  Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman
establish equalities between the correlation functions or distributions of
occupation times of random paths or ensembles of paths and Markovian fields,
such as the discrete Gaussian free field. We extend these results to the case
of real, complex, or quaternionic vector bundles of arbitrary rank over graphs
endowed with a connection, by providing distributional identities between
functionals of the Gaussian free vector field and holonomies of random paths.
As an application, we give a formula for computing moments of a large class of
random, in general non-Gaussian, fields in terms of holonomies of random paths
with respect to an annealed random gauge field, in the spirit of Symanzik's
foundational work on the subject.
