Supergravity p-Branes, NS5-Branes, and Throat Geometries

D-branes were introduced as hypersurfaces on which open strings can end. That is the weak-coupling, open-string definition. But a D-brane also carries stress energy and Ramond—Ramond charge, so it must source the closed-string fields of type II supergravity. At sufficiently large charge, the backreaction is not a small correction: the brane becomes a spacetime geometry.

This is the conceptual bridge from perturbative string theory to holography. Open strings tell us that a stack of $N$ D$p$-branes supports a $(p+1)$-dimensional gauge theory. Closed strings tell us that the same stack sources a curved supergravity solution whose fields are controlled by a harmonic function $H_p(r)$. The two descriptions are not competing stories. They are two limits of the same object.

The main character on this page is the extremal BPS $p$-brane solution. It has a particularly rigid form because supersymmetry fixes the relative strength of gravity, dilaton exchange, and R—R gauge-field exchange. For a stack of coincident branes, the transverse-space harmonic function is

$$H_p(r)=1+{R_p^{7-p}\over r^{7-p}}, \qquad p<7,$$

where $r$ is the distance from the brane in the transverse space $\mathbb R^{9-p}$. The constant $R_p$ is the charge radius. When $r\gg R_p$, the geometry is nearly flat. When $r\ll R_p$, the constant $1$ becomes negligible and the geometry develops a throat.

An extremal D$p$-brane is translationally invariant along $x^0,\ldots,x^p$ and spherically symmetric in the transverse space $\mathbb R^{9-p}$. Its geometry is controlled by a harmonic function $H_p(r)$, and its R—R charge is measured by flux through the linking sphere $S^{8-p}$.

Why harmonic functions appear

Consider an infinite flat $p$-brane in ten dimensions. It fills $p$ spatial directions and time, so the number of transverse spatial directions is

$$n=9-p.$$

The brane is localized at $\vec y=0$ in $\mathbb R^n$. Away from the source, the fields are constrained by the sourceless supergravity equations. Supersymmetry reduces the coupled nonlinear equations to a remarkably simple statement: the function $H_p(\vec y)$ is harmonic in the transverse space,

$$\nabla_{\mathbb R^{9-p}}^2 H_p(\vec y)=0, \qquad \vec y\neq0.$$

For a rotationally invariant solution, $H_p=H_p(r)$ with $r=|\vec y|$. The flat radial Laplacian in $n$ dimensions is

$$\nabla^2 H={1\over r^{n-1}}{d\over dr}\left(r^{n-1}{dH\over dr}\right).$$

For $n>2$, the Green function behaves as $1/r^{n-2}$. Since $n=9-p$, this gives

$$H_p(r)=1+{Q_p\over r^{7-p}}.$$

The constant term is fixed by the boundary condition that the metric approaches ten-dimensional Minkowski space at infinity. The coefficient $Q_p=R_p^{7-p}$ is fixed by charge quantization. The same logic that gives the Coulomb potential in four-dimensional electromagnetism gives the D-brane harmonic function in the transverse dimensions.

For separated parallel BPS branes, the no-force property allows a multi-center solution,

$$H_p(\vec y)=1+\sum_a {Q_a\over |\vec y-\vec y_a|^{7-p}},$$

again for $p<7$. This linear superposition inside a nonlinear theory is one of the most practical signatures of BPS saturation. The branes can sit at arbitrary positions because their attractive NS—NS forces and repulsive R—R forces cancel.

The harmonic function interpolates between an asymptotically flat region, $H_p\to1$, and a near-core region where $H_p\simeq (R_p/r)^{7-p}$. For BPS centers, harmonic functions can be superposed.

There are special cases. For $p=7$, the transverse space is two-dimensional and the harmonic function is logarithmic. For $p=8$, the transverse space is one-dimensional and the solution belongs to massive type IIA supergravity. The pages here will mostly use $p\le6$, with the D3-brane playing the central role later.

