Type IIB on AdS5 x S5

The main idea

The field-theory side of the canonical duality is four-dimensional $\mathcal N=4$ $SU(N)$ super-Yang-Mills. The bulk side is not merely “gravity in five dimensions.” The exact statement is that the CFT is dual to type IIB string theory on

$$\mathrm{AdS}_5\times S^5$$

with $N$ units of Ramond-Ramond five-form flux through the $S^5$.

The useful hierarchy of bulk descriptions is

$$\text{type IIB string theory on }\mathrm{AdS}_5\times S^5 \quad\longrightarrow\quad \text{type IIB supergravity on }\mathrm{AdS}_5\times S^5 \quad\longrightarrow\quad \text{five-dimensional gauged supergravity or Einstein gravity}.$$

Each arrow is an approximation or a consistent truncation, not a change of the duality. The full bulk theory contains massive string excitations, Kaluza-Klein modes on $S^5$, D-branes, black holes, and quantum effects. Classical five-dimensional Einstein gravity is a powerful subsector, but it is not the whole bulk theory.

The canonical bulk background is a product of a noncompact $\mathrm{AdS}_5$ factor and a compact $S^5$ factor with the same radius $L$. The self-dual five-form flux $F_5$ threads the $S^5$ and fixes $L^4=4\pi g_sN\alpha'^2$ in the conventions of this course. The two isometry groups $SO(2,4)$ and $SO(6)_R$ match the conformal and R-symmetry groups of $\mathcal N=4$ SYM.

The most important formulas are

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

so that

$$\boxed{\frac{L^4}{\alpha'^2}=\lambda.}$$

This is the first place where the strong/weak character of AdS/CFT becomes concrete. When $\lambda\gg1$, the AdS radius is large compared with the string length, and the bulk string theory can be approximated by supergravity for observables that do not excite string-scale modes. When $N\gg1$, bulk quantum loops are suppressed. The next page will systematize these statements into a regime-of-validity map.

The ten-dimensional background

In Poincare coordinates for $\mathrm{AdS}_5$, the metric is

$$ds_{10}^2 = \frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right) +L^2d\Omega_5^2, \qquad \mu,\nu=0,1,2,3.$$

Here $d\Omega_5^2$ is the unit-radius round metric on $S^5$. The same radius $L$ multiplies both factors. This equality is not a convention; it is forced by the type IIB equations of motion with self-dual five-form flux.

A coordinate-independent way to write the background is

$$ds_{10}^2 = L^2ds_{\mathrm{AdS}_5,\mathrm{unit}}^2 +L^2d\Omega_{5,\mathrm{unit}}^2.$$

The curvature of the two factors is

$$R_{\mu\nu}^{\mathrm{AdS}_5} = -\frac{4}{L^2}g_{\mu\nu}, \qquad R_{ab}^{S^5} = +\frac{4}{L^2}g_{ab},$$

where Greek indices are along $\mathrm{AdS}_5$ and Latin indices are along $S^5$.

The dilaton and axion are constant:

$$e^\Phi=g_s, \qquad C_0=\text{constant}.$$

The NS-NS two-form and RR two-form vanish in the undeformed background:

$$B_2=0, \qquad C_2=0.$$

The essential field is the RR five-form field strength,

$$F_5={}^\star_{10}F_5.$$

With unit-radius volume forms, it can be written as

$$F_5 = 4L^4 \left( \mathrm{vol}_{\mathrm{AdS}_5,\mathrm{unit}} + \mathrm{vol}_{S^5,\mathrm{unit}} \right).$$

The factor $4$ is tied to the fact that both factors are five-dimensional and have curvature magnitude $4/L^2$. The self-duality condition says that the flux has equal electric and magnetic components: it threads the compact $S^5$ and also has legs along $\mathrm{AdS}_5$.

Type IIB fields: what is turned on?

The bosonic massless fields of type IIB string theory include the metric, the dilaton, the RR axion, two two-form potentials, and the RR four-form potential. For this background, only a small subset is nonzero.

