The geometry of loop spaces II: Characteristic classes

• Characteristic classes on loop spaces are introduced to study diffeomorphism groups of certain 5-manifolds.
• The characteristic classes use the Wodzicki residue of pseudodifferential operators.
• The primary characteristic classes vanish, but the secondary Chern–Simons classes can be nontrivial.

We develop a theory of Chern–Simons classes CS2k−1W∈H2k−1(LM2k−1;R) on the loop space LM of a Riemannian manifold M. These classes are associated to a pair of connections on LM whose connection and curvature forms take values in pseudodifferential operators by [19]. We use the Wodzicki residue of these operators to define and compute the Chern–Simons classes. As an application, we prove that |π1(Diff(M‾))|=∞ for the total space M‾ of circle bundles associated to high multiples of a Kähler class over integral Kähler surfaces.