The extremal D$p$-brane solution

Let $x^\alpha$, $\alpha=0,1,\ldots,p$, be coordinates along the brane, and let $y^m$, $m=p+1,\ldots,9$, be transverse coordinates. Define

$$dx_{\parallel}^2=\eta_{\alpha\beta}dx^\alpha dx^\beta, \qquad dx_{\perp}^2=dy^m dy^m=dr^2+r^2d\Omega_{8-p}^2.$$

In string frame, the extremal D$p$-brane solution is

$$\boxed{ ds_{\rm str}^2=H_p(r)^{-1/2}dx_{\parallel}^2 +H_p(r)^{1/2}dx_{\perp}^2 }$$

with dilaton

$$\boxed{ e^{\Phi}=g_s H_p(r)^{(3-p)/4} }$$

and R—R potential, in a gauge regular at infinity,

$$\boxed{ C_{01\cdots p}=g_s^{-1}\bigl(H_p^{-1}-1\bigr). }$$

The associated R—R field strength is electric with respect to the D$p$-brane,

$$F_{p+2}=dC_{p+1}, \qquad F_{r01\cdots p}=g_s^{-1}\partial_r H_p^{-1}.$$

A magnetic description uses the dual field strength $*F_{p+2}$, whose flux through the linking sphere $S^{8-p}$ measures the brane charge. Schematically,

$$\int_{S^{8-p}} *F_{p+2}=2\kappa_{10}^2\,\mu_p N,$$

where $N$ is the number of branes and

$$\mu_p={1\over (2\pi)^p(\alpha')^{(p+1)/2}}$$

is the R—R charge normalization appearing in the Wess—Zumino coupling $\mu_p\int C_{p+1}$. As on the previous D-brane pages, the $g_s$-independent DBI normalization is $\tau_p=\mu_p$, while the physical tension measured at asymptotic string coupling $g_s$ is

$$T_p={\tau_p\over g_s}={\mu_p\over g_s} ={1\over g_s(2\pi)^p(\alpha')^{(p+1)/2}}.$$

For $p<7$, charge quantization gives

$$\boxed{ R_p^{7-p}=c_p\,g_sN(\alpha')^{(7-p)/2}, \qquad c_p=(2\sqrt\pi)^{5-p}\Gamma\left({7-p\over2}\right). }$$

This formula is worth checking in the two most famous cases:

$$\begin{aligned} D3:&\qquad R_3^4=4\pi g_sN(\alpha')^2,\\ D5:&\qquad R_5^2=g_sN\alpha'. \end{aligned}$$

The D3 case is special because the dilaton is constant. This is the first hint that D3-branes will produce a particularly clean holographic duality: the near-horizon geometry has constant string coupling and, for large $g_sN$, small curvature.

String frame and Einstein frame

The string-frame metric is the metric that appears directly in the sigma-model action of the fundamental string. The Einstein-frame metric is the one in which the gravitational action has the canonical Einstein—Hilbert form. If we normalize the two metrics to agree at infinity, then

$$g_{\mu\nu}^{(E)}=e^{-(\Phi-\Phi_\infty)/2}g_{\mu\nu}^{({\rm str})}, \qquad \Phi_\infty=\log g_s.$$

Using

$$e^{\Phi-\Phi_\infty}=H_p^{(3-p)/4},$$

we obtain

$$\boxed{ ds_E^2 =H_p^{-(7-p)/8}dx_{\parallel}^2 +H_p^{(p+1)/8}dx_{\perp}^2. }$$

This form makes the universal BPS structure more symmetrical. The exponents add up in exactly the way needed for an extremal charged object in ten-dimensional Einstein gravity. The string-frame form is often more useful for worldsheet questions and T-duality, while the Einstein-frame form is often more useful for black-brane thermodynamics and comparisons with lower-dimensional gravity.

The distinction matters physically. For example, the D3-brane is self-dual in type IIB and has constant dilaton, so the string and Einstein frames differ only by a constant. For $p\neq3$, the dilaton varies with $r$, and the local effective string coupling

$$g_{\rm eff}^{\rm string}(r)=e^{\Phi(r)}=g_sH_p(r)^{(3-p)/4}$$

can become large or small near the brane depending on $p$.

Near $r=0$, $H_p\to\infty$. Hence

$$\begin{array}{c|c} \text{brane} & e^\Phi \text{ near the core}\\ \hline p<3 & \text{grows}\\ p=3 & \text{constant}\\ p>3 & \text{decreases} \end{array}$$

This simple table is a useful diagnostic. D1- and D2-brane geometries become strongly coupled in the deep interior; their complete description requires S-duality or M-theory in the appropriate regimes. D4-, D5-, and D6-brane geometries have weak string coupling near the core, but their curvature and ultraviolet behavior still require care.

