Inside the compactified moduli stack of elliptic curves, at the cusp point corresponding to the nodal cubic curve sits the moduli stack of one dimensional tori (Lawson-Naumann 12, def. A.1, A.3). This is equivalent to the quotient stack of the point by the group of order 2
(Lawson-Naumann 12, prop. A.4). Here the -action is the inversion involution on abelian groups.
Using the Goerss-Hopkins-Miller theorem this stack carries an E-∞ ring-valued structure sheaf (Lawson-Naumann 12, theorem A.5); and by the above equivalence this is a single E-∞ ring equipped with a -∞-action. This is KU with its involution induced by complex conjugation, hence essentially is .
Accordingly, the global sections of over are the -homotopy fixed points of this action, hence is KO. This is further amplified in (Mathew 13, section 3) and (Mathew, section 2).
As suggested there and by the main (Lawson-Naumann 12, theorem 1.2) this realizes (at least localized at ) the inclusion as the restriction of an analogous inclusion of tmf built as the global sections of the similarly derived moduli stack of elliptic curves.
Discussion of with its -action as the E-∞ ring-valued structure sheaf of the moduli stack of tori is due to
which is reviewed and amplified further in
Akhil Mathew, section 3 of The homology of (arXiv:1305.6100)
Akhil Mathew, section 2 of The homotopy groups of , talk notes (pdf)
Last revised on November 13, 2020 at 18:28:02. See the history of this page for a list of all contributions to it.