TITLE: Twists of K-theory and tmf
QUESTION [12 upvotes]: 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?

REPLY [5 votes]: You can construct the map $K(\mathbb{Z},3) \to tmf$ as follows: first there is the String orientation of tmf, which you already mention. This is a map 
$$ MString \to tmf$$
Then String is by definition a $K(\mathbb{Z},2)$-fibration over Spin. This yields in particular a map 
$$ K(\mathbb{Z},3) \to MString $$
Then you can construct the map $K(\mathbb{Z},3) \to tmf$ as the composition of the above two maps. In order to extend this constuction you had to find a map $BBU_\otimes \to MString$. I think such a map does not exist apart from the one you describe, but I am not entirely sure.