The D3-brane as the cleanest example

For $p=3$, the solution becomes

$$ds_{\rm str}^2=H_3^{-1/2}dx_{1,3}^2+H_3^{1/2}(dr^2+r^2d\Omega_5^2), \qquad H_3=1+{R^4\over r^4},$$

with

$$R^4=4\pi g_sN(\alpha')^2, \qquad e^\Phi=g_s.$$

In the near-horizon region $r\ll R$, the constant $1$ may be dropped:

$$H_3\simeq {R^4\over r^4}.$$

Then

$$ds^2\simeq {r^2\over R^2}dx_{1,3}^2+{R^2\over r^2}dr^2+R^2d\Omega_5^2.$$

This is $AdS_5\times S^5$ in Poincare coordinates. The detailed holographic interpretation will be developed later, but it is useful to see already what makes the D3-brane special: the same radius $R$ controls both the anti-de Sitter space and the sphere, and the dilaton is constant.

The curvature scale in string units is

$${R^2\over \alpha'}=\sqrt{4\pi g_sN}.$$

Classical supergravity requires

$$R^2\gg \alpha' \qquad\Longleftrightarrow\qquad g_sN\gg1.$$

At the same time, string loops are suppressed when

$$g_s\ll1.$$

Thus the most useful supergravity regime for $N$ D3-branes is

$$N\gg1, \qquad 1\ll g_sN, \qquad g_s\ll1.$$

This is the familiar large-$N$, large ‘t Hooft coupling, weak closed-string coupling window.

The NS-sector dual pair: F1 and NS5

D-branes are R—R charged. The NS—NS sector also has charged extended objects. The fundamental string is electrically charged under $B_2$, while the NS5-brane is magnetically charged under $B_2$.

The fundamental string solution in string frame has the schematic form

$$\begin{aligned} ds_{\rm str}^2&=H_F^{-1}dx_{1,1}^2+dx_{\perp}^2,\\ e^{2\Phi}&=g_s^2H_F^{-1},\\ B_{01}&=H_F^{-1}-1, \end{aligned}$$

where $H_F=1+Q_F/r^6$ in the eight transverse dimensions. This is the NS—NS analogue of an electrically charged brane.

The magnetic dual is the NS5-brane. It fills $x^0,\ldots,x^5$ and has four transverse directions. Its string-frame solution is

$$\boxed{ ds_{\rm str}^2=dx_{1,5}^2+H_5(r)\left(dr^2+r^2d\Omega_3^2\right) }$$

with

$$\boxed{ e^{2\Phi}=g_s^2H_5(r), \qquad H_5(r)=1+{N\alpha'\over r^2}. }$$

The NS—NS three-form flux is magnetic through the transverse three-sphere:

$$\boxed{ {1\over (2\pi)^2\alpha'}\int_{S^3}H_3=N. }$$

Equivalently, if $\omega_3$ is the volume form of the unit $S^3$, normalized by $\int_{S^3}\omega_3=2\pi^2$, then in the near-throat region one may write

$$H_3=2N\alpha'\,\omega_3.$$

This flux is the NS5-brane charge. The harmonic function has the $1/r^2$ form because the transverse space is $\mathbb R^4$.

The NS5-brane is not merely the $p=5$ member of the D$p$ family. Compare it with the D5-brane. For a D5-brane,

$$e^\Phi=g_sH_{D5}^{-1/2},$$

so the string coupling decreases near the core. For an NS5-brane,

$$e^\Phi=g_sH_5^{1/2},$$

so the string coupling grows near the core. In type IIB, these two solutions are related by S-duality. The Einstein-frame description is the natural place to compare them, because the Einstein metric is invariant under the $SL(2,\mathbb Z)$ duality group.

The linear-dilaton throat of NS5-branes

The near-horizon region of $N$ NS5-branes is obtained by taking

$$r\ll \sqrt{N\alpha'}, \qquad H_5(r)\simeq {N\alpha'\over r^2}.$$

Then the transverse part of the string-frame metric becomes

$$H_5(r)\left(dr^2+r^2d\Omega_3^2\right) \simeq N\alpha'\left({dr^2\over r^2}+d\Omega_3^2\right).$$

Introduce a radial coordinate $\rho$ by

$$\rho=\sqrt{N\alpha'}\log\left({r\over \sqrt{N\alpha'}}\right).$$

Then

$${N\alpha'\over r^2}dr^2=d\rho^2,$$

and the near-horizon metric is

$$\boxed{ ds_{\rm str}^2\simeq dx_{1,5}^2+d\rho^2+N\alpha' d\Omega_3^2. }$$

The throat is a cylinder: $\mathbb R_\rho\times S^3$, with the $S^3$ radius fixed by the number of fivebranes. The dilaton is not constant. Since

$$e^{2\Phi}\simeq g_s^2{N\alpha'\over r^2},$$

we get

$$\boxed{ \Phi(\rho)=\Phi_0-{\rho\over\sqrt{N\alpha'}} }$$

up to an additive constant. As $r\to0$, one has $\rho\to-\infty$, so $\Phi\to+\infty$. The throat becomes strongly coupled at its bottom.

