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?

  • 4
    $\begingroup$ 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 $TMF_1(3)$" $\endgroup$
    – Drew Heard
    Commented Apr 7, 2014 at 22:43
  • 1
    $\begingroup$ 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? $\endgroup$
    – nsrt
    Commented Apr 8, 2014 at 8:34
  • 1
    $\begingroup$ 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. $\endgroup$ Commented Apr 10, 2014 at 1:51
  • 1
    $\begingroup$ 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. $\endgroup$ Commented Apr 10, 2014 at 4:02
  • 1
    $\begingroup$ ... and that theory is famously controled by a $\mathrm{SL}_2(\mathbb{Z})$-bundle (being the monodromy bundle of an elliptic fibration). An old argument due to Ashoke Sen (ncatlab.org/nlab/show/F-theory#RelationToOrientifolds) says that the inversion involution inside $SL_2(\mathbb{Z})$ induces the orientifolding, hence that this $SL_2(\mathbb{Z})$-local system is indeed the F-theoretic analog of the $\mathbb{Z}_2$-twist on $\mathrm{KU}$. But now by Mahowald-Rezk and then your work, we have that $SL_2(\mathbb{Z})$ or at least the quotient $SL_2(\hat {\mathbb{Z}})$ is the twist of... $\endgroup$ Commented Apr 11, 2014 at 13:42

1 Answer 1


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 $KU$.)


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