Thursday, May 27, 2010

Wodzicki-Chern Classes

So I took a look today at this new paper on the arXiv, by Steve Rosenberg and collaborators, that considers bundles (typically over infinite-dimensional manifolds) whose structural group is a subgroup of the invertible pseudodifferential operators of order $\le 0$ (on some other manifold).

A natural context for such a situation to arise is where your infinite-dimensional manifold is some (Sobolev-type completion $\mathcal M$ of) Maps(N,M), where N and M are ordinary finite dimensional closed manifolds - loop spaces being the canonical example of this kind of thing. Choosing metrics on N and M allows one to give the tangent bundle $T{\mathcal M}$, etc, the structure described above.

Now suppose we have a bundle with structure group $G\subseteq \Psi_{\le 0}$. We can try to use Chern-Weil theory to generate characteristic classes. In order to do this we need a trace on $\Psi_{\le 0}$ itself (just as standard Chern-Weil theory uses the trace on $M_n$ which is the Lie algebra of $GL(n)$).

The most natural example of such a trace is the famous Wodzicki residue which is in fact a trace on the algebra of all pseudodifferential operators. Inputting this trace into Chern-Weil theory yields characteristic classes called the Wodzicki-Chern classes.

The authors conjecture, and prove in some special cases, that these Wodzicki-Chern classes always vanish. (Of course this would mean that it is open season on secondary invariants related to these classes. The authors reference an earlier paper, "Riemannian geometry of loop spaces", where they discuss some nontrivial examples of Wodzicki-Chern-Simons classes.

In view of the importance of the Dixmier trace/Wodzicki residue in Connes' noncommutative geometry scheme, it would be interersting to know how this paper fits with the Connes picture.

1 comment:

Paul said...

Is there a good theory of elliptic operators on these sorts of infinite dimensional manifolds? If so, can one envision a super-symmetric analogue of the "Wodzicki Residue" and perhaps an infinite dimensional McKean-Singer formula? Perhaps this requires an unreasonable amount of cooperation on the part of the infinite dimensional manifold in question, but maybe it's worth a shot!