Holonomy and Parallel Transport in the Differential Geometry of the Space of Loops and the Groupoid of Generalized Gauge Transformations

The motivation for this paper stems [4] from the need to construct explicit isomorphisms of (possibly nontrivial) principal G-bundles on the space of loops or, more generally, of paths in some manifold M , over which I consider a fixed principal bundle P ; the aforementioned bundles are then pull-backs of P w.r.t. evaluation maps at different points. The explicit construction of these isomorphisms between pulled-back bundles relies on the notion of parallel transport. I introduce and discuss extensively at this point the notion of generalized gauge transformation between (a priori) distinct principal G-bundles over the same base M ; one can see immediately that the parallel transport can be viewed as a generalized gauge transformation for two special kind of bundles on the space of loops or paths; at this point, it is possible to generalize the previous arguments for more general pulled-back bundles. Finally, I discuss how flatness of the reference connection, w.r.t. which I consider holonomy and parallel transport, is related to horizontality of the associated generalized gauge transformation.