Super fiber bundles, connection forms, and parallel transport

TL;DR

This paper rigorously develops super fiber bundle theory, including connections and parallel transport, with applications to modeling fermionic fields and supergravity in mathematical physics.

Contribution

It introduces the concept of relative supermanifolds and constructs parallel transport maps, linking super fiber bundle theory with supergravity applications.

Findings

Constructed parallel transport maps for super connections

Compared super bundle results with existing mathematical literature

Applied methods to supergravity, including super Cartan geometries

Abstract

The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms and their parallel transport, that 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 Poincar\'e supergravity as well as the description of Killing vector fields and Killing spinors of super Riemannian manifolds arising from metric reductive super Cartan geometries.