Field	Meaning	Value in $\mathrm{AdS}_5\times S^5$
$g_{MN}$	ten-dimensional metric	product metric on $\mathrm{AdS}_5\times S^5$
$\Phi$	dilaton	constant, $e^\Phi=g_s$
$C_0$	RR axion	constant, identified with $\theta/(2\pi)$
$B_2$	NS-NS two-form	zero
$C_2$	RR two-form	zero
$C_4$	RR four-form potential	nonzero through $F_5=dC_4+\cdots$
$F_5$	self-dual RR five-form field strength	supports the geometry and carries $N$ units of flux

The relevant complex scalar is the axio-dilaton

$$\tau_{\mathrm{IIB}}=C_0+i e^{-\Phi}.$$

The canonical dictionary identifies it with the complexified Yang-Mills coupling,

$$\tau_{\mathrm{IIB}} \longleftrightarrow \tau_{\mathrm{YM}} = \frac{\theta}{2\pi}+\frac{4\pi i}{g_{\mathrm{YM}}^2}.$$

Thus, in the convention $g_{\mathrm{YM}}^2=4\pi g_s$,

$$C_0=\frac{\theta}{2\pi}, \qquad e^\Phi=g_s=\frac{g_{\mathrm{YM}}^2}{4\pi}.$$

The type IIB $SL(2,\mathbb Z)$ duality acts naturally on $\tau_{\mathrm{IIB}}$. In the boundary theory this is the Montonen-Olive duality of $\mathcal N=4$ SYM. We will not need the full modular structure on this page, but it is a useful reminder that the canonical duality is much more rigid than its two-derivative gravity limit.

Why the five-form supports the product geometry

For the fields turned on above, the type IIB equations reduce schematically to

$$R_{MN} = \frac{1}{96}F_{MPQRS}F_N{}^{PQRS}, \qquad dF_5=0, \qquad F_5={}^\star_{10}F_5,$$

with constant $\Phi$ and $C_0$. One should treat this as an equation-of-motion statement. Because $F_5$ is self-dual, type IIB supergravity is usually written using a pseudo-action and the condition $F_5={}^\star F_5$ is imposed after variation. Imposing self-duality too early in a naive action is a classic way to lose factors of two.

The five-form flux is a Freund-Rubin type ingredient. Its stress tensor gives opposite-sign curvature to the Lorentzian and compact factors:

$$R_{\mu\nu}(\mathrm{AdS}_5) =-\frac{4}{L^2}g_{\mu\nu}, \qquad R_{ab}(S^5) =+\frac{4}{L^2}g_{ab}.$$

A useful mental model is

$$\text{D3-brane charge} \quad\Longrightarrow\quad F_5\text{ flux} \quad\Longrightarrow\quad \mathrm{AdS}_5\times S^5\text{ geometry}.$$

The compact sphere is therefore not a passive internal decoration. It is part of the solution that allows the noncompact spacetime to be anti-de Sitter.

Flux quantization and the origin of $N$

D3-branes couple electrically to the RR four-form potential $C_4$. The corresponding five-form flux is quantized. In the standard D3-brane normalization,

$$N = \frac{1}{2\kappa_{10}^2T_3} \int_{S^5}F_5,$$

where

$$T_3=\frac{1}{(2\pi)^3g_s\alpha'^2}, \qquad 2\kappa_{10}^2=(2\pi)^7g_s^2\alpha'^4.$$

Since

$$\mathrm{Vol}(S^5_{\mathrm{unit}})=\pi^3,$$

we have

$$\int_{S^5}F_5=4L^4\pi^3.$$

Therefore

$$N = \frac{4\pi^3L^4}{(2\pi)^4g_s\alpha'^2} = \frac{L^4}{4\pi g_s\alpha'^2},$$

and hence

$$\boxed{L^4=4\pi g_sN\alpha'^2.}$$

Using the course convention

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

this becomes

$$\boxed{L^4=\lambda\alpha'^2.}$$

The interpretation is immediate:

$$\frac{L^2}{\alpha'}=\sqrt\lambda, \qquad \frac{\ell_s}{L}=\lambda^{-1/4}.$$

Large ‘t Hooft coupling makes the string length small compared with the AdS radius.

