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?
Best Answer
Let $G$ be a finite group. Then one has a symmetric monoidal, stable $\infty$-category of (genuine) $G$-spectra. Inside here is a subcategory obtained by localization (it behaves more like a completion) at the commutative algebra object $\mathbb{D}(G_+)$ where $\mathbb{D}$ denotes Spanier-Whitehead duality. This subcategory is equivalent to the $\infty$-category of spectra equipped with a $G$-action, i.e., $\mathrm{Fun}(BG, \mathrm{Sp})$. Let's call such $G$-spectra Borel-equivariant.
The $\mathbb{Z}/2$-spectrum $KR$ representing Atiyah's $KR$-theory for $\mathbb{Z}/2$-spaces is Borel-equivariant, and it comes from the $\mathbb{Z}/2$-action on complex $K$-theory $KU$ given by complex conjugation. As a result, $KR$ can be constructed as an equivariant spectrum one one knows the conjugation action on $KU$.
Given any faithful $G$-Galois extension of $E_\infty$-ring spectra $R \to R'$ in the sense of Rognes, one can thus construct a genuine $G$-spectrum, which is the Borel-equivariant spectrum that comes from the $G$-action on $R'$. Like $KR$, these genuine $G$-spectra will have trivial Tate constructions. For $TMF$ with a prime $p$ inverted, there are a number of Galois extensions that come from taking covers of the moduli stack of elliptic curves by imposing some sort of level structure. Each of these Galois extensions thus gives a genuine $G$-spectrum that one can think of as analogous to the real $K$-theory $KR$.
(Essentially, the content of this answer is that while there analogs of $KR$ for $TMF$, there is no new information that they contain that you have not cited, since $KR$ is determined entirely in terms of the $\mathbb{Z}/2$-action on $KU$.)