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

- Akhil Mathew,
*The homology of $tmf$*(arXiv:1305.6100)

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?