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\begin{document}

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\section*{Theta characteristic}

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

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

[[!include synthetic differential geometry - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{OverRiemannSurfaces}{Over Riemann surfaces}\dotfill \pageref*{OverRiemannSurfaces} \linebreak
\noindent\hyperlink{AsMetaplecticCorretionOfK&#228;hlerPolarizations}{As metaplectic and Spin structure over (K\"a{}hler-)polarized varieties}\dotfill \pageref*{AsMetaplecticCorretionOfK&#228;hlerPolarizations} \linebreak
\noindent\hyperlink{OverIntermediateJacobians}{Over intermediate Jacobians}\dotfill \pageref*{OverIntermediateJacobians} \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}

For $X$ a [[space]] equipped with a notion of [[dimension]] $dim X \in \mathbb{N}$ and a notion of [[Kähler differential forms]], a \emph{$\Theta$-characteristic of $X$ is a choice of [[square root]] of the [[canonical characteristic class]] of $X$. See there for more details.}

In [[complex analytic geometry]] and at least if the Theta characteristic is [[polarized variety|principally polarizing]] then its [[holomorphic sections]] are called [[theta functions]]. In particular for line bundles over the [[Jacobian variety]] of a [[Riemann surface]] they are called \emph{[[Riemann theta functions]]}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{OverRiemannSurfaces}{}\subsubsection*{{Over Riemann surfaces}}\label{OverRiemannSurfaces}

\begin{prop}
\label{}\hypertarget{}{}
For $\Sigma$ a [[Riemann surface]], the choices of [[square roots]] of the [[canonical bundle]] correspond to the choice of [[spin structures]].

For $X$ of [[genus of a surface|genus]] $g$, there are $2^{2g}$ many choices of square roots of the canonical bundle.

\end{prop}
(\hyperlink{Atiyah}{Atiyah, prop. 3.2}).

\begin{remark}
\label{}\hypertarget{}{}
The first statement remains true in higher dimensions over [[Kähler manifolds]], see at \emph{\href{spin%20structure#OverAKahlerManifold}{Spin structure -- On K\"a{}hler manifolds}}.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The function that sends a square root line bundle to the [[dimension]] of its space of [[holomorphic sections]] $mod \;2$ is a [[quadratic refinement]] of the [[intersection pairing]] on $H^1(X, \mathbb{Z}_2)$.

\end{prop}
This is due to (\hyperlink{Atiyah}{Atiyah, theorem 2}). A motivational survey in broader context of [[quadratic refinements]] of the [[intersection pairing]] in higher dimensions is in (\hyperlink{HopkinsSinger02}{Hopkins-Singer 02, section 2.1}).

\hypertarget{AsMetaplecticCorretionOfK&#228;hlerPolarizations}{}\subsubsection*{{As metaplectic and Spin structure over (K\"a{}hler-)polarized varieties}}\label{AsMetaplecticCorretionOfK&#228;hlerPolarizations}

In the context of [[geometric quantization]] a [[metaplectic structure]] on a [[polarization]] is a [[square root]] of a certain line bundle. In the special case of [[Kähler polarization]] this is a square root precisely of the [[canonical line bundle]] of the underlying [[complex manifold]] and hence is a $\Theta$-characteristic. Also, equivalently this is a [[Spin structure]], see at \emph{\href{http://ncatlab.org/nlab/show/spin%20structure#OverAKahlerManifold}{spin structure -- Over a K\"a{}hler manifold}}. For more on this see at \emph{\href{geometric+quantization#IndexOfDolbeaultDiracOperator}{geometric quantization -- Quantum states as index of Dolbeault-Dirac operator}}.

Notice that generalizing from [[complex analytic geometry]] to [[algebraic geometry]] over other bases, then the analog of a [[Kähler polarization]] is a \emph{[[polarized variety]]}. Hence a choice of [[Theta characteristic]] on a [[polarized variety]] is the analog of a metaplectically corrected K\"a{}hler manifold.

\hypertarget{OverIntermediateJacobians}{}\subsubsection*{{Over intermediate Jacobians}}\label{OverIntermediateJacobians}

A special square root of the canonical bundle on [[intermediate Jacobians]] in dimension $2k+1$ thought of as [[moduli spaces]] of (flat) [[circle n-bundles with connection|circle (2k+1)-bundles with connection]] has a unique [[section]] the [[partition function]] of abelian [[self-dual higher gauge theory]] (see there for details). (\hyperlink{Witten96}{Witten 96}, \hyperlink{HopkinsSinger02}{Hopkins-Singer 02}).

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

[[!include square roots of line bundles - table]]

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

The spaces of choices of $\Theta$-characteristics over [[Riemannian manifolds]] were originally discussed in

\begin{itemize}%
\item [[Michael Atiyah]], \emph{Riemann surfaces and spin structures}, Annales Scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, (1971), Quatri\`e{}me S\'e{}rie 4: 47--62, ISSN 0012-9593, MR0286136

\end{itemize}
See also

\begin{itemize}%
\item M. Bertola, \emph{Riemann surfaces and Theta Functions}, August 2010 (\href{http://www.mathstat.concordia.ca/faculty/bertola/ThetaCourse/ThetaCourse.pdf}{pdf})


\item Gavril Farkas, \emph{Theta characteristics and their moduli} (2012) (\href{http://arxiv.org/abs/1201.2557}{arXiv:1201.2557})



\end{itemize}
The relation of Theta characteristics on [[intermediate Jacobians]] to [[self-dual higher gauge theory]] was first recognized in

\begin{itemize}%
\item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory}, J.Geom.Phys.22:103-133,1997 (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234})

\end{itemize}
and the argument there was made rigorous in

\begin{itemize}%
\item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology, and M-Theory]]}

\end{itemize}
Related arguments revolving around [[characteristic element of a bilinear form|characteristic elements]] for the [[intersection pairing]] appear in

\begin{itemize}%
\item Bjorn Poonen, Eric Rains, \emph{Self cup products and the theta characteristic torsor} (\href{http://arxiv.org/abs/1104.2105}{arXiv:1104.2105})

\end{itemize}
[[!redirects ? characteristic]] [[!redirects ∞-characteristic]] [[!redirects Theta-characteristic]]

[[!redirects ? characteristics]] [[!redirects ∞-characteristics]] [[!redirects Theta characteristics]] [[!redirects Theta-characteristics]]

[[!redirects theta characteristic]] [[!redirects theta characteristics]] [[!redirects theta-characteristic]] [[!redirects theta-characteristics]]



\end{document}