Supermembrane closed four-form ($G_4\propto i{\Pi }^m\wedge {\Pi }^n\wedge d\bar{\theta }{\gamma }_{mn}d\theta$)

The four-dimensional membrane Fierz rearrangement identity $(C{\gamma }_{mn}{)}_{(\alpha \beta }(C{\gamma }^n{)}_{\gamma \delta )}=0$ makes the displayed supertranslation-invariant four-form closed. It is locally the exterior derivative of a three-form potential, which can change by an exact form under target supersymmetry. Pulling the potential back to a closed worldvolume gives an invariant Wess-Zumino brane coupling. The charge magnitude and normalization of this form are fixed together by cancellation of the area variation under kappa symmetry.