TITLE: String Orientation and Level Structures
QUESTION [16 upvotes]: Atiyah, Bott and Shapiro defined orientations of real and complex K-theory that were later refined to maps of ($E_\infty$-ring) spectra 
$$MSpin \to KO$$
and
$$MSpin^c \to KU.$$
Likewise, but more complicated, the Witten genus was refined in a paper by Ando, Hopkins and Rezk to the string orientation
$$MString \to tmf$$
into connective $tmf$. This has also a "complex" version: for every (even periodic) elliptic spectrum, there is an orientation
$$MU\langle 6\rangle \to E,$$
as already shown earlier in Elliptic Spectra, the Witten Genus and the Theorem of the Cube.
This provides "string orientations" for many variants of $TMF$ with level structure; for example, $TMF(n)$ is an elliptic spectrum for $n\geq 3$. But it does not constitute a fully satisfactory theory of string orientations for topological modular forms with level structures for the following two reasons:

It is not obvious that the map $MU\langle 6\rangle \to TMF(n)$ factor over its connective version $tmf(n)$. The latter is understood to be the connective cover of the "compactified" $Tmf(n)$ defined first by Goerss and Hopkins and then in greater generality by Hill and Lawson. 
Many level structures do not define elliptic spectra. For example, the spectra $TMF_0(n)$ are not complex oriented in general. 

So my question is: Is there a notion of "string bordism with level structure" and corresponding string orientations, producing for example a map into $tmf_0(3)$?
Note: The composition $MString \to tmf \to tmf_0(3)$ is obviously not what I want.

REPLY [10 votes]: Dylan's nice paper at https://arxiv.org/abs/1507.05116 answers this question.