Closed three-form on four-dimensional superspace ($H_3={\Pi }^m\wedge d\bar{\theta }{\gamma }_{m}d\theta$)

The contraction of invariant vector and spinor frames makes $H_3$ invariant under the Super-Poincaré group. Odd-coordinate differentials commute in the graded exterior algebra. The four-dimensional Fierz rearrangement identity $(C{\gamma }^m{)}_{(\alpha \beta }(C{\gamma }_m{)}_{\gamma \delta )}=0$ consequently gives $d{H}_3=i(d\bar{\theta }{\gamma }^{m}d\theta )(d\bar{\theta }{\gamma }_{m}d\theta )=0$. Its three-form degree is distinct from the four-form needed in the four-dimensional supermembrane Wess-Zumino coupling.