The near-horizon D3-brane metric

The same geometry appears directly from the extremal D3-brane solution. In string frame,

$$ds^2 = H(r)^{-1/2}\eta_{\mu\nu}dx^\mu dx^\nu +H(r)^{1/2}\left(dr^2+r^2d\Omega_5^2\right),$$

where

$$H(r)=1+\frac{L^4}{r^4}.$$

In the near-horizon region $r\ll L$,

$$H(r)\simeq\frac{L^4}{r^4}.$$

The metric becomes

$$ds^2 \simeq \frac{r^2}{L^2}\eta_{\mu\nu}dx^\mu dx^\nu + \frac{L^2}{r^2}dr^2 +L^2d\Omega_5^2.$$

Now define

$$z=\frac{L^2}{r}.$$

Then

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right) +L^2d\Omega_5^2.$$

This is the Poincare patch of $\mathrm{AdS}_5$ times a round $S^5$. The low-energy open-string description gives $\mathcal N=4$ SYM. The low-energy near-horizon closed-string description gives type IIB string theory on $\mathrm{AdS}_5\times S^5$. Maldacena’s conjecture identifies these two interacting descriptions after the common decoupled flat-space sector is removed.

Symmetry matching

The bosonic isometry group of the background is

$$\mathrm{Isom}(\mathrm{AdS}_5\times S^5) =SO(2,4)\times SO(6).$$

The first factor is the conformal group of four-dimensional Minkowski space:

$$SO(2,4) \simeq \mathrm{Conf}(\mathbb R^{1,3}).$$

The second factor is the isometry group of the five-sphere:

$$SO(6) \simeq SU(4)_R,$$

which is the R-symmetry group of $\mathcal N=4$ SYM.

The supersymmetry also matches. The D3-brane worldvolume theory has $16$ Poincare supercharges. At the conformal point it also has $16$ conformal supercharges. The total number of fermionic generators is therefore $32$, matching the maximal supersymmetry of type IIB string theory on $\mathrm{AdS}_5\times S^5$.

Together these generators form

$$PSU(2,2|4),$$

which is the superconformal symmetry group of $\mathcal N=4$ SYM. This is one of the strongest elementary checks of the canonical duality.

What does the $S^5$ mean in the CFT?

The CFT does not live on $S^5$. It lives on the conformal boundary of $\mathrm{AdS}_5$, for example $\mathbb R^{1,3}$ in the Poincare patch or $\mathbb R_t\times S^3$ in global coordinates.

The $S^5$ is an internal compact space. Its angular momenta are R-charges. A particle moving on $S^5$ transforms under $SO(6)$, and the dual CFT operator transforms under the corresponding $SO(6)_R$ representation.

A rough dictionary is:

Bulk structure	CFT meaning
$\mathrm{AdS}_5$ position	spacetime and energy-scale data
$S^5$ angular momentum	$SO(6)_R$ representation and R-charge
five-form flux quantum $N$	rank of the gauge group
radius $L$ in string units	’t Hooft coupling $\lambda$
ten-dimensional graviton multiplet	protected single-trace operators
massive string oscillator states	generic unprotected single-trace operators

This is why the $S^5$ should not be discarded. Even when a calculation can be done in a five-dimensional truncation, the ten-dimensional origin determines which fields exist, which symmetries they carry, and which CFT operators they source.

Kaluza-Klein modes and protected operators

Fields on $\mathrm{AdS}_5\times S^5$ can be expanded in spherical harmonics on $S^5$. For a scalar field,

$$\phi(x,y)=\sum_{k,I}\phi_{k,I}(x)Y_{k,I}(y),$$

where $x$ denotes AdS coordinates and $y$ denotes sphere coordinates. Scalar spherical harmonics obey

$$-\nabla^2_{S^5}Y_{k,I} =\frac{k(k+4)}{L^2}Y_{k,I}, \qquad k=0,1,2,\ldots.$$

From the five-dimensional point of view, angular momentum on $S^5$ appears as mass in AdS. Thus one ten-dimensional field becomes an infinite tower of five-dimensional fields.

This tower is not a small correction. Since the radius of $S^5$ equals the radius of $\mathrm{AdS}_5$,

$$m_{\mathrm{KK}}\sim\frac{1}{L}.$$

The KK modes are order-one in AdS units. What becomes heavy at large $\lambda$ are the genuine massive string oscillator modes:

$$\Delta_{\mathrm{string}} \sim m_{\mathrm{string}}L \sim \frac{L}{\ell_s} =\lambda^{1/4}.$$

This distinction is crucial. KK modes are supergravity modes. String oscillator modes are not.

Many KK modes are dual to protected short multiplets of $\mathcal N=4$ SYM. A central family is the chiral primary tower

$$\mathcal O_k^{(I_1\cdots I_k)} = \operatorname{Tr}\left(\Phi^{(I_1}\cdots\Phi^{I_k)}\right)-\text{traces}, \qquad \Delta=k.$$

The parentheses denote the symmetric traceless projection under $SO(6)_R$. The dimensions of these operators are protected, so they can be matched between weakly coupled SYM and strongly coupled supergravity.

