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Here is a question, probably for Charles (Rezk), if he sees it.
The string orientation of tmf $M String \to tmf$ 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 $\frac{1}{2}p_1 - c_2 = 0$.
In the article
this twisted Witten genus on $String^c$-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 $M String^c \longrightarrow tmf$, namely the hocolim over the left half of the diagram
$\array{ && B String^c \\ & \swarrow && \searrow^{\mathrlap{p}} \\ \ast & \mathrlap{\swArrow_{\sigma^c}} & \downarrow^{\mathrlap{p^\ast(\frac{1}{2}p_1 - c2)}} & \mathrlap{B Spin \times B SU} \\ & \searrow && \swarrow_{\frac{1}{2}p_1 - c_2} \\ && B^3 U(1) \\ && \downarrow^{\mathrlap{\rho}} \\ && B GL_1(tmf) \\ && \downarrow \\ && tmf Mod } \,,$where $\sigma^c$ denotes the homotopy that exhibits $B String^c$ as the homotopy fiber of $\frac{1}{2}p_1 - c_2$, and where $\rho$ 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
$\underset{\to}{\lim} \sigma^c : M String^c \to tmf$induces 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
$\array{ && B String^c \simeq B String // SU \\ & \swarrow && \searrow^{\mathrlap{p_{Spin}}} \\ (B SU \simeq \ast//SU )& \mathrlap{\swArrow_{\sigma^c}} & \downarrow^{\mathrlap{p_{Spin}^\ast(\frac{1}{2}p_1)}} & \mathrlap{B Spin } \\ & {}_{\mathllap{c_2}}\searrow && \swarrow_{\frac{1}{2}p_1} \\ && B^3 U(1) \\ && \downarrow^{\mathrlap{\rho}} \\ && B GL_1(tmf) \\ && \downarrow \\ && tmf Mod }$and hence land in $SU$-equivariant elliptic cohomology.
( Here I am using the pasting of homotopy pullbacks
$\array{ B String &\to& B String^c &\to& B Spin \\ \downarrow && \downarrow && \downarrow \\ \ast &\to& B SU &\to& B^3 U(1) }$in order to identify $B String^c \simeq B String // SU$. )
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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