten-dimensional supersymmetric string theory, on $\mathrm{AdS}_5\times S^5$. Its massless low-energy fields form IIB supergravity. Approximating the full string theory by that gravitational field theory requires the scale comparisons below.

Fix the coupling before using the dictionary

Use Hermitian generators and the ordinary fundamental trace:

$$\operatorname{Tr}_N(t^at^b)=\frac12\delta^{ab}, \qquad F=da-i\,a\wedge a, \qquad S_{\mathrm{gauge}}=-\frac1{2g_{\mathrm{YM}}^2} \int d^4x\,\operatorname{Tr}_N(F_{\mu\nu}F^{\mu\nu}). \tag{N19.1}$$

The coupling is absorbed into the connection $a$. In components the last coefficient is $-1/(4g_{\mathrm{YM}}^2)$, and the ‘t Hooft coupling is $\lambda=g_{\mathrm{YM}}^2N$. For the geometric formulas use Lorentzian signature $(-,+,\ldots,+)$ and the curvature convention in which a round sphere has positive Ricci curvature.

The string description introduces a length $\ell_s=\sqrt{\alpha'}$ and a dimensionless interaction parameter $g_s=e^{\phi_0}$, the exponential of the constant dilaton. A fundamental string has tension $T_{\mathrm F}=1/(2\pi\alpha')$. A D3-brane is an object with three spatial dimensions on which open strings can end. Its slowly varying gauge field has a Dirac–Born–Infeld action. For a diagonal Abelian field, the relevant input is

$$S_{\mathrm{DBI}}=-T_3\int d^4x\, \sqrt{-\det(\eta_{\mu\nu}+2\pi\alpha'F_{\mu\nu})}, \qquad T_3=\frac1{(2\pi)^3g_s\alpha'^2}. \tag{N19.2}$$

This is the local flat-metric form of the probe normalization in Maldacena’s (2.6); the dimensionful version is Aharony et al., (3.12). Only its quadratic term is needed. We do not assume an all-orders non-Abelian determinant prescription.

To obtain that term, let $M^\mu{}_\nu=2\pi\alpha'F^\mu{}_\nu$. Antisymmetry gives $\operatorname{tr}_{\mathrm{spacetime}}M=0$ and $\operatorname{tr}_{\mathrm{spacetime}}M^2=-(2\pi\alpha')^2F_{\mu\nu}F^{\mu\nu}$. Expanding $\exp[\frac12\operatorname{tr}\log(1+M)]$ therefore gives

$$\begin{aligned} \sqrt{-\det(\eta+2\pi\alpha'F)} &=1+\frac{(2\pi\alpha')^2}{4}F_{\mu\nu}F^{\mu\nu} +O\bigl((\alpha'F)^4\bigr),\\ S_{\mathrm{DBI}}^{(2)} &=-\frac1{8\pi g_s}\int d^4x\,\operatorname{Tr}_N F^2. \end{aligned} \tag{N19.3}$$

Summing diagonal entries fixes the ordinary-trace normalization for a stack of branes. Comparing with (N19.1) yields

$$\boxed{g_{\mathrm{YM}}^2=4\pi g_s,\qquad \lambda=4\pi g_sN.} \tag{N19.4}$$

The original paper’s coupling paragraph uses $g_{\mathrm{MAL}}^2=2\pi g_s$: its named gauge coupling is smaller by a factor of two in the square. Equation (N19.4), also explicit in Aharony et al., (3.8), is the translation appropriate to (N19.1).

For completeness, write the axion in a period-one convention as $C_0$. The same dictionary is $\tau_{\mathrm{YM}}=\theta/(2\pi)+4\pi i/g_{\mathrm{YM}}^2=C_0+i/g_s$; the review’s axion $\chi$ is $2\pi C_0$. We take $\theta=C_0=0$ below. A change of electric-magnetic description also acts on boundary global data; the coupling formula alone does not identify every line spectrum.

Recover the two equal radii

A coincident stack of $N$ D3-branes carries $N$ quantized units of the self-dual Ramond–Ramond five-form flux. This is an integer charge, not a continuous parameter to adjust independently of $N$. The constant-dilaton supergravity solution has string-frame metric

$$\begin{aligned} ds_s^2&=H^{-1/2}\eta_{\mu\nu}dx^\mu dx^\nu +H^{1/2}(dr^2+r^2d\Omega_5^2),\\ H(r)&=1+\frac{L^4}{r^4}, \qquad L^4=4\pi g_sN\alpha'^2. \end{aligned} \tag{N19.5}$$

Here $r$ is the radius in the six transverse directions, $d\Omega_5^2$ is the unit-sphere metric, and $L$ is a length. The coefficient fixed by D3 charge is an input from the brane solution, Maldacena, (2.2); it cannot be obtained by dimensional analysis alone.

