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\begin{document}

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\section*{moduli space of connections}

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

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

[[!include differential cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{compactness}{Compactness}\dotfill \pageref*{compactness} \linebreak
\noindent\hyperlink{over_complex_manifolds__complex_varieties}{Over complex manifolds / complex varieties}\dotfill \pageref*{over_complex_manifolds__complex_varieties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{FlatConnectionsOverATorus}{Flat connections over a torus}\dotfill \pageref*{FlatConnectionsOverATorus} \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}

A [[moduli space]] of [[connections on bundles]] over some prescribed [[space]].

Often one considers [[flat connections]] only, see at \emph{[[moduli space of flat connections]]}.

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

\hypertarget{compactness}{}\subsubsection*{{Compactness}}\label{compactness}

If $\Sigma$ is a [[compact topological space|compact]] [[smooth manifold]], then the moduli space of flat connections over $\Sigma$ is compact.

\hypertarget{over_complex_manifolds__complex_varieties}{}\subsubsection*{{Over complex manifolds / complex varieties}}\label{over_complex_manifolds__complex_varieties}

Over a [[complex manifold]]/[[complex variety]], the [[Koszul-Malgrange theorem]] identifies holomorphic [[flat connections]] on [[complex vector bundles]] with [[holomorphic vector bundles]]. This identifies the moduli space of flat connections as a [[complex manifold]] with (a non-abelian version of) the first Griffiths [[intermediate Jacobian]]. See at \emph{\href{http://nlab.mathforge.org/nlab/show/intermediate+Jacobian#ExamplePicard}{intermediate Jacobian -- Examples -- k=0}}.

More specifically over a [[Riemann surface]] \emph{[[Narasimhan–Seshadri theorem]]} identifies the [[moduli spaces of flat connections]] with that of certain [[complex manifold|complex]] spaces of [[stable vector bundle|stable]] [[holomorphic vector bundles]]. This space appears as the [[phase space]] for [[Chern-Simons theory]] over that surface. See there for more.

More generally, the [[Donaldson-Uhlenbeck-Yau theorem]] similarly gives a [[Kähler structure]] on the moduli space of flat connections also over higher dimensional K\"a{}hler manifolds (\hyperlink{ScheinostSchottenloher96}{Scheinost-Schottenloher 96, corollary 1.16}).

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

\hypertarget{FlatConnectionsOverATorus}{}\subsubsection*{{Flat connections over a torus}}\label{FlatConnectionsOverATorus}

Let $G$ be a [[compact Lie group]]. Assume either that $G$ is [[simply connected]] or is a [[torus]] (what we really need below is that any two commuting elements in $G$ sit jointly in one [[maximal torus]]).

The moduli space of $G$ [[flat connections]] on a 2-dimensional [[torus]] $A \simeq S^1 \times S^1$ (e.g. underlying a complex [[elliptic curve]]) has the following description:

first, the [[moduli stack]] of flat connections is

\begin{displaymath}
\begin{aligned}
    [\Pi(A), \mathbf{B}G] 
    & \simeq [B [A, S^1], \mathbf{B}G]
    \\
    & \simeq Hom_{Grp}(\mathbb{Z} \times \mathbb{Z}, G)//_{ad} G
  \end{aligned}
\end{displaymath}
(see also the discussion at \emph{\href{http://ncatlab.org/nlab/show/group+character#CharactersAndFundamentalGroupsOfTori}{characters and fundamental groups of tori}}).

Here a single [[flat connection]] is just a choice of pair of two commuting elements in $G$, and $G$ acts on that by [[conjugation]]. Now any two commuting elements can be taken to sit in a [[maximal torus]] $T \hookrightarrow G$, and up to [[conjugation]] we can take this to be one fixed maximal torus. This means that the moduli space is actually

\begin{displaymath}
\pi_0 \left(
    Hom_{Grp}([A,S^1], G)//_{ad} G
  \right)
  \simeq
  Hom_{Grp}([A, S^1], T)/W
  \,,
\end{displaymath}
where $W$ is the [[Weyl group]] $W = N_G(T)/T$.