A few universal entries are:

Bulk mode	Boundary operator	Protected dimension
five-dimensional graviton	stress tensor $T_{\mu\nu}$	$4$
gauge fields from $S^5$ isometries	R-currents $J_\mu^{[IJ]}$	$3$
dilaton zero mode	$\operatorname{Tr}F^2+\cdots$	$4$
axion zero mode	$\operatorname{Tr}F\wedge F+\cdots$	$4$
selected $S^5$ metric and $F_5$ fluctuations	chiral primary multiplets	$k$

The word “selected” matters. Some scalar modes arise from mixtures of the $S^5$ metric and the four-form potential. One should not identify every spherical harmonic of every ten-dimensional field with a simple $\operatorname{Tr}\Phi^k$ operator without diagonalizing the coupled linearized equations.

The five-dimensional viewpoint

Many calculations use a five-dimensional action such as

$$S_5 = \frac{1}{16\pi G_5} \int d^5x\sqrt{-g} \left(R+\frac{12}{L^2}\right)+\cdots.$$

The equation of motion is

$$R_{\mu\nu}=-\frac{4}{L^2}g_{\mu\nu},$$

so pure $\mathrm{AdS}_5$ is a solution. This five-dimensional description is the simplest way to compute stress-tensor correlators, black-brane thermodynamics, hydrodynamic transport, and holographic entanglement in the classical limit.

But there are two different ideas here.

First, a Kaluza-Klein reduction on $S^5$ rewrites ten-dimensional supergravity as five-dimensional gravity coupled to infinitely many fields.

Second, a consistent truncation keeps only a finite subset of fields such that every solution of the lower-dimensional theory uplifts to a solution of the ten-dimensional equations. Type IIB on $S^5$ admits a maximally supersymmetric consistent truncation to five-dimensional $\mathcal N=8$ gauged supergravity with gauge group $SO(6)$, and many smaller truncations are used in practical holography.

A consistent truncation is powerful, but it is not justified by the KK modes being parametrically heavy. They are not. It is justified by symmetry and the special structure of the equations.

The Newton constant and $N^2$

The ten-dimensional Newton constant is

$$G_{10}=8\pi^6g_s^2\alpha'^4.$$

Reducing on a round $S^5$ of radius $L$ gives

$$G_5=\frac{G_{10}}{\mathrm{Vol}(S^5)} =\frac{G_{10}}{\pi^3L^5}.$$

Therefore

$$\frac{L^3}{G_5} =\frac{\pi^3L^8}{G_{10}}.$$

Using

$$L^4=4\pi g_sN\alpha'^2,$$

we find

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

This is the gravitational origin of the $N^2$ scaling of the boundary matrix theory. For Einstein gravity in $\mathrm{AdS}_5$, the holographic central charges are

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

at leading large $N$. The exact $SU(N)$ result is

$$a=c=\frac{N^2-1}{4}.$$

The classical geometry captures the leading $N^2$ term and misses the $O(1)$ correction, as expected.

Regime of validity, preview

The background contains three useful length scales:

$$\ell_s=\sqrt{\alpha'}, \qquad \ell_{10}\sim G_{10}^{1/8}, \qquad L.$$

The ratio between the AdS radius and the string length is controlled by $\lambda$:

$$\frac{\alpha'}{L^2}\sim\lambda^{-1/2}.$$

The ratio between the AdS scale and the five-dimensional Newton constant is controlled by $N$:

$$\frac{G_5}{L^3}\sim\frac{1}{N^2}.$$

Thus the cleanest classical type IIB supergravity regime is

$$N\gg1, \qquad \lambda\gg1,$$

with weak string coupling in the chosen IIB frame when

$$g_s=\frac{\lambda}{4\pi N}\ll1.$$

The conceptual hierarchy is:

CFT limit	Bulk meaning
$N\to\infty$	suppresses bulk quantum loops
$\lambda\to\infty$	suppresses finite string-length corrections
fixed protected sector	often captured by supergravity KK modes
generic unprotected high-spin operators	require massive string states at finite $\lambda$

The next page will turn this preview into a detailed map of the $(N,\lambda)$ plane.