In the near-horizon region $r\ll L$, drop the one in $H$. With $z=L^2/r$,

$$ds_s^2=\frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right) +L^2d\Omega_5^2, \qquad \boxed{\frac{L^4}{\alpha'^2}=\lambda.} \tag{N19.6}$$

The first term is the Poincaré-patch AdS metric of the bridge lesson. Both factors have radius $L$. Their curvatures have opposite signs; their radii are not imaginary.

The decoupling limit underlying this construction is $\alpha'\to0$ with $U=r/\alpha'$, $g_s$, and $N$ fixed. In that limit $H-1=\lambda/(\alpha'^2U^4)$ dominates, while

$$\frac{ds_s^2}{\alpha'}= \frac{U^2}{\sqrt\lambda}\eta_{\mu\nu}dx^\mu dx^\nu +\sqrt\lambda\,\frac{dU^2}{U^2} +\sqrt\lambda\,d\Omega_5^2. \tag{N19.7}$$

Here $U$ has mass dimension one. The finite string-unit metric shows why sending $\alpha'$ to zero in the construction does not discard all excited strings in the resulting AdS theory. The meaningful radius is $L/\ell_s=\lambda^{1/4}$; $\alpha'$ is not an additional dimensional coupling of the conformal boundary theory.

The complete background also contains the constant dilaton and the nonzero self-dual five-form; the two-form backgrounds vanish. The metric by itself is not a vacuum solution of ten-dimensional pure Einstein gravity. The full Lorentzian IIB background preserves 32 real supersymmetries. They match the boundary theory’s sixteen Poincaré charges and sixteen conformal charges, not thirty-two Poincaré charges. An $\mathcal N=1$ notation for the Yang–Mills fields displays only four of the sixteen Poincaré charges. The $SO(4,2)$ AdS isometries and the $SO(6)$ sphere rotations match the conformal and R-symmetry algebras; spinors require the corresponding covers. See Minahan, §3.

The brane construction initially gives $U(N)$. Its central vector multiplet is free and decouples from the interacting local $SU(N)$ sector described by the bulk. Globally, $U(N)=[SU(N)\times U(1)]/\mathbb Z_N$, so this is not a claim that all bundles and lines factor into two independent theories. The metric and flux integer specify only part of the duality’s data; boundary conditions for additional bulk fields encode further global choices. The line-operator lesson explains why that qualification matters. The decoupled-center distinction is explicit in the review’s §3.1.

Normalize the gravitational expansion

A frame is a choice of metric variable in the action. To compare Newton’s constant to the radius just obtained, use

$$g_{E,MN}=e^{-(\phi-\phi_0)/2}g_{s,MN}, \qquad 16\pi G_{10}=(2\pi)^7g_s^2\alpha'^4. \tag{N19.8}$$

This Einstein-frame convention, including the constant normalization, is Aharony et al., (1.17). Since $\phi=\phi_0$ here, the background Einstein and string metrics agree. Using $e^{-\phi/2}g_s$ instead would rescale the radius and the numerical Newton constant together; mixing those two choices would give a wrong power of $g_s$.

The unit five-sphere has volume $\pi^3$. For a metric fluctuation independent of the sphere coordinates, the coefficient of its five-dimensional Ricci scalar follows by integrating the ten-dimensional Einstein term:

$$\begin{aligned} \frac1{16\pi G_{10}}\int d^{10}x\sqrt{-g_{10}}\,R_{10} &\supset \frac{\pi^3L^5}{16\pi G_{10}} \int d^5x\sqrt{-g_5}\,R_5,\\ G_5&=\frac{G_{10}}{\pi^3L^5},\qquad \boxed{\frac{G_5}{L^3}=\frac{\pi}{2N^2}.} \end{aligned} \tag{N19.9}$$

The last equality uses $G_{10}=8\pi^6g_s^2\alpha'^4$ and $L^8=16\pi^2g_s^2N^2\alpha'^4$. Newton’s constant has length dimension eight in ten dimensions and three in five, so both sides pass a dimensional check.

Rescale five-dimensional coordinates by $L$. The Einstein action has overall coefficient $L^3/(16\pi G_5)=N^2/(8\pi^2)$. Canonically normalizing fluctuations then makes their interaction expansion an expansion in inverse powers of $N$. This is the gravitational counterpart of the connected color expansion; it supplies a normalization, not a proof of the duality.

