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Let $a=0,\dots ,p$ label directions tangent to the D-brane and $i=p+1,\dots ,25$ label transverse directions. The endpoints obey Dirichlet conditions transversely,and the boundary variation of the metric and constant B-field terms gives mixed tangential conditionsup to the orientation sign at the two ends. A worldvolume gauge potential contributes $2\pi {\alpha }^{\prime }{F}_{ab}$ in the same place, so the gauge-invariant condition contains $\mathcal{F}=b+2\pi {\alpha }^{\prime }F$.

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.