To see what this ought to be, restrict attention to $X = Y \times (*/\mathbb{Z}_2)$ and the ordinary “non-elliptic” case where $\tilde \tau$ is constant on the inclusion of the nodal curve. The fiber of $\mathcal{O}^{top}_{ell}$ over $\tilde \tau$ then is just $ko \simeq ku^{\mathbb{Z}_2}$ And so in this case the charges of the M2 are found to sit in $ko$

$\array{ X \times E && \stackrel{L_{M2}}{\longrightarrow} && ko \\ & {}_{\mathllap{\tilde \tau}}\searrow && \swarrow \\ && \{nodal\} }$Accordingly, after double dimensional reduction/transgression along the circle, the string will take charges in $\Sigma^{-1} ko$.

And this is indeed the right answer for type II backgrounds, that’s the point around p. 13 in Freed 12.

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All this looks good. So we need to see how it all integrates into one integrated bouquet. At this point we have this:

$\array{ \vdots \\ \downarrow \\ X \times_{\tilde \tau} E &\stackrel{\mathbf{L}_{WZW}^{M2} + i \alpha}{\longrightarrow}& \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn} \\ \downarrow \\ X &\stackrel{\tilde \tau}{\longrightarrow}& \overline{\mathcal{M}_{ell}} }$This D0-M2 stage is directly analogous to the M2-M5 stage which we analyzed in The WZW term of the M5-brane (schreiber). If we are to proceed as we did there, then we have to take the above diagram and rewrite it as

$\array{ \vdots \\ \downarrow \\ X \times_{\tilde \tau} E && \stackrel{\mathbf{L}_{WZW}^{M2} + i \alpha}{\longrightarrow}&& \mathbf{B}^3 (\mathbb{C}/\mathbb{Z})_{conn} \\ \downarrow && && \downarrow \\ X && \longrightarrow && Q \\ & {}_{\mathllap{\tilde \tau}}\searrow && \swarrow \\ && \overline{\mathcal{M}_{ell}}(\mathbb{C}) }$with $Q$ something like a $K(\mathbb{Z},4)$-fibration over $\overline{\mathcal{M}_{ell}}$. And then we should check if $\mathbf{L}_{WZW}^{M2} + i \alpha$ descends after transgression (double dimensional reduction) along the fiber to give a map

$X \longrightarrow \overline{\mathcal{M}_{ell}} \tilde \times K(\mathbb{Z},3)$that map would then be the combined twisting class of F-theory, combining the elliptic parameterization with the B-field twist for K-theory.

Something like this.

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I am maybe beginning to see how to obtain the F-theory picture from super-cohesion. Here is some brainstorming.

Let’s consider the M-theory bouquet, but starting down in IIA:

$\array{ \mathfrak{m}5\mathfrak{brane} \\ \downarrow \\ \mathfrak{m}2\mathfrak{brane} &\longrightarrow& \mathbb{R}[6] \\ \downarrow \\ {{\mathfrak{D}0\mathfrak{brane}} \atop {= \mathbb{R}^{10,1\vert \mathbf{32}}}} &\longrightarrow& \mathbb{R}[3] \\ \downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} &\stackrel{}{\longrightarrow}& \mathbb{R}[1] }$In Lie integrating and globalizing this, we should really be complexifying the coefficients.

Recall the reason for this for the M2-brane (from *Structure Theory for Higher WZW Terms (schreiber)*):

Considering its WZW term as complexified $\mathbf{L}_{WZW}^{M2} \colon \mathbb{R}^{10,1\vert \mathbf{32}} \longrightarrow \mathbf{B}^3(\mathbb{C}/\mathbb{Z})_{conn}$ gives us the freedom to add a closed 3-form $\alpha$, and if we take $\alpha$ to be the associative 3-form on $\mathbb{R}^7 \hookrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}$ then we get the neat statement that

globalizing $\mathbf{L}_{WZW}^{M2} + i \alpha$ over an 11-dimensional supermanifold $X$ is precisely the structure of making $X$ a solution to the equations of motion of 11d supergravity and fibered by $G_2$-manifolds together with the classical anomaly cancellation data that makes the M2 be globally defined on this background, hence is precisely the data of M-theory on G2-manifolds;

Since the thus globalized $\alpha$ is a calibration on $X$, it follows that the volume holonomy of the globalized $\mathbf{L}_{WZW}^{M2} + i \alpha$ gives precisely the membrane instanton contributions.

So let’s think about what it means to complexify the coefficients of the other stages in the bouquet.

Consider the D0-brane. Its WZW term is the RR-field which in the literature is often written $C_1$. Under T-duality, this is identified with the WZW-term $C_0$ of the D(-1)-brane in type IIB. Now, the complexification of $C_0$ is well-known and famous: this is the axio-dilaton $\tau \coloneqq C_0 + i g_2$, where $g_s$ is the string coupling constant. The proposal of F-theory is to think of $\tau$ as varying over spacetime, defining an elliptic fibration over 10d spacetime.

But we may be more systematic about this: passing back via T-duality, we find that the complexified D0-brane WZW term is identified with the axio-dilaton. But now we *know* that this WZW term induces an extension of spacetime by Lie extension/D0-brane condensation.

To summarize this: we know

from first principles that the complexified D0 WZW term induces an extension of 10d spacetime by fibers locally modeled on $\mathbb{C}$;

from comparison with the string literature we see that this is nothing but that type IIA pre-image of the F-theory elliptic fibration.

So we may conclude that the integrated version of the D0-brane cocycle is to be identified with the map

$\mathbf{L}_{WZW}^{D0} + i \tilde g_s \colon X \longrightarrow \overline{\mathbb{H}//SL(2,\mathbb{Z})}$to the compactified complex moduli stack of elliptic curves which classifies the elliptic fibration.

(Maybe we may fill in more details in this step. What if we used $\mathbb{C}//GL(2,\mathbb{Z})$ instead? which is more natural from the perspective of integrating our $\mathbb{C}[1]$-valued Lie algebra cocycle)

Once at this point, it makes sense to rewrite this as

$\mathbf{L}_{WZW}^{D0} + i \tilde g_s \colon X \longrightarrow \overline{\mathcal{M}_{ell}}(\mathbb{C}) \,.$Now by GHL this moduli stack of course carries the bundle $\mathcal{O}^{top}_{ell}$ of elliptic spectra, and so we may say that the D0-brane cocycle twists elliptic cohomology on $X$. This is pretty much the proposal by Kriz-Sati. Notice that when the cocycle map hits (only) the point of the nodal curve, then this statement comes out precisely matching the known data:

physically (see at F-branes – table) this is the phase of F-theory with ordinary type IIB D-branes around which carry their ordinary D-brane charge in K-theory;

mathematically, the restriction of $\mathcal{O}^{top}_{ell}$ to the nodal curve is $KU$ with its $\mathbb{Z}_2$-action sitting over $\ast/\mathbb{Z}_2$ (see the table here), hence mathematically this gives indeed KR-theory (nlab) cohomology in exactly the right way as needed for general type II backgrounds (see at orientifold (nlab)).

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