The theory of bundle gerbes with connection has long been known to provide us with the most adequate geometric and cohomological tools for a rigorous canonical description, rigid- and gauge-symmetry as well as duality analysis and equivariant geometric quantisation of the loop mechanics on metric target manifolds determined by a natural generalisation of the standard lagrangean model of propagation of a charged massive point-like particle in background gravitational and electromagnetic fields, known as the two-dimensional bosonic σ-model. Up to now, no extension of the grebe-theoretic description to σ-models with supersymmetry has been worked out although proposals of the relevant geometric structures (involving, i.a., Chern–Simons 2-gerbes based on spin groups) exist. This is a most unsatisfactory situation also from the physical vantage point as it is among supersymmetric σ-models that we find those describing (super)string theory on anti-de Sitter (AdS) metric target spaces, and the latter has proven instrumental in the study of realistic systems of strongly interacting fields with colour, such as the quark–gluon plasmas, beyond the perturbative paradigm.
In my talk, I shall present an original proposal of the gerbe theory behind the so-called Green–Schwarz super-σ-models that capture the dynamics of extended solitonic objects — the superparticle, the superstring and various supermembranes — on supertargets with the structure of a homogeneous space of a super-Lie group, with emphasis on the super-Minkowski space. The proposal is based on geometrisation of elementary Chevalley–Eilenberg cocycles on the relevant super-Lie algebras through the construction of so-called extended superspaces (due to de Azcárraga et al.) and yields supersymmetry-equivariant n-gerbes with connection over the supertargets. The latter are also quasi-equivariant with respect to the local supersymmetry of the respective super-σ-models (Siegel's κ-symmetry) responsible for the removal of the spurious fermionic degrees of freedom. Time permitting, I shall give an outlook on the super n-gerbe-theoretic approach to loop dynamics on the AdS backgrounds of Мецаев and Цейтлин.