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.
- Pub Date:
- November 2009
- Mathematics - Differential Geometry
- 45 pages, 1 figure. v2: parts of the appendix have been moved to the main text, many small improvements. v3: references updates