I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How exactly is this map constructed? Could it be that there is an extension to higher twists, i.e. is there an extension to $BBU_{\otimes} \to TMF$?

EDIT: I know that $BBU_{\otimes}$ splits off $K(\mathbb{Z},3)$ as a factor. Therefore there is of course a map $BBU_{\otimes} \to TMF$, which factors over $K(\mathbb{Z},3)$. The corresponding classes of TMF, however, only see the ordinary third-cohomology twists. So, I reshould restate the second question as something like: Is there an extension $BBU_{\otimes} \to TMF$, which "sees" higher twists?

`$BBU_\otimes$`

. All I can say is that one of the most natural places that possess those kinds of orientations is $K(ku)$, the algebraic K-theory of the complex K-theory spectrum $ku$. Ausoni-Rognes showed that $K(ku)$ possesses chromatic features similar to elliptic cohomology, and Baas-Dundas-Richter-Rognes showed that $K(ku)$ can be interpreted in terms of 2-vector bundles. So far as I know there isn't a direct comparison to elliptic cohomology theories. – Tyler Lawson Nov 14 '11 at 17:59