2003/01/17 by Sylvie Paycha, Paycha, Sylvie, Steven Rosenberg +1
Mathematics · Medicine · Physics and Astronomy · #Advanced Operator Algebra Research #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #math-ph #math.AP #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.math/0301185
arxiv created 2003/02/21 · arxiv updated 2009/11/30
We construct Chern-Weil classes on infinite dimensional vector bundles with structure group contained in the algebra \cl[≤ 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[≤ 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.