  We derive a stochastic representation for determinants of Laplace-type
operators on vectors bundles over manifolds. Namely, inverse powers of those
determinants are written as the expectation of a product of holonomies defined
over Brownian loop soups. Our results hold over compact manifolds of dimension
2 or 3, in the presence of mass or a boundary. We derive a few consequences,
including some regularity as a function of the operator and the conformal
invariance of the zeta function on surfaces.
  Our second main result is the rigorous construction of a stochastic gauge
theory minimally coupling a scalar field to a prescribed random smooth gauge
field, which we prove obeys the so-called Symanzik identities. Some of these
results are continuous analogues of the work of A. Kassel and T. L\'evy in the
discrete.
