= Solution
A <D-brane> is a localized RR-charged source whose long-distance fields solve <supergravity>. The extremal charged $p$-brane solutions carry the same RR charges, <brane tension> and preserved supersymmetry as the open-string construction. For the standard asymptotically flat cases $p<7$, the <D-brane supergravity solution> in the <string frame> is
$$
ds_s^2=H^{-1/2}ds^2(\mathbb R^{1,p})+H^{1/2}dy_\perp^2,\qquad
e^\Phi=g_sH^{(3-p)/4},\qquad H=1+\frac Q{r^{7-p}}.
$$
The function $H$ is a positive <harmonic function> away from the source, in $9-p$ transverse dimensions. In an electric gauge, $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$, with its orientation chosen to match the source. Magnetic branes use the dual flux; for the <D3-brane>, the five-form includes both electric and magnetic pieces and is self-dual. <Killing spinors> obey one brane projector, leaving sixteen supersymmetries. Aligned multicenter harmonic functions describe several branes with no static binding force.
To check this identification directly, insert an equally oriented static probe into the solution. Put $\mu_p=(2\pi)^{-p}\alpha'^{-(p+1)/2}$, so the asymptotic tension is $\mu_p/g_s$. The induced-volume factor is $H^{-(p+1)/4}$; combined with $e^{-\Phi}$, its <DBI action> density is $-\mu_p g_s^{-1}H^{-1}$. Its <Wess-Zumino brane coupling> contributes $+\mu_p g_s^{-1}(H^{-1}-1)$. Hence the total static Lagrangian density is the constant $-\mu_p/g_s$, independently of position. This gives the <parallel D-brane no-force identity> in the spacetime description.
The supergravity background is the back-reacted, coarse long-distance description, while the boundary-condition description exposes the light open-string degrees of freedom. Controlled use of supergravity requires curvature small in string units and suitable local coupling. Singular cores and the special codimension-two or domain-wall cases are not automatically resolved by the elementary harmonic ansatz.
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