Common mistakes

Mistake 1: saying the dual is pure gravity on $\mathrm{AdS}_5$

Pure Einstein gravity is a useful subsector. The full bulk theory is type IIB string theory on $\mathrm{AdS}_5\times S^5$.

Mistake 2: treating the $S^5$ as negligible

The $S^5$ has the same radius as $\mathrm{AdS}_5$. There is no large hierarchy that makes all KK modes heavy compared with the AdS scale. Ignoring the sphere is justified only for suitable neutral observables or within a consistent truncation.

Mistake 3: confusing KK modes with string modes

KK masses scale like $1/L$ and remain order one in AdS units. Massive string modes scale like $1/\ell_s$ and have dimensions of order $\lambda^{1/4}$ at large $\lambda$.

Mistake 4: forgetting flux quantization

The integer $N$ is not merely a continuous parameter of a classical solution. It is the quantized five-form flux through $S^5$, and it is also the rank of the gauge group.

Mistake 5: identifying bulk fields with elementary SYM fields

The elementary fields $A_\mu$, $\lambda^A$, and $\Phi^I$ are not gauge-invariant local observables. Bulk fields are dual to gauge-invariant operators, typically single-trace operators such as $T_{\mu\nu}$, $J_\mu^{[IJ]}$, $\operatorname{Tr}F^2$, and protected scalar composites.

Summary

The canonical bulk background is specified by

$$ds_{10}^2 =L^2ds_{\mathrm{AdS}_5}^2+L^2d\Omega_5^2,$$ $$F_5=4L^4 \left( \mathrm{vol}_{\mathrm{AdS}_5}+\mathrm{vol}_{S^5} \right), \qquad F_5={}^\star_{10}F_5,$$

and constant axio-dilaton

$$\tau_{\mathrm{IIB}}=C_0+i e^{-\Phi}=\tau_{\mathrm{YM}}.$$

Flux quantization gives

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

and dimensional reduction gives

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

The bosonic symmetry match is

$$SO(2,4)\times SO(6)_R,$$

and the full superisometry is

$$PSU(2,2|4).$$

The $S^5$ geometrizes R-symmetry and produces KK towers dual to protected operator multiplets. Classical five-dimensional gravity is a controlled approximation to the full type IIB dual, not the full duality itself.

Exercises

Exercise 1: Recover the Poincare-patch AdS metric

Starting from the near-horizon D3-brane metric

$$ds^2 =\frac{r^2}{L^2}\eta_{\mu\nu}dx^\mu dx^\nu +\frac{L^2}{r^2}dr^2 +L^2d\Omega_5^2,$$

use $z=L^2/r$ to show that the first two terms become the Poincare-patch metric on $\mathrm{AdS}_5$.

Solution

From

$$z=\frac{L^2}{r},$$

we have

$$r=\frac{L^2}{z}, \qquad dr=-\frac{L^2}{z^2}dz.$$

Then

$$\frac{r^2}{L^2}=\frac{L^2}{z^2},$$

and

$$\frac{L^2}{r^2}dr^2 =\frac{L^2}{L^4/z^2}\frac{L^4}{z^4}dz^2 =\frac{L^2}{z^2}dz^2.$$

Therefore

$$ds^2 =\frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right) +L^2d\Omega_5^2.$$

The first bracket is the Poincare patch of $\mathrm{AdS}_5$.

Exercise 2: Derive the radius-flux relation

Assume

$$\int_{S^5}F_5=4\pi^3L^4,$$

and use

$$N =\frac{1}{2\kappa_{10}^2T_3} \int_{S^5}F_5, \qquad T_3=\frac{1}{(2\pi)^3g_s\alpha'^2}, \qquad 2\kappa_{10}^2=(2\pi)^7g_s^2\alpha'^4.$$

Show that

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

Solution

First compute

$$2\kappa_{10}^2T_3 =(2\pi)^7g_s^2\alpha'^4 \frac{1}{(2\pi)^3g_s\alpha'^2} =(2\pi)^4g_s\alpha'^2.$$

Therefore

$$N =\frac{4\pi^3L^4}{(2\pi)^4g_s\alpha'^2} =\frac{4\pi^3L^4}{16\pi^4g_s\alpha'^2} =\frac{L^4}{4\pi g_s\alpha'^2}.$$

Solving for $L^4$ gives

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

Using $g_{\mathrm{YM}}^2=4\pi g_s$ then gives $L^4/\alpha'^2=\lambda$.

