= Solution
Let $a=0,\ldots,p$ label directions tangent to the <D-brane> and $i=p+1,\ldots,25$ label transverse directions. The endpoints obey Dirichlet conditions transversely,
$$
\boxed{\delta X^i=0\quad\text{or equivalently}\quad
X^i|_{\partial\Sigma}=x_0^i},
$$
and the boundary variation of the metric and constant B-field terms gives mixed tangential conditions
$$
\boxed{\eta_{ab}\partial_\sigma X^b
+b_{ab}\partial_\tau X^b=0
\quad\text{on }\partial\Sigma},
$$
up to the orientation sign at the two ends. A worldvolume gauge potential contributes $2\pi\alpha'F_{ab}$ in the same place, so the gauge-invariant condition contains $\mathcal F=b+2\pi\alpha'F$.
From the brane perspective, the pullback of the background B-field is therefore indistinguishable locally from a constant worldvolume electromagnetic field strength, modulo its two-form gauge symmetry and a compensating transformation of the brane gauge field. This is the <open-string boundary condition in a B-field>; sufficiently general constant $b$ also induces the familiar noncommutative deformation of D-brane endpoint coordinates.
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