Chern-Simons classes on loop spaces and diffeomorphism groups
by Steve Rosenberg
Abstract: The loop space LM of a Riemannian manifold M is itself an interesting infinite dimensional manifold. LM has a family of Riemannian metrics indexed by a Sobolev parameter. We can construct characteristic classes for LM by using the Wodzicki residue instead of the usual matrix trace. The Pontrjagin classes of LM vanish, but the secondary or Chern-Simons classes may be nonzero. A similar approach applies to diffeomorphism groups of manifolds.