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The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action is $KO$ and the corresponding $\mathbb{Z}_2$-equivariant cohomology theory is KR-theory ("real K-theory").

A pleasant conceptual account of this state of affairs that puts it into the bigger context of chromatic homotopy theory was given in the appendix of

  • Tyler Lawson, Niko Naumann, Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2 (arXiv:1203.1696)

which is usefully amplified and further expanded on in section 3 of

and section 2 of

  • Akhil Mathew, The homotopy groups of $TMF$, talk notes (pdf)

Namely, $KU$ equipped with its involution appears, via Goerss-Hopkins-Miller, as the $E_\infty$-structure sheaf on the "moduli stack of 1-dimension tori", which is just $\mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2$.

The main theorem (1.2) of Lawson-Naumann above is that the inclusion $KO \to KU$ obtained this way as the image under forming global sections of the canonical double cover of $\mathcal{M}_{\mathbb{G}_m}$ is (at prime 2) the restriction of a similar inclusion of topological modular forms $tmf \to tmf_1(3)$.

Discussion of such finite covers of the moduli stack of elliptic curves goes back to a result by Mahowald-Rezk and a similar realization of $tmf$ as a homotopy fixed-point spectrum induced by a cover of elliptic curves with "level N structure" is in

  • Vesna Stojanoska, Duality for Topological Modular Forms (arXiv:1105.3968)

this time the group acting is $GL_2(\mathbb{Z}_2)$ (for $N= 2$).

Now I suppose if I dig through all this a bit more it will all become clear, but right now the following question seems as obvious as its explicit statement seems to be missing from all of the above:

Question. What exactly is to the real-oriented cohomology theory $KR$ as $KO$ is to $tmf$?

Actually I have a more concrete question that I am trying to understand, but that requires the following 2-sentence introduction:

the homotopy fiber of $B String \to B O$ should be equivalent to $Pic(KU)_{\leq 3}$. By the argument in section 8 of

  • Matthew Ando, Andrew J. Blumberg, David Gepner, Twists of K-theory and TMF (arXiv:1002.3004)

this should induce a homomorphism of connective spectra of the form

$$ T: Pic(KU)_{\leq 3} \longrightarrow GL_1(tmf) \,. $$

On the left we have a quotient map $Pic(KU)_{\leq 3} \to (Pic(KU)_{\leq 3})//\mathbb{Z}_2$ induced by the above involution, hence essentially passing to KR-theory.

Question. In which sense would $T$ naturally descend along this quotient? Specifically, is there a ring spectrum $Q$ with $\mathbb{Z}_2$-action whose homotopy fixed points is $tmf$ and such that $T$ naturally descends to $$ (Pic(KU)_{\leq 3})//\mathbb{Z}_2 \longrightarrow GL_1(Q)//\mathbb{Z}_2 $$

?

Or else, if all this is misled: what IS the natural way to bring in the $\mathbb{Z}_2$-action on the right, compatible with KR-theory?

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3  
What about the remark in the end of the introduction of Mahowald-Rezk? Hu and Kriz (math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf‎) have constructed real oriented theories based on $E(n)$, which are usually denoted $ER(n)$. Now $ER(1) = KO_{(2)}$ and $ER(2)$ is very much like $TMF_0(3)$. They then make the comment "Presumably the construction of [HK01] can be carried out to construct a “real” version of $TMF_1(3)$" –  Drew Heard Apr 7 at 22:43
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We (Gerd Laures and I) have now used this construction in arxiv:1403.7301. There were choices involved, however. For us the equivariant homotopy type was not so important. So my question would be: Does Hill-Lawson's paper arxiv:1312.7394 together with Elmendorf's theorem fix the equivariant homotopy type? –  nsrt Apr 8 at 8:34
    
Drew and Martin, thanks, so indeed TMF_0(3) is the Z/2Z homotopy fixed points of TMF_1(3); but what about TMF itself? And the compatibility with the action on KU? (Maybe I am missing somehting obvious here...) I was guessing that for TMF itself the construction by Lawson-Naumann is going in the right direction. It bothered me that it only worked 2-locally, but maybe that shouldn't bother me too much. Hm... –  Urs Schreiber Apr 8 at 15:45
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Urs, so far as I know there is not a natural double cover of tmf exactly like you request. One reason is that most of the natural etale covers of the moduli stack come from subgroups of GL_2(Z/N). The natural negation on elliptic curves corresponds to the element -1, which is a square and hence inside any index 2 subgroup. –  Tyler Lawson Apr 10 at 1:51
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Another possible answer might be that, in certain circumstances, we might end up with $Tmf$ (with some primes inverted) as the $G$-fixed points of some extension, and $G$ might act on some finite product of things related to $K$-theory rather than a single term. These kinds of situations do exist (corresponding to "evaluation at the cusps"). However, this doesn't even come close to answering your question about the relationship between Pic and GL_1. –  Tyler Lawson Apr 10 at 4:02
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