Exercise 3: Why a pure five-dimensional truncation is subtle

Explain why the statement

$$m_{\mathrm{KK}}\sim\frac{1}{L}$$

means that Kaluza-Klein modes on $S^5$ are not parametrically heavier than the AdS scale. Why can five-dimensional effective actions nevertheless be useful?

Solution

The AdS curvature scale is also $1/L$. If

$$m_{\mathrm{KK}}L\sim O(1),$$

then KK modes are light in AdS units. There is no hierarchy of the form

$$m_{\mathrm{KK}}L\gg1.$$

Thus one cannot justify throwing away all KK modes by a simple low-energy expansion below the internal scale.

Five-dimensional effective actions are still useful for two reasons. First, some calculations only source fields in a closed sector of the equations, such as the metric and a few matter fields. Second, special consistent truncations exist: every solution of the lower-dimensional truncated theory uplifts to a solution of the ten-dimensional theory. In such cases the truncation is justified by consistency and symmetry, not by a large KK mass gap.

Exercise 4: A minimal scalar harmonic on $S^5$

Consider a ten-dimensional massless scalar obeying

$$\Box_{10}\phi=0$$

on $\mathrm{AdS}_5\times S^5$. Expand

$$\phi(x,y)=\phi_k(x)Y_k(y),$$

where

$$-\nabla_{S^5}^2Y_k=k(k+4)Y_k.$$

Find the effective five-dimensional mass $m_k^2L^2$ and the corresponding standard-quantization dimension $\Delta$ in $d=4$.

Solution

The ten-dimensional Laplacian separates as

$$\Box_{10} =\Box_{\mathrm{AdS}_5} +\frac{1}{L^2}\nabla_{S^5,\mathrm{unit}}^2.$$

Substitution gives

$$\left( \Box_{\mathrm{AdS}_5} -\frac{k(k+4)}{L^2} \right)\phi_k=0.$$

Thus

$$m_k^2L^2=k(k+4).$$

For a scalar in $\mathrm{AdS}_5$, the mass-dimension relation is

$$\Delta(\Delta-4)=m^2L^2.$$

So

$$\Delta(\Delta-4)=k(k+4),$$

whose solutions are

$$\Delta=k+4, \qquad \Delta=-k.$$

The standard positive-dimension choice is

$$\Delta=k+4.$$

For $k=0$, this gives $\Delta=4$, as expected for the dilaton zero mode dual to an exactly marginal operator. This exercise also shows why one must be careful: the chiral primaries with $\Delta=k$ do not arise from a minimally coupled scalar in this simple way; they come from specific mixed metric and five-form fluctuations.

Exercise 5: Central charge scaling from dimensional reduction

Use

$$G_{10}=8\pi^6g_s^2\alpha'^4, \qquad G_5=\frac{G_{10}}{\pi^3L^5}, \qquad L^4=4\pi g_sN\alpha'^2$$

to show that

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

Then show that the holographic result

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

gives $a=c=N^2/4$ at leading order.

Solution

First,

$$\frac{L^3}{G_5} =\frac{L^3\pi^3L^5}{G_{10}} =\frac{\pi^3L^8}{G_{10}}.$$

Since

$$L^8=(4\pi g_sN\alpha'^2)^2 =16\pi^2g_s^2N^2\alpha'^4,$$

we get

$$\frac{L^3}{G_5} =\frac{\pi^3\,16\pi^2g_s^2N^2\alpha'^4}{8\pi^6g_s^2\alpha'^4} =\frac{2N^2}{\pi}.$$

Therefore

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

The exact $SU(N)$ result is $(N^2-1)/4$, so the missing $-1/4$ is an $O(1)$ correction beyond classical supergravity.

Further reading

• J. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity. The original D-brane decoupling and near-horizon argument.
• O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large N Field Theories, String Theory and Gravity. The classic review of the correspondence and its parameter dictionary.
• E. D’Hoker and D. Freedman, Supersymmetric Gauge Theories and the AdS/CFT Correspondence. Detailed lecture notes on the canonical example and its supergravity side.
• H. Kim, L. Romans, and P. van Nieuwenhuizen, Mass Spectrum of Chiral Ten-Dimensional N=2 Supergravity on $S^5$. The classic computation of the type IIB supergravity Kaluza-Klein spectrum on $S^5$.