The near-horizon NS5 geometry is a cylinder $\mathbb R_\rho\times S^3$ times the flat six-dimensional worldvolume. The $S^3$ carries $N$ units of $H_3$ flux, while the dilaton varies linearly along the throat and grows toward the core.

This geometry is qualitatively different from the D3-brane throat. The D3 throat is anti-de Sitter and has constant coupling. The NS5 throat is a linear-dilaton background with a strong-coupling end. Its decoupled theory is not an ordinary local quantum field theory but a six-dimensional nonlocal theory known as little string theory.

Exact worldsheet description of the NS5 throat

The NS5 near-horizon throat is one of the rare curved string backgrounds that has an exact worldsheet CFT description. The geometry is

$$\mathbb R^{5,1}\times \mathbb R_\phi\times S^3,$$

and the $S^3$ with $N$ units of $H_3$ flux is described by a supersymmetric $SU(2)$ Wess—Zumino—Witten model. In a common convention one writes the throat CFT as

$$\mathbb R^{5,1}\times \mathbb R_\phi\times SU(2)_N.$$

There is a standard level-shift subtlety: the supersymmetric $SU(2)_N$ theory can be represented as a bosonic $SU(2)$ WZW model at level $N-2$ plus three free fermions. This is why different books sometimes quote nearby-looking levels. The physical integer $N$ is the number of NS5-branes and the number of units of $H_3$ flux.

The central charge works beautifully. The supersymmetric $SU(2)_N$ sector has

$$c_{SU(2)_N}^{\rm susy}={9\over2}-{6\over N},$$

while the linear-dilaton boson plus its fermionic partner contributes

$$c_{\rm lin}={3\over2}+{6\over N},$$

in the common $\alpha'=2$ worldsheet convention. Their sum is

$$c_{SU(2)_N}^{\rm susy}+c_{\rm lin}=6,$$

which is precisely the central charge of four transverse supersymmetric coordinates. Adding the six flat worldvolume directions gives the critical type II worldsheet theory.

This exact CFT viewpoint explains why the NS5-brane throat is more than a supergravity approximation. The sphere radius and $H_3$ flux are not arbitrary classical decorations; they are encoded by the WZW level. The linear dilaton is also not a guess: it is required by conformal invariance.

Validity of supergravity

A supergravity solution is useful only when both stringy and quantum corrections are small. There are two local conditions:

$$\alpha'\mathcal R\ll1, \qquad e^\Phi\ll1.$$

The first condition suppresses higher-derivative corrections. The second suppresses string loops. For D$p$-branes, the curvature radius is controlled by a charge scale $R_p$, so large $g_sN$ often helps. But the dilaton may still run, and for $p\neq3$ different radial regions can require different dual descriptions.

For NS5-branes, the sphere radius in the throat is

$$R_{S^3}^2=N\alpha'.$$

Thus large $N$ suppresses curvature corrections. However,

$$e^\Phi\simeq g_s{\sqrt{N\alpha'}\over r}$$

in the throat, so string loops become important near $r=0$ no matter how small $g_s$ is at infinity. In type IIA, the strong-coupling core is better described by lifting to M-theory, where NS5-branes become M5-branes. In type IIB, S-duality maps the NS5-brane to a D5-brane, which may provide a weakly coupled description in the appropriate regime.

The practical lesson is not that the supergravity solutions are unreliable. Rather, they are reliable in their proper domains. The best string-theory calculations often proceed by patching together dual descriptions, each valid in a different region of moduli space or spacetime.

What to remember

The essential facts are as follows.

