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Infinite Dimensional Chern-Simons Theory

Steven Rosenberg·Fabian Torres-Ardila
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Abstract

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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Steven Rosenberg, Fabian Torres-Ardila. 2004-11-08. Infinite Dimensional Chern-Simons Theory. https://arxiv.org/abs/math/0411161

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