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\begin{document}

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\section*{cubical structure on a line bundle}

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

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

[[!include cohomology - contents]]

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{relation_to_orientations_in_complexoriented_cohomology_theory}{Relation to orientations in complex-oriented cohomology theory}\dotfill \pageref*{relation_to_orientations_in_complexoriented_cohomology_theory} \linebreak
\noindent\hyperlink{on_the_11dimensional_chernsimons_term}{On the 11-dimensional Chern-Simons term}\dotfill \pageref*{on_the_11dimensional_chernsimons_term} \linebreak
\noindent\hyperlink{related_concept}{Related concept}\dotfill \pageref*{related_concept} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{cubical structure} on a [[complex line bundle]] over an [[abelian group]] is a certain trivialization of a certain induced line bundle on the 3-fold Cartesian product (``cube'') of the group which is constructed in a kind of cubical generalization of the [[polarization identity]] formula for [[quadratic forms]].

Over [[formal groups]] associated with [[complex oriented cohomology theories]] cubical structures encode [[orientation in generalized cohomology]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
Given a [[circle group]]-[[principal bundle]]/[[complex line bundle]] $\mathcal{L}$ on an [[abelian group]] $A$, write $\Theta(\mathcal{L})$ for the line bundle on $G^3$ which is given by the formula

\begin{displaymath}
\Theta(\mathcal{L})_{x,y,z}
  =
  \mathcal{L}_{x+y+z} \otimes \mathcal{L}_{x+y}^{-1} \otimes \mathcal{L}_{x+z}^{-1} \otimes \mathcal{L}_{y+z}^{-1} \otimes \mathcal{L}_x \otimes \mathcal{L}_y \otimes \mathcal{L}_z \otimes \mathcal{L}_0^{-1}
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{CubicalStructure}\hypertarget{CubicalStructure}{}
A \emph{cubical structure} on $\mathcal{L}$ is a trivializing [[section]] $s$ of $\Theta(\mathcal{L})$ such that

\begin{enumerate}%
\item $s(0,0,0) = 1$


\item $s(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}) = s(x_1, x_2, x_3)$


\item $s(w+x,y,z) s(w,x,z) = s(w,x + y, z) s(x,y,z)$



\end{enumerate}
for all elements of $A$ as indicated, and for all [[permutations]] $\sigma$ of three elements. Here the equalities are equalities of section after applying the canonical [[isomorphisms]] of [[complex lines]] on both sides.

\end{defn}
(\hyperlink{Breen83}{Breen 83}, \hyperlink{Hopkins94}{Hopkins 94, section 4}, \hyperlink{AndoHopkinsStrickland01}{Ando-Hopkins-Strickland 01, def. 2.40})

\begin{remark}
\label{}\hypertarget{}{}
The canonical isomorphsms hidden in def. \ref{CubicalStructure} are:

\begin{enumerate}%
\item $\mathcal{L}_0^{\otimes 3} \otimes (\mathcal{L}_0^{-1})^{\otimes 3} \to 1$ the canonical map exhibiting $\mathcal{L}_0^{-1}$ as the inverse ([[dual object]]) of $\mathcal{L}_0$:


\item etc.



\end{enumerate}
\end{remark}
There is the following further refinement.

\begin{defn}
\label{}\hypertarget{}{}
In the situation of def. \ref{CubicalStructure}, if the line bundle $\mathcal{L}$ is equipped with a natural ``symmetry''

\begin{displaymath}
t \colon \mathcal{L}_x \stackrel{\simeq}{\longrightarrow} \mathcal{L}_{-x}
\end{displaymath}
then a \emph{$\Sigma$-structure} on $\mathcal{L}$ is a cubical structure, def. \ref{CubicalStructure}, such that in addition

\begin{displaymath}
s(x,y , -x-y) = 1
 \,.
\end{displaymath}
\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{relation_to_orientations_in_complexoriented_cohomology_theory}{}\subsubsection*{{Relation to orientations in complex-oriented cohomology theory}}\label{relation_to_orientations_in_complexoriented_cohomology_theory}

For $E$ a [[multiplicative cohomology theory|multiplicative]] [[weakly periodic cohomology theory|weakly periodic]] [[complex orientable cohomology theory]] then $Spec E^0(B U\langle 6\rangle)$ is naturally equivalent to the space of cubical structures on the trivial line bundle over the [[formal group]] of $E$.

In particular, [[homotopy classes]] of maps of [[E-infinity ring]] spectra $MU\langle 6\rangle \to E$ from the [[Thom spectrum]] to $E$, and hence universal $MU\langle 6\rangle$-[[orientation in generalized cohomology|orientations]] (see there) of $E$ are in natural bijection with these cubical structures.

(\hyperlink{Hopkins94}{Hopkins 94, theorem 6.1, 6.2}, \hyperlink{AndoHopkinsStrickland01}{Ando-Hopkins-Strickland 01, corollary 2.50})

This way for instance the [[string orientation of tmf]] has been constructed. See there for more on this.

\hypertarget{on_the_11dimensional_chernsimons_term}{}\subsubsection*{{On the 11-dimensional Chern-Simons term}}\label{on_the_11dimensional_chernsimons_term}

The [[higher dimensional Chern-Simons theory|11-dimensiona Chern-Simons]] [[action functional]] in [[11-dimensional supergravity]] gives a line bundle $L$ on the space of [[supergravity C-fields]] whose $\Theta^3(L)$ is the transgression of the [[cup product in ordinary differential cohomology]] of three factors. It seems that each trivialization of the class of the [[supergravity C-field]] induces a ``cubical'' trivialization of $\Theta^3(L)$ as above, and hence a cubical structure on $L$. See at \emph{[[cubical structure in M-theory]]} for more on this.

\hypertarget{related_concept}{}\subsection*{{Related concept}}\label{related_concept}

\begin{itemize}%
\item [[theta characteristic]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

An early reference discussing the relation with [[theta functions]] is

\begin{itemize}%
\item [[Lawrence Breen]], \emph{Fonctions th\^e{}ta et th\'e{}or\`e{}me du cube}, Springer Lecture Notes in Mathematics \textbf{980} (1983). (\href{http://www.ams.org/mathscinet-getitem?mr=823233}{MR0823233}).

\end{itemize}
In relation to [[orientation in generalized cohomology]] cubical structures have been prominently discussed in

\begin{itemize}%
\item [[Michael Hopkins]], \emph{Topological modular forms, the Witten genus, and the theorem of the cube}, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z\"u{}rich, 1994) (Basel), Birkh\"a{}user, 1995, 554--565. MR 97i:11043 (\href{http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf}{pdf})


\item [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], \emph{Elliptic spectra, the Witten genus and the theorem of the cube}, Invent. Math. 146 (2001) 595--687 MR1869850 (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf}{pdf})



\end{itemize}
[[!redirects cubical structures on a line bundle]] [[!redirects cubical structures on line bundles]]

[[!redirects cubical structure on a complex line bundle]] [[!redirects cubical structures on a complex line bundle]]



\end{document}