21829 articles – 15616 references  [version française]
 HAL: hal-00129467, version 1
 arXiv: math.DG/0301185
 Chern-Weil Constructions on $\Psi$DO Bundles
 (2003)
 We construct Chern-Weil classes on infinite dimensional vector bundles with structure group contained in the algebra $\cl[\leq 0](M, E)$ 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 $\cl[\leq 0] (M, E)$ 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.
 1: Laboratoire de Mathématiques CNRS : UMR6620 – Université Blaise Pascal - Clermont-Ferrand II
 Subject : Mathematics/Differential GeometryMathematics/Analysis of PDEsMathematics/Mathematical Physics