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Here is a question, probably for Charles (Rezk), if he sees it.
The string orientation of tmf MString→tmf refines the un-twisted Witten genus on manifolds with String-structure. More generally, the Witten genus on a Spin-manifold is twisted by a complex vector bundle (“heterotic string”) and is a modular form for “String^c-structure”, characterized by 12p1−c2=0.
In the article
this twisted Witten genus on Stringc-structures is re-considered, and on p.2 an obvious question is mentioned: does the twisted Witten genus also have a “topological” lift to a map of spectra?
Now, there is indeed an obious map of spectra MStringc⟶tmf, namely the hocolim over the left half of the diagram
BStringc↙↘p*⇙σc↓p*(12p1−c2)BSpin×BSU↘↙12p1−c2B3U(1)↓ρBGL1(tmf)↓tmfMod,where σc denotes the homotopy that exhibits BStringc as the homotopy fiber of 12p1−c2, and where ρ is the twisting map exhibiting the plain string orientation of tmf as in Ando-Blumberg-Gepner 10.
My exercise is to check if that map
lim→σc:MStringc→tmfinduces on homotopy groups the twisted Witten genus, correctly.
While I am slowly chewing on this, I thought I’d ask if anyone has considered this before. Quite likely this is clear to experts such as Charles.
Or rather, possibly the push should be rather along the left half of
BStringc≃BString//SU↙↘pSpin(BSU≃*//SU)⇙σc↓p*Spin(12p1)BSpinc2↘↙12p1B3U(1)↓ρBGL1(tmf)↓tmfModand hence land in SU-equivariant elliptic cohomology.
( Here I am using the pasting of homotopy pullbacks
BString→BStringc→BSpin↓↓↓*→BSU→B3U(1)in order to identify BStringc≃BString//SU. )
I suppose that the twisted Witten genus should land in equivariant tmf this way is something that Matthew Ando has been suggesting, though I am not sure if I have seen the place where this is stated explicitly.
I am not an expert on this question. You should try Matt.
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