Field-strength force on a charged brane ($\delta {\int }_W{X}^{\ast }A={\int }_W{X}^{\ast }({\iota }_{V}F)+{\int }_{\partial W}{X}^{\ast }({\iota }_{V}A)$)

For $F=dA$ and an embedding variation $V$, the pullback Lie derivative and Cartan's magic formula give the displayed variation. Consequently the interior force from the Wess-Zumino brane coupling depends only on the gauge-invariant field strength. A change $A\mapsto A+d\Lambda$ changes the action only on the boundary, by the Generalized Stokes theorem; boundaries require the appropriate fixed boundary data or additional boundary coupling.