Supermembrane action ($S=-T_2\int \sqrt{-\det h}+q{T}_2\int Z^{\ast }{\mathcal{A}}_3$)

The first term uses the induced worldvolume metric from the supervielbein; the second is the Wess-Zumino brane coupling to the super-three-form. Its fermionic dependence is retained before bosonic truncation. The relative charge magnitude equals the brane tension, while $q=\pm 1$ sets the orientation and matching kappa symmetry projector sign. In an on-shell eleven-dimensional supergravity superspace, this action has kappa symmetry.