Apparently a gauge field is really a phase-shift field. In Yang-Mills theories there's a phase-shift 'operator' defined on particles from the Standard Model, that for instance, transform protons into neutrons and back. But apart from that, we have fiber bundles, where a fiber can be a symmetry group, like SU(2). In an article by Bernstein and Phillips, the idea of a bundle of directions is used, such that each fibre is the interval [0,2π), but this only (obviously) makes sense if there is a reference direction. So then they present the example of a sphere (manifold), which can't have a reference direction assigned everywhere, but a hemisphere can, in fact the equator has a natural east/west direction on it. Now, this fiber over each point on a hemisphere is what the authors call a "circle of directions", but it lies on a tangent plane so what they really mean is the tangent space of a hemisphere, where at each point one and only one plane is tangent. Then they explain parallel transport around this space in terms of moving through the bundle of fibers over the half sphere, which is called 'lifting a path' from the base space to the total space (the tangent space encoding 'directions' on the manifold). Then lifting a path and parallel transport are equivalent, but first the fiber bundle needs more structure, it needs an associated bundle of gradients over each point in the manifold. So they say (I think).