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Traces and characteristic classes on loop spaces. (English) Zbl 1064.58023
Wurzbacher, Tilmann (ed.), Infinite dimensional groups and manifolds. Based on the 70th meeting of theoretical physicists and mathematicians at IRMA, Strasbourg, France, May 2004. Berlin: de Gruyter (ISBN 3-11-018186-X/pbk). IRMA Lectures in Mathematics and Theoretical Physics 5, 185-212 (2004).
Let \({\mathcal{C}}l^*_0(M,E)\) be the group of zeroth order invertible classical pseudo-differential operators acting on a finite rank vector bundle \(E\) over a closed manifold \(M\). \({\mathcal{C}}l^*_0(M,E)\) is the structure group of geometric bundles naturally associated to loop spaces of Riemannian manifolds.
In this paper the authors construct Chern-Weil classes on infinite dimensional vector bundles with structure group \({\mathcal{C}}l^*_0(M,E)\). They introduce two types of traces in infinite dimensions, each of which can be interpreted as generalizations of the ordinary trace on matrices. They use (i) traces build from the leading symbol, and (ii) a linear map which considers all terms in the asymptotic expansion of a heat kernel regularized trace. For a specific bundle on loop spaces, the first approach yields non-vanishing Chern classes in all degrees. The second approach produces connection independent cohomology classes under stringent conditions. For the tangent bundle to a loop group, the first method gives a vanishing first Chern class, while the second method recovers the first Chern class investigated by Freed, and explains why this class is not connection independent.
For the entire collection see [Zbl 1050.22003].

58J40 Pseudodifferential and Fourier integral operators on manifolds
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
57R20 Characteristic classes and numbers in differential topology
53C05 Connections, general theory