Object	Harmonic function	String-frame behavior	Charge
D$p$	$H_p=1+R_p^{7-p}/r^{7-p}$	$ds^2=H_p^{-1/2}dx_{\parallel}^2+H_p^{1/2}dx_\perp^2$	electric under $C_{p+1}$
D3	$H_3=1+R^4/r^4$	constant dilaton	self-dual $F_5$ charge
F1	$H_F=1+Q_F/r^6$	$ds^2=H_F^{-1}dx_{1,1}^2+dx_\perp^2$	electric under $B_2$
NS5	$H_5=1+N\alpha'/r^2$	linear-dilaton throat	magnetic under $B_2$

The D$p$ solutions teach us that branes are not merely boundary conditions for open strings. They are charged gravitating objects. The NS5 solution teaches us that exact worldsheet CFTs can describe highly nontrivial curved backgrounds. Together, these examples prepare the ground for black branes, entropy, absorption, and the decoupling limits that lead to AdS/CFT.

Exercises

Exercise 1: the radial harmonic function in $\mathbb R^{9-p}$

Let $n=9-p$ and suppose $H=H(r)$ is harmonic away from the origin in $\mathbb R^n$:

$${1\over r^{n-1}}{d\over dr}\left(r^{n-1}{dH\over dr}\right)=0.$$

For $n>2$, show that

$$H(r)=h_0+{Q\over r^{n-2}}.$$

Then rewrite the exponent in terms of $p$.

Solution

The radial equation says

$${d\over dr}\left(r^{n-1}{dH\over dr}\right)=0.$$

Therefore

$$r^{n-1}{dH\over dr}=A$$

for a constant $A$, so

$${dH\over dr}=A r^{1-n}.$$

For $n\neq2$,

$$H(r)=h_0+{A\over 2-n}r^{2-n}.$$

Renaming $Q=A/(2-n)$ gives

$$H(r)=h_0+{Q\over r^{n-2}}.$$

Since $n=9-p$, the exponent is

$$n-2=7-p.$$

Thus the D$p$-brane harmonic function has the form

$$H_p(r)=1+{Q_p\over r^{7-p}}.$$

Exercise 2: D3-brane near-horizon geometry

Starting from

$$ds^2=H^{-1/2}dx_{1,3}^2+H^{1/2}(dr^2+r^2d\Omega_5^2), \qquad H=1+{R^4\over r^4},$$

show that the near-horizon region $r\ll R$ is $AdS_5\times S^5$ with common radius $R$.

Solution

For $r\ll R$,

$$H\simeq {R^4\over r^4}.$$

Then

$$H^{-1/2}\simeq {r^2\over R^2}, \qquad H^{1/2}\simeq {R^2\over r^2}.$$

Substituting into the metric gives

$$ds^2\simeq {r^2\over R^2}dx_{1,3}^2+{R^2\over r^2}dr^2+R^2d\Omega_5^2.$$

The first two terms are the Poincare-patch metric on $AdS_5$ of radius $R$, and the last term is the metric on a round $S^5$ of the same radius. Hence the near-horizon geometry is

$$AdS_5\times S^5.$$

Exercise 3: Einstein-frame exponents for D$p$-branes

Use

$$ds_{\rm str}^2=H_p^{-1/2}dx_{\parallel}^2+H_p^{1/2}dx_{\perp}^2, \qquad e^{\Phi-\Phi_\infty}=H_p^{(3-p)/4},$$

and

$$g_E=e^{-(\Phi-\Phi_\infty)/2}g_{\rm str}$$

to derive

$$ds_E^2=H_p^{-(7-p)/8}dx_{\parallel}^2+H_p^{(p+1)/8}dx_{\perp}^2.$$

Solution

The Weyl factor is

$$e^{-(\Phi-\Phi_\infty)/2} =\left(H_p^{(3-p)/4}\right)^{-1/2} =H_p^{(p-3)/8}.$$

Multiplying the parallel part of the string-frame metric gives

$$H_p^{(p-3)/8}H_p^{-1/2} =H_p^{(p-3)/8-4/8} =H_p^{(p-7)/8} =H_p^{-(7-p)/8}.$$

Multiplying the transverse part gives

$$H_p^{(p-3)/8}H_p^{1/2} =H_p^{(p-3)/8+4/8} =H_p^{(p+1)/8}.$$

Therefore

$$ds_E^2=H_p^{-(7-p)/8}dx_{\parallel}^2+H_p^{(p+1)/8}dx_{\perp}^2.$$

Exercise 4: the local string coupling near a D$p$-brane

For the D$p$-brane solution,

$$e^\Phi=g_sH_p^{(3-p)/4}.$$

Assuming $H_p\to\infty$ near the core, determine for which values of $p$ the local string coupling grows, stays constant, or decreases.

