Four-dimensional supermembrane

The four-dimensional minimal supersymmetric membrane has embedding coordinates $(X^m,{\theta }^{\alpha })$ with four real odd components. Its area term uses the supertranslation-invariant superspace coframe, and a Wess-Zumino brane coupling integrates a superspace three-form potential with closed four-form curvature. Equal charge magnitude and tension permit kappa symmetry. After worldvolume reparametrization and kappa fixing, one transverse scalar and one three-dimensional Majorana fermion remain. This gives the long-wavelength description of the translation and fermion zero modes of a supersymmetric domain wall; it is distinct from restricting the eleven-dimensional M2-brane to a four-dimensional target.

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.