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 a mass or a boundary. We derive a few consequences, including some continuity of these determinants as a function of the operator, and the conformal invariance of the determinant on surfaces. This expression allows us to construct the scalar field minimally coupled 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. Lévy in the discrete.