Topological "observables" in semiclassical field theories

We give a geometrical set-up for the semiclassical approximation to euclidean field theories having families of minima (instantons) parametrized by suitable moduli spaces M. The standard examples are of course Yang-Mills theory and non-linear Ï -models. The relevant space here is a family of measure spaces N Ì â M, with standard fibre a distribution space, given by a suitable extension of the normal bundle to M in the space of the smooth fields. Over N Ì there is a probability measure dÎ1⁄4 given by the twisted product of the (normalized) volume element on M and the family of gaussian measures with covariance given by the tree propagator CÏ in the background of an instanton Ï ÎμlunateM. The space of "observables", i.e., measurable functions on ( N Ì , dÎ1⁄4), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on M. The expectation value of these topological "observables" does not depend on the covariance; it is therefore exact at all orders in perturbation theory and can moreover be computed in the topological regime by setting the covariance to zero. © 1992.