Holonomy and parallel transport in the differential geometry of loop spaces and generalized gauge transformations

摘要

The motivation of this paper stems from the need to construct explicit isomorphismsnof u0001possibly nontrivialu0002 principal G-bundles on the space of loops or, morengenerally, of paths in some manifold M, over which we consider a fixed principalnbundle P; the aforementioned bundles are then pull backs of P with respect tonevaluation maps at different points. The explicit construction of these isomorphismsnbetween pulled-back bundles relies on the notion of parallel transport.We introducenand discuss extensively at this point the notion of generalized gauge transformationnbetween u0001a prioriu0002 different principal G-bundles over the same base M, and we seenimmediately that the parallel transport is a generalized gauge transformation forntwo special kinds of bundles on the space of loops or paths; at this point, we mayntry and generalize the previous arguments for more general pulled-back bundles.nFinally, we discuss how flatness of the reference connection, with respect to whichnwe consider parallel transport, is related to horizontality of the associated generalizedngauge transformation.