The sphere’s nonconstant harmonics have masses of order $1/L$. They have not all disappeared from IIB supergravity at the AdS energy scale. Equation (N19.9) identifies the metric-sector coefficient; it does not replace the full dual by pure five-dimensional Einstein gravity. Nor does its classical $N^2$ coefficient determine the order-one correction that distinguishes $N^2-1$ from $N^2$.

Separate three questions about control

The derived parameters answer different questions:

Question	Dimensionless parameter	Condition giving suppression
Is the geometry weakly curved in string units?	$\alpha'/L^2=\lambda^{-1/2}$	$\lambda\gg1$
Are additional string handles suppressed in this weak-coupling frame?	$g_s^2=[\lambda/(4\pi N)]^2$	$g_s\ll1$
Are low-energy gravitational loops at scale $1/L$ suppressed?	$G_5/L^3=\pi/(2N^2)$	$N\gg1$

The string worldsheet action provides another way to see the first row: its area coefficient in radius units is $T_{\mathrm F}L^2=\sqrt\lambda/(2\pi)$. Small $\lambda$ leaves a string-scale geometry even at infinite $N$.

These are expansion parameters, not claims that every observable has a nonzero correction at the first displayed power. Maximal supersymmetry protects the background and some observables; actual corrections require the appropriate interaction and observable. Also, a gravitational loop at wavelength $L$ and a string loop at string-scale energies are not interchangeable estimates.

A sufficient window for a weakly coupled string description whose low-energy leading approximation is classical supergravity is $1\ll\lambda\ll4\pi N$. A conventional planar limit with $\lambda$ fixed suppresses handles, but does not improve curvature. Taking $\lambda\to\infty$ at fixed $N$ improves string-unit curvature while making $g_s$ large and leaving $G_5/L^3$ fixed. A duality transformation may offer a different description, but requires its own parameter and global-data check.

Worked example. Choose $N=10^5$ and $\lambda=10^4$. Then $g_{\mathrm{YM}}^2=0.1$, $g_s=1/(40\pi)\simeq0.00796$, $L/\ell_s=10$, and $G_5/L^3\simeq1.57\times10^{-10}$. The gauge coupling is numerically small while the planar expansion is strongly coupled. The curvature parameter is $0.01$; this alone is not a one-percent error guarantee for every observable. Increasing $N$ at the same $\lambda$ further suppresses quantum effects but leaves that curvature parameter unchanged.

There is a separate geometric approximation in dropping the one in $H$. For a radial segment $0<r_1<r_2\ll L$, its full D3 proper length is $\mathcal D=\int_{r_1}^{r_2}(1+L^4/r^4)^{1/4}dr$, whereas its throat value is $L\log(r_2/r_1)$. Concavity gives $0\leq(1+x)^{1/4}-1\leq x/4$, and hence the explicit bound

$$0\leq\mathcal D-L\log\frac{r_2}{r_1} \leq\frac{r_2^4-r_1^4}{16L^3}. \tag{N19.10}$$

For $r_1=L/20$, $r_2=L/10$, the relative error is at most $8.46\times10^{-6}$. This bound controls the near-horizon approximation to the stated classical metric. It estimates neither omitted string effects nor quantum loops.

Exercises

1. Change the coupling convention and reassess curvature

A colleague writes the same connection and ordinary trace as in (N19.1), but defines $\bar g$ by $S=-\int\operatorname{Tr}_N F^2/(4\bar g^2)$ and $\bar\lambda=\bar g^2N$. Derive the dictionary in those symbols. Separately verify that the ten-dimensional scalar curvature of (N19.6) vanishes, while $R_{MNPQ}R^{MNPQ}$ does not. Does zero scalar curvature remove the need for large coupling?

Hint 1

Match the action coefficients before comparing named couplings. For a $d$-dimensional constant-sectional-curvature space, $R=d(d-1)K$.

Hint 2

The two factors have $d=5$ and $K=\mp1/L^2$. Their mixed Riemann components vanish, while each squared Riemann tensor contributes $2d(d-1)K^2$.

Solution

Equality of actions gives $g_{\mathrm{YM}}^2=2\bar g^2$. Thus $\bar g^2=2\pi g_s$, $\lambda=2\bar\lambda$, and

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

The Newton ratio $\pi/(2N^2)$ is unchanged. Replacing the action’s trace by a normalized trace would require a further conversion; none was made here.

The Ricci scalars are $-20/L^2$ and $+20/L^2$, giving $R_{10}=0$. However, contraction of $R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})$ gives $40/L^4$ for each factor, including the Lorentzian AdS factor. Therefore

$$R_{MNPQ}R^{MNPQ}=\frac{80}{L^4}, \qquad \alpha'^2R_{MNPQ}R^{MNPQ}=\frac{80}{\lambda}.$$

The cancellation of the scalar traces does not make the product flat. Curvature control concerns the full tensors and the terms in a given effective action. This calculation does not assert a nonzero higher-derivative correction to this maximally supersymmetric solution.

