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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 ${\cal M}$. The standard examples are of course Yang-Mills theory and non-linear $σ$-models. The relevant space here is a family of measure spaces $\tilde {\cal N} \ra {\cal M}$, with standard fibre a distribution space, given by a suitable extension of the normal bundle to ${\cal M}$ in the space of smooth fields. Over $\tilde {\cal N}$ there is a probability measure $dμ$ given by the twisted product of the (normalized) volume element on ${\cal M}$ and the family of gaussian measures with covariance given by the tree propagator $C_ϕ$ in the background of an instanton $ϕ\in {\cal M}$. The space of ``observables", i.e. measurable functions on ($\tilde {\cal N}, \, dμ$), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on ${\cal M}$. 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.