Spin goups of super metrics and a Theorem of Rogers

TL;DR

This paper characterizes super Riemannian metrics, their isometry groups, and introduces super spin groups, extending Rogers' theorem to super metrics that are body reducible, within the $G^{ty}$ framework.

Contribution

It derives canonical forms of super Riemannian metrics, computes their local isometry groups, and generalizes Rogers' theorem to super metrics, establishing super Lie group structures.

Findings

Canonical forms of super Riemannian metrics

Classification of local isometry groups

Extension of Rogers' theorem to super metrics

Abstract

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. 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 G is a super Lie group with a conventional super Lie algebra as its corresponding super Lie algebra. We work exclusively in the $G^{\infty}$ category.