Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps

Abstract

We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth and satisfy a "fusion" property with respect to triples of paths. Our bijections are established by explicit group isomorphisms: transgression and regression. Restricted to smooth, finite-dimensional manifolds, our results extend previous work of J. W. Barrett.

45 pages, 1 figure. v2: parts of the appendix have been moved to the main text, many small improvements. v3: references updates; v3 is the published version

Cited by in corpus (13)

• A Construction of String 2-Group Models using a Transgression-Regression Technique
• Spin structures on loop spaces that characterize string manifolds
• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• Adjusted Parallel Transport for Higher Gauge Theories
• String geometry vs. spin geometry on loop spaces
• Transgressive loop group extensions
• String Connections and Chern-Simons Theory
• Higher Gauge Theory
• Equivalence of string and fusion loop-spin structures
• Simplicial cochain algebras for diffeological spaces
• Gerbes in Geometry, Field Theory, and Quantisation
• Constructing the Virasoro groups using differential cohomology
• The Clifford Algebra Bundle on Loop Space