  A Riemannian metric on a manifold M induces a family of Riemannian metrics on
the loop space LM depending on a Sobolev space parameter s. The connection and
curvature forms of these metrics take values in pseudodifferential operators.
We develop a theory of Wodzicki-Chern-Simons classes using the s=0, 1
connections and the Wodzicki residue. These classes distinguish the smooth
homotopy type of some circle actions on M = S^2 x S^3, and imply that the
fundamental group of Diff(M) is infinite.
