It is "well known" that

- on the hand there is a "cubical line bundle" governing the fine structure of the Chern-Simons term in 11-dimensional supergravity/M-theory;
- on the other hand just such "cubical lines" on elliptic curves induce the elliptic cohomology refinement of the partition function of the heterotic string.

I provide a review of that with pointers to the literature below, to be self-contained. First though my question:

it is natural to speculate that these two "cubical structures" are in fact "the same", or at least closely related. In fact it seems to me that standard F-theory lore gives a way to relate them in some detail (this, too, I spell out below). But I am not really sure yet about the full story. My question is:

has this or anything like this been considered/worked out anywhere?

Here now more details on and pointers to what I have in mind here:

**Cubical Structure in M-Theory**

It is well known that when the higher Chern-Simons term in 11-dimensional supergravity is compactified on a 4-sphere to yield the 7-dimensional Chern-Simons theory which inside AdS7/CFT6 is dual to the M5-brane 6d (2,0)-superconformal QFT, the cup product square in ordinary differential cohomology that enters its definition is to receive a quadratic refinement. This was originally argued in (Witten 97) and then formalized and proven in (Hopkins-Singer 02).

What though is the situation up in 11 dimensions before compactifying to 7-dimensions?

In (DFM 03, section 9) it is claimed that the full 11-dimensional Chern-Simons term evaluated on the supergravity C-field (with its flux quantization correction, see there) indeed carries a *cubic refinement*.

More precisely, and slightly paraphrasing, the transgression \(\int_X \mathrm{CS}_{11}(\hat C)\) of the 11-dimensional Chern-Simons term of 11d SuGra to 10d spacetime *X* is a complex line bundle on the moduli space CField(*X*) of supergravity C-fields \(\hat C\) is claimed to be such that its “cubical line” \(\Theta^3\left(\int_X \mathrm{CS}_{11}(\hat C)\right)\) (in the notation at *cubical structure on a line bundle*) is the line bundle on the space of triples of C-field configurations which is given by the transgression of the three-fold cup product in ordinary differential cohomology,

\(\Theta^3\left(\int_X \mathrm{CS}_{11}(\hat -)\right) \simeq \int_X (\hat -)_1 \cup (\hat -)_2 \cup (\hat-)_3\)

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In the context of “F-theory compactifications” of M-theory, one considers C-fields on an elliptic fibration which are “factorizable fluxes”, in that their underlying cocycle \(\hat C\) in ordinary differential cohomology is the cup product of a cocycle \(\hat C_{fib }\) on the fiber with one \(\hat C_{b}\) on the base

\(\hat C = \hat C_{b} \cup \hat C_{fib}\)

In approaches like (GKP 12 (around p. 19), KMW 12) the C-field is factored as a cup product of a degree-2 cocycle on the elliptic fiber with a degree-2 class in the Calabi-Yau-base. This makes the component of the C-field on the elliptic fiber a complex line bundle (with connection). Notice that the space of complex line bundles on an elliptic curve is dual to the elliptic curve itself.

On the other hand in e.g. (DFM 03, p.38) the factorization is taken to be that of two degree-3 cocycles in the base (which are then identified with the combined degree-3 RR-field/B-field flux coupled to the (p,q)-string) with, respectively, the two canonical degree-1 cocycles \(\hat t_i\) on the elliptic fiber which are given by the two canonical coordinate functions \(t_i\) (speaking of a framed elliptic curve). In this case the fiber-component of thesupergravity C-field “is” the elliptic curve-fiber,

\(\hat C = \hat B_{NS} \cup \hat t_1 + \hat B_{RR}\cup \hat t_2\)

or equivalently: each point in the moduli space of *H*-flux in 10d induces an identification of the *G*-flux with the elliptic curve this way.

This is maybe noteworthy in that when the C-field is identified with the compactification elliptic curve in this way, then the formula for \(\Theta^3\left(\int_X \mathrm{CS}_{11}(\hat C)\right)\) as above is exactly that appearing in the definition of a cubical structure on a line bundle over an elliptic curve. But a “cubical” trivialization of \(\Theta^3(\mathcal{O}(-\{0\}))\) over a given elliptic curve is what in (Hopkins 02, AHS01) is used to induce the sigma-orientation of the corresponding elliptic cohomology theory and in totality the string-orientation of tmf. But that is the refinement of the Witten genus, hence of the partition function of the heterotic string.

Now, by the above M-theoretic equivalence, the cubical trivialization is also given by a trivialization of the topological class of the C-field. This is one way (or is at least closely related) to the trivialization of the anomaly line bundle which “sets the quantum integrand” of M-theory.

So there is a curious coincidence of concepts here, which might want to become a precise identification:

on the one hand there is naturally a cubical structure on a line bundle on the Chern-Simons line bundle over the moduli space of supergravity C-fields which for F-theory compactifications and factorizable flux configurations induces in particular a cubical structure on a line bundle over the compactification elliptic curve. On the other hand, the latter are the structures that enter the refined construction of the Witten genus via the string orientation of tmf.

Has this been related further anywhere?