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A complete and natural geometric and physical interpretation of thetangential gauge supersymmetry, also known as $\kappa$-symmetry, of a largeclass of Green-Schwarz(-type) super-$\sigma$-models for the super-$p$-brane ina homogeneous space of a (supersymmetry) Lie supergroup is established in theconvenient setting of the topological Hughes-Polchinski formulation of thesuper-$\sigma$-model and illustrated on a number of physical examples. Thesupersymmetry is identified as an odd superdistribution in the tangent sheaf ofthe supertarget of the super-$\sigma$-model, generating - through its weakderived flag - the vacuum foliation of the supertarget. It is also demonstratedto canonically lift to the vacuum restriction of the extended Hughes-Polchinski$p$-gerbe associated with the superbackground of the field theory, and that inthe form of a canonical linearised equivariant structure thereon, canonicallycompatible with the residual global supersymmetry of the vacuum.

The square root of the vacuum I. Equivariance for the $\kappa$-symmetry superdistribution