2. Compare limits with the same increasing rank

Let $N$ run through positive integers and set $\lambda(N)=cN^a$, with $c>0$ fixed. Classify when curvature corrections and string handles both become parametrically suppressed. Compare $a=0$, $a=1/2$, $a=1$, and $a=2$. Explain why a small gravitational loop parameter alone cannot establish a useful string-frame expansion.

Hint 1

Write the three parameters as powers of $N$ before taking a limit.

Hint 2

At $a=1$, $g_s$ is constant. “Small if chosen small” and “tends to zero in this limit” are different statements.

Solution

The parameters are $c^{-1/2}N^{-a/2}$, $g_s=cN^{a-1}/(4\pi)$, and $\pi/(2N^2)$. Curvature becomes parametrically small for $a>0$, while $g_s\to0$ for $a<1$. Thus $0<a<1$ makes both vanish.

For $a=0$, handles vanish, but curvature remains $c^{-1/2}$; it can be small only by choosing a large fixed $c$. For $a=1/2$, curvature falls as $N^{-1/4}$ and $g_s$ as $N^{-1/2}$. For $a=1$, curvature vanishes and $g_s=c/(4\pi)$ remains fixed; a small chosen $c/(4\pi)$ is a separate control assumption. For $a=2$, $g_s$ grows without bound, so perturbative string genus counting in this frame fails.

In all four families the gravitational parameter at scale $1/L$ vanishes. That fact does not guarantee either small string-scale curvature or weak string-scale interactions, as the first and last families demonstrate. These comparisons diagnose approximations; they do not disprove the duality outside those approximations.

3. Split the stack and inspect each throat

Change the vacuum by separating the branes into groups of $N_1$ and $N_2$, with $N_1+N_2=N$. Use the sourced classical harmonic function

$$H(\mathbf r)=1+ \frac{4\pi g_s\alpha'^2N_1}{|\mathbf r-\mathbf r_1|^4} +\frac{4\pi g_s\alpha'^2N_2}{|\mathbf r-\mathbf r_2|^4}.$$

Near either group, find its local radius and gravitational parameter. At fixed total $\lambda\gg1$ and $g_s\ll1$, compare an equal split with separating just one brane. State both the geometric dominance condition and the approximation that fails for the one-brane throat.

Hint 1

At distance $\rho=|\mathbf r-\mathbf r_i|$ from group $i$, the other group’s contribution is approximately constant when $\rho\ll|\mathbf r_1-\mathbf r_2|$.

Hint 2

Apply the radius and Newton dictionary to $N_i$, with the same $g_s$. The original large total $N$ is not the flux integer of the smaller throat.

Solution

Writing the separation as $d$, the $i$th throat dominates when $L_i^4/\rho^4\gg1+L_j^4/d^4$ and $\rho\ll d$, where

$$L_i^4=4\pi g_sN_i\alpha'^2,\qquad \lambda_i=g_{\mathrm{YM}}^2N_i=\lambda\frac{N_i}{N}, \qquad \frac{G_{5,i}}{L_i^3}=\frac{\pi}{2N_i^2}.$$

For an equal split, $\lambda_i=\lambda/2$ and $N_i=N/2$. Each local radius is $2^{-1/4}L$ and its gravitational parameter is four times the unsplit value. Both throats remain controlled if their own ranks and couplings are large.

For $N_1=1$, $\lambda_1=4\pi g_s$. In the parametrically weak-string-coupling limit this is small, while the formal gravitational ratio is $\pi/2$, not small. Approaching that center more closely improves harmonic-function dominance but cannot turn its local flux into a large integer. The classical solution is then not a controlled approximation in that throat; the separated object is better treated as a brane probe when appropriate.

The split is a Coulomb-branch vacuum with additional scales and reduced unbroken gauge symmetry, not the original conformal vacuum. For an $SU(N)$ parent the block subgroup is $S(U(N_1)\times U(N_2))$; it is not two independent unrestricted $U(N_i)$ theories. The harmonic-function construction and its regime are given in Maldacena, (2.10) and Aharony et al., (3.11).

The dictionary now turns a proposed calculation into explicit questions: which vacuum and boundary data are fixed, what are $N$ and $\lambda$, and which correction is being neglected? The source-response rule from the bridge supplies the observable map; these parameters determine when its classical bulk evaluation is controlled.