Solution

The behavior is controlled by the exponent

$${3-p\over4}.$$

If $p<3$, the exponent is positive, so $e^\Phi$ grows as $H_p\to\infty$. If $p=3$, the exponent vanishes, so

$$e^\Phi=g_s$$

is constant. If $p>3$, the exponent is negative, so $e^\Phi$ decreases near the core.

Thus

$$\begin{array}{c|c} p<3 & \text{coupling grows}\\ p=3 & \text{coupling is constant}\\ p>3 & \text{coupling decreases} \end{array}$$

This is why D3-branes are especially clean, while D1/D2 systems and D4/D5/D6 systems often require dual descriptions in different regimes.

Exercise 5: the NS5 linear-dilaton throat

Starting from

$$ds^2=dx_{1,5}^2+H_5(r)(dr^2+r^2d\Omega_3^2), \qquad H_5(r)=1+{N\alpha'\over r^2},$$

show that for $r\ll\sqrt{N\alpha'}$ the metric becomes

$$ds^2\simeq dx_{1,5}^2+d\rho^2+N\alpha' d\Omega_3^2$$

with

$$\rho=\sqrt{N\alpha'}\log\left({r\over\sqrt{N\alpha'}}\right).$$

Then derive the linear dilaton.

Solution

In the near-horizon region,

$$H_5(r)\simeq {N\alpha'\over r^2}.$$

Therefore

$$H_5(r)(dr^2+r^2d\Omega_3^2) \simeq N\alpha'\left({dr^2\over r^2}+d\Omega_3^2\right).$$

The coordinate definition gives

$$d\rho=\sqrt{N\alpha'}{dr\over r},$$

so

$$N\alpha'{dr^2\over r^2}=d\rho^2.$$

Thus

$$ds^2\simeq dx_{1,5}^2+d\rho^2+N\alpha'd\Omega_3^2.$$

For the dilaton,

$$e^{2\Phi}=g_s^2H_5\simeq g_s^2{N\alpha'\over r^2}.$$

Using

$$r=\sqrt{N\alpha'}\,e^{\rho/\sqrt{N\alpha'}},$$

we find

$$e^{2\Phi}\simeq g_s^2 e^{-2\rho/\sqrt{N\alpha'}}.$$

Taking the logarithm gives

$$\Phi(\rho)=\Phi_0-{\rho\over\sqrt{N\alpha'}},$$

where $\Phi_0$ is an additive constant.

Exercise 6: NS5 charge from $H_3$ flux

Let $\omega_3$ be the volume form on the unit three-sphere, normalized as

$$\int_{S^3}\omega_3=2\pi^2.$$

Show that

$$H_3=2N\alpha'\omega_3$$

has $N$ units of NS5-brane charge under the convention

$${1\over(2\pi)^2\alpha'}\int_{S^3}H_3=N.$$

Solution

Compute the flux:

$$\int_{S^3}H_3 =2N\alpha'\int_{S^3}\omega_3 =2N\alpha'(2\pi^2) =4\pi^2N\alpha'.$$

Since

$$(2\pi)^2\alpha'=4\pi^2\alpha',$$

we get

$${1\over(2\pi)^2\alpha'}\int_{S^3}H_3 ={1\over4\pi^2\alpha'}(4\pi^2N\alpha')=N.$$

Thus the flux is precisely $N$ units.

Exercise 7: multi-center harmonic functions and no-force

Explain why

$$H_p(\vec y)=1+\sum_a {Q_a\over |\vec y-\vec y_a|^{7-p}}$$

is a natural candidate for several parallel BPS D$p$-branes. What physical property of BPS branes makes this possible?

Solution

Away from all centers, each term

$${Q_a\over |\vec y-\vec y_a|^{7-p}}$$

is harmonic in the transverse space. Since the Laplace equation is linear, the sum is also harmonic away from the sources. The constants $Q_a$ encode the individual brane charges.

The nontrivial point is that this harmonic superposition solves the full BPS supergravity equations, not merely a linearized approximation. Supersymmetry enforces a balance between forces: gravitational and dilaton attraction are exactly canceled by R—R repulsion for parallel branes of the same orientation. Therefore the branes can be placed at arbitrary positions $\vec y_a$ without acceleration. This no-force condition is the physical reason the multi-center harmonic function is allowed.