2001/05/31 by Romain Attal, Attal, Romain
Computer Science · Mathematics · Physics and Astronomy · #18F20 #53C05 #53C29 #55U10 #Black Holes and Theoretical Physics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Topological and Geometric Data Analysis #math-ph #math.MP #msc:18F20 #msc:53C05 #msc:53C29 #msc:55U10
paper · pdf · doi:10.48550/arxiv.math-ph/0105050
18 pages ; v2 : interpretation of abelianisation changed and typos corrected
openalex publication_date 2001/05/31 · arxiv created 2001/10/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend the usual notion of parallel transport along a path to triangulated surfaces. A homotopy of paths is lifted into a fibered category with connection and this defines a functor between the fibers above the boundary paths. These "sweeping functors" transport fiber bundles with connection along a surface whereas usual connections transport a group element along a path. We show that to get rid of the parametrization, we must use Abelian degrees of freedom. In the general, non-Abelian case, we conjecture that the smooth limit of this construction provides us with representations of the group of diffeomorphisms of the swept surface. Applications to gauge theories are proposed.