Green-Schwarz超膜的等变Cartan-Eilenberg超胚I。超级闵可夫斯基案
Equivariant Cartan-Eilenberg supergerbes for the Green-Schwarz superbranes I. The super-Minkowskian case
摘要：
提出了Green-Schwarz型超$σ$ -模型的一个明确的Gerbe理论描述，并详细研究了具有Lie超群的齐次空间结构的目标的基本结构性质，如目标超流形的等距性和$κ$ -对称性. Cartan-Eilenberg超- $(p+2)$ -上循环的（超）几何化方案确定了超- $p$ -膜力学的拓扑内容，并确保其$κ$ -对称性，其动机是其玻色子前辈的成功，并由其提供的直觉指导，是基于李超代数的Chevalley-Eilenberg上同调中存在非平凡超2-上圈时李超群的（超）中心扩张的思想，这两个上同调之间的差距正在弥合的经典Chevalley-Eilenberg建设的超级变种。在此，我们首先详细研究了超庞加莱群的基本齐性空间，即超闵可夫斯基时空，从而系统地实现了这一计划。超闵可夫斯基时空的简单性使我们能够直接识别超几何机制，并使我们能够顺利地开发出可用于更复杂情况的形式化工具。
英文摘要：
An explicit gerbe-theoretic description of the super-$σ$-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 $κ$-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 $κ$-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-Poincaré 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.
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