The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms, and their parallel transport, which ties together various approaches. We begin with a detailed introduction to super fiber bundles. We then introduce the concept of so-called relative supermanifolds as well as bundles and connections defined in these categories. Studying these objects turns out to be of utmost importance in order to, among other things, model anticommuting classical fermionic fields in mathematical physics. We then construct the parallel transport map corresponding to such connections and compare the results with those found by other means in the mathematical literature. Finally, applications of these methods to supergravity will be discussed, such as the Cartan geometric formulation of Poincaré supergravity as well as the description of Killing vector fields and Killing spinors of super Riemannian manifolds arising from metric reductive super Cartan geometries.

中文翻译：

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超纤束、连接形式和并行传输

目前的工作对超纤维丛理论、连接形式及其平行传输提供了数学上严格的说明，将各种方法联系在一起。我们首先详细介绍超纤维丛。然后我们介绍所谓的相对超流形的概念以及在这些类别中定义的束和连接。研究这些物体对于在数学物理中对反对易经典费米子场进行建模是极其重要的。然后我们构建对应于这种连接的并行传输图，并将结果与​​通过数学文献中的其他方法找到的结果进行比较。最后，将讨论这些方法在超重力中的应用，