Moreover, by [[Pontryagin duality]] this may be re-expressed as

\begin{displaymath}
\cdots \simeq Hom_{Grp}([T,S^1], A)/W
  \,.
\end{displaymath}
where now $[T,S^1]$ is the [[character group]] of the [[maximal torus]].

In this form the moduli space of flat connections appears prominently for instance in the discussion of [[equivariant elliptic cohomology]]. But beware that the above interpretation in [[algebraic geometry]] is at least more subtle, see (\hyperlink{Lurie15}{Lurie 15}).

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

\begin{itemize}%
\item [[Donaldson-Uhlenbeck-Yau theorem]]


\item [[Tamagawa numbers]]


\item [[Hitchin connection]]



\end{itemize}
[[!include moduli spaces -- contents]]

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

Original articles include

\begin{itemize}%
\item [[Michael Atiyah]], [[Raoul Bott]], \emph{The Yang-Mills equations over Riemann surfaces}, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (\href{http://www.jstor.org/stable/37156}{jstor}, \href{http://math.stackexchange.com/a/295505/58526}{lighning summary})


\item Nan-Kuo Ho, Chiu-Chu Melissa Liu, \emph{On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces}, Canad. J. Math. 56(2004), 1228-1236 (\href{https://cms.math.ca/10.4153/CJM-2004-053-3}{publisher})



\end{itemize}
Reviews and lecture notes include, for the case of flat connections

\begin{itemize}%
\item Remi Janner, \emph{Notes on the moduli space of flat connections}, 2005 (\href{https://www.researchgate.net/publication/265223952_Notes_on_the_moduli_space_of_flat_connections}{pdf})


\item Daan Michiels, \emph{Moduli spaces of flat connections}, Master Thesis Leuven 2013 (\href{http://www.staff.science.uu.nl/~Schat001/Daan_thesis.pdf}{pdf})


\item [[Jörg Teschner]], \emph{Quantization of moduli spaces of flat connections and Liouville theory}, proceedings of the International Congress of Mathematics 2014 (\href{http://arxiv.org/abs/1405.0359}{arXiv:1405.0359})


\item [[Vladimir Fock]], [[Alexander Goncharov]], \emph{Symplectic double for moduli spaces of G-local systems on surfaces} (\href{http://arxiv.org/abs/1410.3526}{arXiv:1410.3526})



\end{itemize}
and for the case of general and [[logarithmic connections]]

\begin{itemize}%
\item Indranil Biswas, V.Munoz, \emph{Moduli spaces of connections on a Riemann surface} (\href{http://www.mat.ucm.es/~vmunozve/Moduli.pdf}{pdf})

\end{itemize}
Detailed discussion of moduli space of flat connections also on higher dimensional base spaces is in

\begin{itemize}%
\item Peter Scheinost, [[Martin Schottenloher]], pp. 154 (11 of 76) of \emph{Metaplectic quantization of the moduli spaces of flat and parabolic bundles}, J. reine angew. Mathematik, 466 (1996) (\href{https://eudml.org/doc/153753}{web})

\end{itemize}
For more references see at \emph{[[Hitchin connection]]}.

Discussion in [[algebraic geometry]] includes

\begin{itemize}%
\item [[Jacob Lurie]], \emph{\href{http://mathoverflow.net/a/226716/381}{MO comment}} 2015

\end{itemize}
[[!redirects moduli spaces of connections]]

[[!redirects moduli space of flat connections]] [[!redirects moduli spaces of flat connections]]

[[!redirects moduli stack of connections]] [[!redirects moduli stacks of connections]]

[[!redirects moduli stack of flat connections]] [[!redirects moduli stacks of flat connections]]



\end{document}