Equivariant Cartan-Eilenberg supergerbes for the Green-Schwarz superbranes I. The super-Minkowskian case

TL;DR

This paper develops a gerbe-theoretic framework for super-$\sigma$-models of the Green-Schwarz type, focusing on super-Minkowski space, to better understand their topological and symmetry properties, including $\kappa$-symmetry.

Contribution

It introduces a systematic supergeometric approach to describe super-$\sigma$-models using equivariant Cartan-Eilenberg supergerbes, starting with the super-Minkowski case.

Findings

Explicit gerbe-theoretic description of super-$\sigma$-models.

Analysis of equivariance and $\kappa$-symmetry properties.

Foundational study for extending to more complex supermanifolds.

Abstract

An explicit gerbe-theoretic description of the super-$\sigma$-models of the Green-Schwarz type is proposed and its fundamental structural properties, such as equivariance with respect to distinguished isometries of the target supermanifold and $\kappa$-symmetry, are studied at length for targets with the structure of a homogeneous space of a Lie supergroup. The programme of (super)geometrisation of the Cartan-Eilenberg super-$(p+2)$-cocycles that determine the topological content of the super-$p$-brane mechanics and ensure its $\kappa$-symmetry, motivated by the successes of and guided by the intuitions provided by its bosonic predecessor, is based on the idea of a (super)central extension of a Lie supergroup in the presence of a nontrivial super-2-cocycle in the Chevalley-Eilenberg cohomology of its Lie superalgebra, the gap between the two cohomologies being bridged by a super-variant of the classic Chevalley-Eilenberg construction. A systematic realisation of the programme is herewith begun with a detailed study of the elementary homogeneous space of the super-Poincar\'e group, the super-Minkowskian spacetime, whose simplicity affords straightforward identification of the supergeometric mechanisms and unobstructed development of formal tools to be employed in more complex circumstances.