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\section*{cubical structure in M-theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{relation_to_ftheory_and_the_topological_witten_genus}{Relation to F-theory and the topological Witten genus}\dotfill \pageref*{relation_to_ftheory_and_the_topological_witten_genus} \linebreak
\noindent\hyperlink{RelationToSDualityAnd3FormFlux}{Relation to S-duality and 3-form flux}\dotfill \pageref*{RelationToSDualityAnd3FormFlux} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

It is well known that when the [[higher dimensional Chern-Simons theory|higher Chern-Simons term]] in [[11-dimensional supergravity]] is [[KK-compactification|compactified]] on a 4-sphere to yield the [[7-dimensional Chern-Simons theory]] which inside \href{AdS-CFT#AdS7CFT6}{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 (\href{http://ncatlab.org/nlab/show/7d+Chern-Simons+theory#Witten97}{Witten 97}) and then formalized and proven in ([[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]]).

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

In (\href{supergravity+C-field#DFM}{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 \emph{cubic refinement}.

More precisely, and slightly paraphrasing, the [[fiber integration in ordinary differential cohomology|transgression]] $\int_X CS_{11}(\hat C)$ of the [[higher dimensional Chern-Simons theory|11-dimensional Chern-Simons term]] of [[11-dimensional supergravity|11d SuGra]] to 10d spacetime $X$ is a [[complex line bundle]] on the [[moduli space]] $CField(X)$ of [[supergravity C-field|supergravity C-fields]] $\hat C$ is claimed to be such that its ``cubical line'' $\Theta(\int_X CS_{11}(\hat C))$ (in the notation at \emph{[[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]],

\begin{displaymath}
\Theta\left(\int_X CS_{11}\left(-\right)\right)
  \simeq
  \int_X (-)_1 \cup (-)_2 \cup (-)_3
  \,.
\end{displaymath}
\hypertarget{relation_to_ftheory_and_the_topological_witten_genus}{}\subsection*{{Relation to F-theory and the topological Witten genus}}\label{relation_to_ftheory_and_the_topological_witten_genus}

In the context of ``[[F-theory]] compactifications'' of [[M-theory]], one considers [[supergravity C-field|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 in ordinary differential cohomology|cup product]] of a cocycle $\hat C_{fib}$ on the fiber with one $\hat C_b$ on the base

\begin{displaymath}
\hat C \coloneqq \hat C_{b} \cup \hat C_{fib}
 \,.
\end{displaymath}
In approaches like (\href{http://ncatlab.org/nlab/show/F-theory#GKP12}{GKP 12 (around p. 19)}, \href{http://ncatlab.org/nlab/show/F-theory#KMW12}{KMW 12}) the [[supergravity C-field|C-field]] is factored as a [[cup product in ordinary differential cohomology|cup product]] of a degree-2 cocycle on the elliptic fiber with a degree-2 class in the [[Calabi-Yau space|Calabi-Yau]]-base. This makes the component of the C-field on the elliptic fiber a [[complex line bundle]] (with [[connection on a bundle|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. (\href{supergravity+C-field#DFM}{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 the [[supergravity C-field]] ``is'' the [[elliptic curve]]-fiber,

\begin{displaymath}
\hat C \coloneqq \hat B_{NS} \cup \hat t_1 + \hat B_{RR} \cup \hat t_2
\end{displaymath}
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 [[supergravity C-field|C-field]] is identified with the compactification [[elliptic curve]] in this way, then the formula for $\Theta\left(\int_X CS_{11}(\hat C)\right)$ as \hyperlink{Idea}{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 $\Theta(\mathcal{O}(-\{0\}))$ over a given elliptic curve is what in (\href{http://ncatlab.org/nlab/show/string+orientation+of+tmf#Hopkins02}{Hopkins 02}, \href{http://ncatlab.org/nlab/show/string+orientation+of+tmf#AndoHopkinsStrickland01}{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 fact that $\Theta\left(CS_{11}(-)\right) \simeq \int_X (-)_1 \cup (-)_2 \cup (-)_3$ a cubical trivialization of $\Theta(L)$ 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]].

\hypertarget{RelationToSDualityAnd3FormFlux}{}\subsection*{{Relation to S-duality and 3-form flux}}\label{RelationToSDualityAnd3FormFlux}

The refined perspective on [[perturbative string theory|perturbative]] [[type II string theory]] is that (see also at \emph{[[orientifold]]}) the [[B-field]] is a [[cocycle]] in ([[twisted cohomology|twisted]]) [[ordinary differential cohomology]], while the [[RR-field]] is a [[cocycle]] in [[differential K-theory]] (im fact [[KR-theory]]). This is however not compatible with [[non-perturbative effect|non-perturbative]] [[S-duality]], which mixes the degree- components here.

In (\href{supergravity+C-field#DFM}{DFM 03, section 9.3}) it was argued that the cubical structure on the 11d CS term alleviates this problem, even though at face value it does not really solve it. But see at \emph{\href{S-duality#CohomologicalNatureOfTypeIIFieldsUnderSDuality}{S-duality -- Cohomological nature of type II fields}} for more on this.



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