arXiv math/04111612004-11-08math.DG

Infinite Dimensional Chern-Simons Theory

Steven Rosenberg, Fabian Torres-Ardila

英文摘要：

We extend finite dimensional Chern-Simons theory to certain infinite dimensional principal bundles with connections, in particular to the frame bundle $FLM\to LM$ over the loop space of a Riemannian manifold $M$. Chern-Simons forms are defined roughly as in finite dimensions with the invariant polynomials replaced by appropriate Wodzicki residues. This produces odd dimensional $\R/\Z$-valued cohomology classes on $LM$ if $M$ is parallelizable. We compute an example of a metric on the loop space of $S^3\times S^1$ for which the three dimensional Chern-Simons class is nontrivial.

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