Equivariant Cartan-Eilenberg supergerbes II. Equivariance in the super-Minkowskian setting

This is a continuation of a programme, initiated in Part I [arXiv:1706.05682], of geometrisation, compatible with the SUSY present, of the Green-Schwarz $(p+2)$-cocycles coupling to the topological charges carried by $p$-branes on reductive homogeneous spaces of SUSY groups described by GS(-type) super-$\sigma$-models. In the present part, higher-geometric realisations of the various SUSYs - both global and local - of these field theories are studied at length in the form of - respectively - families of gerbe isomorphisms indexed by the global-SUSY group and equivariant structures with respect to SUSY actions amenable to gauging. The discussion, employing an algebroidal analysis of the small gauge anomaly, leads to a novel definition of a supersymmetric equivariant structure on the Cartan-Eilenberg super-$p$-gerbe of Part I with respect to actions of distinguished normal subgroups of the SUSY group. This is exemplified by the ${\rm Ad}_\cdot$-equivariant structure on the GS super-$p$-gerbes for $\,p\in\{0,1\}\,$ over the super-Minkowski space, whose existence conforms with the classical results for the bosonic counterparts of the corresponding super-$\sigma$-models. The study also explores the fundamental gauge SUSY of the GS super-$\sigma$-model aka $\kappa$-symmetry. Its geometrisation calls for a transcription of the field theory to the dual topological Hughes-Polchinski formulation. Natural conditions for the transcription are identified and illustrated on the example of the super-Minkowskian model of Part I. In the dual formulation, the notion of an extended HP $p$-gerbe unifying the metric and topological degrees of freedom of the GS super-$\sigma$-model is advanced. Its compatibility with $\kappa$-symmetry is ensured by the existence of a linearised equivariant structure. The results reported herein lend strong structural support to the geometrisation scheme postulated in Part I.