The geometrodynamics of topologically charged extended objects, in particular that of loops and paths, in the homogeneous space ${\rm AdS}_5\times\mathbb{S}^5$ of the supersymmetry Lie supergroup ${\rm SU}(2,2|4)$ has long been known to play an important role in modern attempts, based on the so-called AdS/CFT correspondence, at understanding the non-perturbative quantum mechanics of realistic strongly coupled systems with gauge symmetry, such as, e.g., the quark gluon plasma. An important feature of the dynamics is an asymptotic transtiion into its fairly well-understood counterpart on the super-Monikowski space under the Inonu-Wigner contraction $\mathfrak{su}(2,2|4)/\mathfrak{so}(4,1)\times\mathfrak{so}(5))\to \mathfrak{siso}(9,1|32)/so(9,1)$. A rigorous treatment of the gauge field coupling to the topological charge carried by these objects, leading through a supersymmetry-equivariant Dirac-type geometrisation of the corresponding class in the Cartan-Eilenberg cohomology of ${\rm SU}(2,2|4)$, paves the way to a geometric quantisation of the dynamics and a systematic construction of supersymmetric defects central to the AdS/CFT correspondence, and so offers hope for an in-depth elucidation of the higher geometry behind the holographic principle.

In my talk, I shall recapitulate the construction of the so-called super $\sigma$-model on a homogeneous space of a supersymmetry Lie supergroup $\,G\,$ associated with a distinguished super-cocycle $\,\chi\,$ that employs integrable supercentral extensions of the Lie superalgebra of $\,G\,$ induced by the super-cocycle through the standard correspondence between the Cartan-Eilenberg cohomolgy of the Lie (super)group and the Chevalley-Eilenberg cohomoogy of its Lie (super)algebra. An intrincate topological interpretation of the ensuing supersymmetry extension in terms of the Kostelecky--Rabin winding charge shall be given, and due emphasis shall be laid upon the issues of (weak) $\kappa$-equivariance of the geometrisation and its compatibility with the Inonu--Wigner contraction on the base supermanifold. The general construction shall be illustrate with the examples of the super 1-gerbe associated with the super 3-cocycle of Metsaev and Tseytlin, and the super-0-gerbe associated with Zhou's super 2-cocyle.