Two-dimensional parallel transport : combinatorics and functoriality
Abstract
Description
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.
18 pages ; v2 : interpretation of abelianisation changed and typos corrected
18 pages ; v2 : interpretation of abelianisation changed and typos corrected