Kappa symmetry (${\delta }_{\kappa }\Theta =(1+{\Gamma }_{\kappa })\kappa$)

A local fermionic gauge symmetry of supersymmetric brane actions. Since the kappa symmetry projector has half rank, the symmetry removes half the embedding fermions. The remaining physical fermions pair with the bosonic worldvolume degrees of freedom.

With ${\Gamma }_{\kappa }^2=1$, the two complementary linear projections select its two eigenspaces. A bosonic brane preserves a background Killing spinor precisely when its target-space supersymmetry variation can be compensated by kappa symmetry, giving ${\Gamma }_{\kappa }ϵ=ϵ$. This condition must hold at every worldvolume point; several branes require a common solution. The full superspace construction is described in Superspace Geometry for Supermembrane Backgrounds.