= Solution
For $N$ parallel type-II <D-branes>, an oriented open string has endpoint labels $i,j$, its <Chan-Paton factors>. At coincidence, the massless states form $N\times N$ matrices, giving adjoint fields of $U(N)$. Tangential polarizations produce the <Yang-Mills theory> gauge field; transverse polarizations produce $9-p$ adjoint <scalar fields>. The leading low-energy action is the dimensional reduction of <ten-dimensional super Yang-Mills theory> to $p+1$ dimensions. Expanding the <Dirac-Born-Infeld action> gives the gauge and scalar kinetic terms, while dimensional reduction supplies the <Hermitian commutator potential in D-brane Yang-Mills theory>:
$$
V=-\frac1{4g_{\rm YM}^2}\sum_{I,J}\operatorname{Tr}[\phi^I,\phi^J]^2\ge0.
$$
The sign is important: for Hermitian $\phi$, the commutator is anti-Hermitian. Zero potential requires commuting scalars, whose simultaneous eigenvalues are brane positions through the <D-brane scalar-position normalization>
$$
Y_i^I=2\pi\alpha'\langle\phi_i^I\rangle.
$$
This makes the <Higgs mechanism> geometrical. Separating branes gives masses to strings joining different positions; bringing them together gives <Coincident-D-brane gauge enhancement>.
Two parallel <D3-branes> in <type IIB superstring theory> give <four-dimensional N=4 super Yang-Mills theory> with gauge group $U(2)$, containing a gauge field, six real adjoint <scalar fields> and four adjoint <Weyl fermions>. On its <Coulomb branch>, choose
$$
\langle\phi^I\rangle=\frac1{2\pi\alpha'}\begin{pmatrix}Y_1^I&0\\0&Y_2^I\end{pmatrix}.
$$
For distinct position vectors, $U(2)$ breaks to $U(1)\times U(1)$. The off-diagonal vector multiplets are the two orientations of a string joining the branes, and
$$
\boxed{m_W=\frac{|Y_1-Y_2|}{2\pi\alpha'}.}
$$
The same mass follows from the scalar covariant-derivative term, $m_W^2=\sum_I(\phi_1^I-\phi_2^I)^2$. The overall $U(1)$ describes center-of-mass motion; the relative $SU(2)$ sector is Higgsed to its Cartan $U(1)$. At coincidence the off-diagonal states become massless and the full $U(2)$ gauge symmetry is restored.
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