2021/01/31 by Konstantin Eder
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Bundle #Connection (principal bundle) #Fiber bundle #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric (unit) #Parallel transport #Spinor #Supergravity #Supermanifold #Vector bundle #Vector field #gr-qc #hep-th #math-ph #math.DG #math.MP
paper · pdf · doi:10.1063/5.0044343
published as Journal of Mathematical Physics 62, 063506 (2021) · v3: published version, 63 pages
openalex created_date 2021/01/18 · openalex publication_date 2021/06/01 · arxiv created 2021/06/04 · arxiv updated 2021/06/07 · openalex updated_date 2026/08/05
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.