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A charged brane couples to an appropriate differential-form gauge potential through its pullback. Under $A\mapsto A+d\Lambda$, a closed worldvolume changes the action by a boundary term, by Stokes theorem. Reversing the brane charge reverses this coupling, which matters in probe-force and supersymmetry calculations.

The worldvolume is the history of a brane. An embedding $X:\Sigma \to \mathcal{M}$ maps its local coordinates into target spacetime. A string worldvolume is a worldsheet.