Equivariant prequantisation of the super-0-brane in ${\rm AdS}_2\times\mathbb{S}^2$ - a toy model for supergerbe theory on curved spaces

Abstract

The paper is another step towards a realisation of the goal, advanced in articles 1706.05682 [hep-th] and 1808.04470 [hep-th], of a systematic supersymmetry-equivariant geometrisation of physically distinguished Green-Schwarz super-(p+2)-cocycles defining classes in the supersymmetry-invariant refinement of the de Rham cohomology of homogeneous spaces of (supersymmetry) Lie supergroups, associated with reductive decompositions of their Lie superalgebras. It deals with a correlated geometrisation of a pair of such super-(p+2)-cocycles on spaces in correspondence under a blow-up transformation dual to the \.In\"on\"u-Wigner contraction that relates the respective (supersymmetry) Lie superalgebras, the latter correspondence being taken as the organising principle of the geometrisation procedure that exploits the link between the Cartan-Eilenberg cohomology of the supersymmetry group and the Chevalley-Eilenberg cohomology of its Lie superalgebra, alongside a cohomological classification of central extensions thereof. A general scheme of a correlated geometrisation compatible with the contraction is laid out and illustrated on the nontrivial example of a pair of consistent super-0-brane backgrounds: the super-Minkowski space $sMink^{3,1|8}$ with the standard N=2 Green-Schwarz super-2-cocycle on it and the super-$AdS_2 \times S^2$ space with Zhou's super-2-cocycle on it, asymptoting to the former in the limit of an infinite common radius of the $AdS_2$ and the $S^2$ in the body $AdS_2\times S^2$ of the supertarget. The geometrisation yields the respective supersymmetry-equivariant super-0-gerbes, i.e., the prequantum bundles of the associated Green-Schwarz super-$\sigma$-models in the Nambu-Goto formulation. Upon passing to the equivalent Hughes-Polchinski formulation of the models, the relevant extended super-0-gerbes are verified to possess a weak $\kappa$-equivariant structure.