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\begin{document}

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\section*{Hitchin functional}

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

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - 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{in_6_dimensions}{In 6 dimensions}\dotfill \pageref*{in_6_dimensions} \linebreak
\noindent\hyperlink{in_7_dimensions}{In 7 dimensions}\dotfill \pageref*{in_7_dimensions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_6_dimensions_2}{In 6 dimensions}\dotfill \pageref*{in_6_dimensions_2} \linebreak
\noindent\hyperlink{in_7_dimensions_2}{In 7 dimensions}\dotfill \pageref*{in_7_dimensions_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

What are now called \emph{Hitchin functionals} are [[functions]]/[[functionals]] on spaces of [[differential n-forms]] on [[manifolds]] of [[dimension]] 6 or 7, for $n = 3$ and/or $n =4$. They are such that their [[critical points]] characterize holomorphic 3-forms of [[complex manifolds]] in dimension 6, as [[associative 3-forms]] of [[G2-manifolds]] in dimension 7.

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

\hypertarget{in_6_dimensions}{}\subsubsection*{{In 6 dimensions}}\label{in_6_dimensions}

Let $X$ be a [[closed manifold|closed]] [[orientation|oriented]] [[smooth manifold]] of [[dimension]] 6.

For $\Omega \in \wedge^3 \Gamma(T^* X)$ a [[differential n-form|differential 3-form]] on $X$, write

\begin{itemize}%
\item ${\vert \lambda \Omega \vert} \in (\wedge^6 \Gamma(T^* X))^{\otimes 2}$


\item $\sqrt{\vert \lambda \Omega \vert} \in \wedge^6 \Gamma(T^* X)$.



\end{itemize}
The \emph{Hitchin function} on 3-forms is the function

\begin{displaymath}
\wedge^3 \Gamma(T^* X) \to \mathbb{R}
\end{displaymath}
which sends 3-form to the [[integration of differential forms]] of this 6-form over $X$

\begin{displaymath}
\Omega \mapsto \int_X \sqrt{\vert \lambda (\Omega) \vert}
  \,.
\end{displaymath}
\hypertarget{in_7_dimensions}{}\subsubsection*{{In 7 dimensions}}\label{in_7_dimensions}

Let $X$ be 7-dimensional.

Let $\Omega \in \wedge^3 \Gamma(T^* X)$ be a [[stable differential form]]. This determines a [[Riemannian metric]] $\g_\Omega$ on $X$. Write $\star_\Omega$ for the corresponding [[Hodge star operator]]. The \emph{Hitchin functional} now is the function that takes stable forms to

\begin{displaymath}
\int_X \Omega \wedge \star_\Omega \Omega
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{in_6_dimensions_2}{}\subsubsection*{{In 6 dimensions}}\label{in_6_dimensions_2}

A differential 3-form $\Omega$ such that $\lambda(\Omega) \lt 0$ is a [[critical point]] of the above functional precisely if there is the structure of a [[complex manifold]] on $X$ such that $\Omega$ is the real part of a non-vanishing holomorphic 3-form.

This is (\hyperlink{Hitchin00}{Hitchin, theorem 13}).

\hypertarget{in_7_dimensions_2}{}\subsubsection*{{In 7 dimensions}}\label{in_7_dimensions_2}

(\ldots{})

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

\begin{itemize}%
\item [[Calabi-Yau manifold]], [[G2-manifold]]


\item [[topological M-theory]]



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

The original articles are

\begin{itemize}%
\item [[Nigel Hitchin]], \emph{The geometry of three-forms in six and seven dimensions}, (\href{http://arxiv.org/abs/math/0010054}{arXiv:math/0010054})

\end{itemize}
\begin{itemize}%
\item [[Nigel Hitchin]], \emph{Stable forms and special metrics} (\href{http://arxiv.org/abs/math/0107101}{arXiv:math/0107101})

\end{itemize}
Reviews with a perspective on the role of the functionals in [[physics]]/[[gravity]]/[[string theory]]/[[topological M-theory]] include

\begin{itemize}%
\item [[Robbert Dijkgraaf]], [[Sergei Gukov]], [[Andrew Neitzke]], [[Cumrun Vafa]], \emph{Topological M-theory as Unification of Form Theories of Gravity} (\href{http://arxiv.org/abs/hep-th/0411073v2}{arXiv:hep-th/0411073v2})

\end{itemize}
[[!redirects Hitchin functionals]]

[[!redirects Hitchin's functional]] [[!redirects Hitchin's functionals]]



\end{document}