Wess-Zumino brane coupling ($q{T}_p{\int }_{\Sigma }{X}^{\ast }{A}_{p+1}$)

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.

A $(p+1)$-form gauge potential couples electrically through its pullback to a $(p+1)$-dimensional worldvolume, whose object has $p$ spatial dimensions. Its field strength has degree $p+2$ and its magnetic dual potential has degree $d-p-3$, hence magnetic spatial dimension $d-p-4$. Electric and magnetic fluxes are integrated over linking spheres of dimensions $d-p-2$ and $p+2$, respectively. This dimensional dictionary applies when the indicated objects and dual descriptions exist; top forms need separate treatment and carry no ordinary propagating polarization.

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.

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.