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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:

  1. 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.
  2. 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.

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    $\begingroup$ tmf_0(2) should admit an $MSO$ orientation (the Ochanine genus). I'm trying to write this down, actually, as an exercise in understanding the Ando-Hopkins-Rezk stuff. $\endgroup$ – Dylan Wilson Dec 9 '14 at 22:03
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    $\begingroup$ The paper "The Jacobi orientation and the two-variable elliptic genus" (arxiv.org/abs/math/0605554; by Ando, French and Ganter) is potentially relevant. $\endgroup$ – Neil Strickland Dec 10 '14 at 19:04
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    $\begingroup$ I believe the paper "HP$^2$-bundles and elliptic homology" by Kreck and Stolz (Acta 171 (1993) pp. 231-261) should be mentioned too as highly relevant $\endgroup$ – მამუკა ჯიბლაძე Feb 12 '18 at 10:13
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Dylan's nice paper at https://arxiv.org/abs/1507.05116 answers this question.

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For $tmf_0(3)$, there should exist an orientation from the bordism spectrum of manifolds with $BString(2)$-structure, where $BString(2)$ is the homotopy fiber of $p_1:BSpin\to K(Z,4)$. The composition with $MString\to MString(2)$ would however be the map you do not want.

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    $\begingroup$ What do you mean by "should"? $\endgroup$ – Lennart Meier Dec 9 '14 at 22:53

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