Seminarium KMMF "Teoria Dwoistości"

The geometrodynamics of topologically charged extended objects, in particular that of loops and paths, in the homogeneous space AdS_5 x |S^5 of the supersymmetry Lie supergroup 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 transition into its fairly well-understood counterpart on the super-Minkowski space under the İnönü-Wigner contraction su(2,2|4)/(so(4,1) x so(5)) -> 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 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-σ-model on a homogeneous space of a supersymmetry Lie supergroup G associated with a distinguished super-cocycle χ in the Cartan-Eilenberg cohomology of the latter, and the ensuing non-linear realisation of supersymmetry, as well as its gauged linearised variant - Siegel's κ-symmetry. I shall also discuss at length a recently proposed scheme of equivariant geometrisation of χ that employs integrable supercentral extensions of the Lie superalgebra of G induced by the super-cocycle through the standard correspondence between the Cartan-Eilenberg cohomology of the Lie (super)group and the Chevalley-Eilenberg cohomology of its Lie (super)algebra. An intricate topological interpretation of the ensuing supersymmetry extension in terms of the Kostelecký-Rabin winding charge shall be given, and due emphasis shall be laid upon the issues of (weak) κ-equivariance of the geometrisation and its compatibility with the İnönü-Wigner contraction on the base supermanifold. The general construction shall be illustrated 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-cocycle.