Two-dimensional parallel transport : combinatorics and functoriality

Details Two-dimensional parallel transport : combinatorics and functoriality Two-dimensional parallel transport : combinatorics and functoriality

2001-05-31

arxiv.org/abs/math-ph/0105050v2

Physics

Mathematical Physics

18 pages ; v2 : interpretation of abelianisation changed and typos corrected

Scientific paper

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.

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