Gauge invariance of an open-brane coupling ($S=q{\int }_W{X}^{\ast }C+q{\int }_{\partial W}{X}^{\ast }B$)

For an oriented brane worldvolume $W$, a gauge transformation $C\mapsto C+d\Lambda$ changes the Wess-Zumino brane coupling by $q{\int }_{\partial W}{X}^{\ast }\Lambda$, by the Generalized Stokes theorem. If the brane boundary lies on a manifold carrying a form $B$ of one lower degree, adding the displayed boundary term and transforming $B\mapsto B-\Lambda$ cancels this variation. The sign uses the induced boundary orientation. A closed worldvolume needs no boundary term; noncompact cases require appropriate support or decay conditions.