Infinite Dimensional Chern-Simons Theory

2004/11/08 by Steven Rosenberg, Rosenberg, Steven, Fabian Torres-Ardila +1
Mathematics · #58B20 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:58B20

paper · pdf · doi:10.48550/arxiv.math/0411161

arxiv created 2004/11/08 · arxiv updated 2009/12/01

Abstract

We extend finite dimensional Chern-Simons theory to certain infinite dimensional principal bundles with connections, in particular to the frame bundle FLM→ 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 S³× S¹ for which the three dimensional Chern-Simons class is nontrivial.

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