Spin groups of super metrics and a theorem of Rogers

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• 作者：Ronald Fulp ^{fulp@math.ncsu.edu}
• 关键词：G&infin ; G&infin ; -supermanifolds ; Super Riemannian metrics ; Canonical forms ; Local isometry groups ; Super Lie groups ; Conventional super Lie algebras
• 刊名：Journal of Geometry and Physics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：111
• 期：Complete
• 页码：40-53
• 全文大小：475 K
• 卷排序：111

文摘

We derive the canonical forms of super Riemannian metrics and the local isometry groups of such metrics. For certain super metrics we also compute the simply connected covering groups of the local isometry groups and interpret these as local spin groups of the super metric. Super metrics define reductions OSgOSg of the relevant frame bundle. When principal bundles S˜g exist with structure group the simply connected covering group G̃ of the structure group of OSg,OSg, representations of G̃ define vector bundles associated to S˜g whose sections are “spinor fields” associated with the super metric g.g. Using a generalization of a Theorem of Rogers, which is itself one of the main results of this paper, we show that for super metrics we call body reducible, each such simply connected covering group G̃ is a super Lie group with a conventional super Lie algebra as its corresponding super Lie algebra.Some of our results were known to DeWitt (1984) using formal Grassmann series and others were known by Rogers using finitely many Grassmann generators and passing to a direct limit. We work exclusively in the category of G∞G∞ supermanifolds with G∞G∞ mappings. Our supernumbers are infinite series of products of Grassmann generators subject to convergence in the ℓ_{1ℓ1 norm introduced by Rogers (1980, 2007).}