Chern-Weil Constructions on ΨDO Bundles

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We construct Chern-Weil classes on infinite dimensional vector bundles with structure group contained in the algebra [≤ 0](M, E) of non-positive order classical pseudo-differential operators acting on a finite rank vector bundle E over a closed manifold M. Mimicking the finite dimensional Chern-Weil construction, we replace the ordinary trace on matrices by linear functionals on [≤ 0] (M, E) built from the leading symbols of the operators. The corresponding Chern classes vanish for loop groups, but a weighted trace construction yields a non-zero class perviously constructed by Freed. For loop spaces, the structure group reduces to a gauge group of bundle automorphisms, and we produce non-vanishing universal Chern classes in all degrees, using a universal connection theorem for these bundles.