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\begin{document}

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\section*{Cayley form}

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

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{invariance}{Invariance}\dotfill \pageref*{invariance} \linebreak
\noindent\hyperlink{as_a_calibration}{As a calibration}\dotfill \pageref*{as_a_calibration} \linebreak
\noindent\hyperlink{GrassmannianOfCayley4Planes}{Grassmannian of Cayley 4-planes}\dotfill \pageref*{GrassmannianOfCayley4Planes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_string_theorymtheory}{In string theory/M-theory}\dotfill \pageref*{in_string_theorymtheory} \linebreak


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

The \emph{Cayley 4-form} $\Phi$ (\hyperlink{HarveyLawson82}{Harvey-Lawson 82}) is a certain [[differential n-form|differential 4-form]] on the [[Cartesian space|real 8-dimensional space]] $\mathbb{R}^8$.

\begin{displaymath}
\Phi \;\in\; \Omega^4(\mathbb{R}^n)
\end{displaymath}
which constitutes an exceptional [[calibration]] of $\mathbb{R}^4$ with its [[Euclidean space|Euclidean geometry]].

More generally, a [[Spin(7)-manifold]] carries a globalization of this 4-form [[calibration]], then also called a Cayley-4-form.

$\backslash$linebreak

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

(\hyperlink{HarveyLawson82}{Harvey-Lawson 82, Def. 1.21})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{invariance}{}\subsection*{{Invariance}}\label{invariance}

The [[stabilizer subgroup]] inside [[general linear group|GL(8)]] of the Cayley 4-from under the [[action]] given by [[pullback of differential forms]] is the [[subgroup]] [[Spin(7)]] inside [[SO(8)]].

(\hyperlink{HarveyLawson82}{Harvey-Lawson 82, Prop. 1.36})

\hypertarget{as_a_calibration}{}\subsubsection*{{As a calibration}}\label{as_a_calibration}

The Cayley 4-form constitutes a [[calibration]] of the [[Euclidean space]] $\mathbb{R}^8$ (\hyperlink{HarveyLawson82}{Harvey-Lawson 82})

\hypertarget{GrassmannianOfCayley4Planes}{}\subsubsection*{{Grassmannian of Cayley 4-planes}}\label{GrassmannianOfCayley4Planes}

A [[calibrated submanifold]] for $\Phi$ is also called a \emph{Cayley 4-plane} (not to be confused with the [[Cayley plane]]).

(\hyperlink{HarveyLawson82}{Harvey-Lawson 82, Def. 1.23})

The space ([[moduli space]]) of Cayley 4-planes, denoted $CAY$ (\hyperlink{BryantHarvey89}{Bryant-Harvey 89, (2.19)}) or $CAYLEY$ (\hyperlink{GluckMackenzieMorgan95}{Gluck-Mackenzie-Morgan 95, (5.20)}), is hence a [[topological subspace]] of the [[Grassmannian]] of all 4-planes in 8-dimensions:

\begin{displaymath}
CAY 
  \subset
  Gr(4,8)
\end{displaymath}
This is of [[codimension]] 4 (\hyperlink{HarveyLawson82}{Harvey-Lawson 82, below (5)}). In fact, this space is [[homeomorphism|homeomorphic]] to the [[coset space]] of [[Spin(7)]] by [[Spin(4).Spin(3)]] = [[Spin(3).Spin(3).Spin(3)]] = [[Sp(1).Sp(1).Sp(1)]]:

\begin{displaymath}
CAY \;\simeq\; Spin(7)/\big( Spin(4) \cdot Spin(3)\big)
\end{displaymath}
(\hyperlink{HarveyLawson82}{Harvey-Lawson 82, Theorem 1.38}, see also \hyperlink{BryantHarvey89}{Bryant-Harvey 89, (3.19)}, \hyperlink{GluckMackenzieMorgan95}{Gluck-Mackenzie-Morgan 95, (5.20)})

Moreover, the [[coset space]] of [[Spin(6)]] by \href{Spn.Sp1#SO4}{Spin(3).Spin(3)} $\simeq$ [[SO(4)]]

\begin{displaymath}
CAY_{sL} 
   \;\simeq\; 
  Spin(6)/\big( Spin(3) \cdot Spin(3)\big)
    \;\simeq\;
  SU(4)/SO(4)
\end{displaymath}
is the Grassmannian of those Cayley 4-planes which are also [[special Lagrangian submanifolds]] (\hyperlink{BBMOOY96}{BBMOOY 96, p. 7 (8 of 17)}).

See also at \emph{\href{Spn.Sp1#SpinGrassmannians}{Spin Grassmannians}}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[8-manifold]]

\end{itemize}
[[!include special holonomy table]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Reese Harvey]], [[H. Blaine Lawson]], \emph{Calibrated geometries}, Acta Math. Volume 148 (1982), 47-157 (\href{https://projecteuclid.org/euclid.acta/1485890157}{Euclid:1485890157})


\item [[Robert Bryant]], [[Reese Harvey]], \emph{Submanifolds in Hyper-Kähler Geometry}, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 1-31 (\href{https://www.jstor.org/stable/1990911}{jstor:1990911})


\item Herman Gluck, Dana Mackenzie, Frank Morgan, \emph{Volume-minimizing cycles in Grassmann manifolds}, Duke Math. J. Volume 79, Number 2 (1995), 335-404 (\href{https://projecteuclid.org/euclid.dmj/1077285156}{euclid:1077285156})


\item Liviu Ornea, [[Paolo Piccinni]], \emph{Cayley 4-frames and a quaternion-Kähler reduction related to $Spin(7)$}, Proceedings of the International Congress of Differential Geometry in the memory of A. Gray, held in Bilbao, Sept. 2000 (\href{https://arxiv.org/abs/math/0106116}{arXiv:math/0106116})


\item [[Mikhail Katz]], [[Steven Shnider]], \emph{Cayley 4-form, comass, and triality isomorphisms} (\href{http://arxiv.org/abs/0801.0283}{arXiv:0801.0283})



\end{itemize}
\hypertarget{in_string_theorymtheory}{}\subsubsection*{{In string theory/M-theory}}\label{in_string_theorymtheory}

In [[string theory]]/[[M-theory]]:

\begin{itemize}%
\item [[Katrin Becker]], [[Melanie Becker]], [[David Morrison]], [[Hirosi Ooguri]], Y. Oz, Z. Yin, \emph{Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds}, Nucl. Phys. B480:225-238, 1996 (\href{https://arxiv.org/abs/hep-th/9608116}{arXiv:hep-th/9608116})

\end{itemize}
[[!redirects Cayley forms]]

[[!redirects Cayley 4-form]] [[!redirects Cayley 4-forms]]

[[!redirects Cayley 4-plane]] [[!redirects Cayley 4-planes]]



\end{document}