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

Abstract

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