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\section*{Hitchin connection}

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

\hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization}

[[!include geometric quantization - contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \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 $G$ a [[compact Lie group]] and $\Sigma$ a [[surface]], the \emph{Hitchin connection} (\hyperlink{Hitchin90}{Hitchin 90}) is a [[projectively flat connection]] over the [[moduli space of Riemann surfaces]] $\mathcal{M}_\Sigma$ whose [[fiber]] over a [[complex structure]] on $\Sigma$ is a [[space of quantum states]] which [[quantization of Chern-Simons theory]] for [[gauge group]] $G$ assigns to the induced [[Kähler polarization]] on the [[moduli space of flat connection]] $Loc_G(\Sigma)$ (the [[phase space]] of [[Chern-Simons theory]]), hence whose fiber is the space of [[holomorphic sections]] of the [[prequantum line bundle]] of the CS theory with respect to the induced K\"a{}hler structure on $Loc_G(\Sigma)$ (induced, for abelian $G$, via the \href{intermediate+Jacobian#WeilIntermediateJacobian}{Weil complex structure} and in the general nonabelian case via the [[Narasimhan-Seshadri theorem]]/[[Donaldson-Uhlenbeck-Yau theorem]]).

Therefore the existence and (projective) flatness of the Hitchin connection exhibits the relative independence of the [[geometric quantization]] of [[Chern-Simons theory]] from the choice of [[polarization]].

The Hitchin connection is akin to the \emph{[[Knizhnik-Zamolodchikov connection]]}, which is built by thinking [[AdS3-CFT2 and CS-WZW correspondence|holographic dually]] of the vector bundle of [[conformal blocks]] of the corresponding $G$-[[Wess-Zumino-Witten model]] [[2d CFT]] on $\Sigma$.

A more axiomatic characterization of such projectively flat connections in terms of [[modular functors]] is due to (\hyperlink{Segal88}{Segal 88, prop. 5.4}). See (\hyperlink{Segal88}{Segal 88, around (5.9)}) for discussion of how to turn these projectively flat connections into genuine [[flat connections]].

Reviews of the Hitchin connection include (\hyperlink{Lauridsen10}{Lauridsen 10, section 2}).

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

\begin{itemize}%
\item [[quantization of Chern-Simons theory]]


\item [[AdS3-CFT2 and CS-WZW correspondence]]


\item [[modular functor]]



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

The original article is

\begin{itemize}%
\item [[Nigel Hitchin]], \emph{Flat connections and geometric quantization}, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (\href{http://projecteuclid.org/euclid.cmp/1104200841}{Euclid})

\end{itemize}
(What is now called the Hitchin connection appears in theorem 3.6 there, its expression in local coordinates is around (3.15). That for abelian gauge group $U(1)$ the classical [[Riemann theta functions]] constitute the covariantly constant sections of the Hitchin connection in these coordinates is remark 4.12.)

More axiomatic/abstract discussion of these projectively flat connections in terms of [[modular functors]] is in

\begin{itemize}%
\item [[Graeme Segal]], section 5 of \emph{The definition of conformal field theory} , preprint, 1988; also in [[Ulrike Tillmann]] (ed.) \emph{Topology, geometry and quantum field theory} , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. (\href{https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf}{pdf})

\end{itemize}
More explicit expressions are discussed in

\begin{itemize}%
\item Bert van Geemen, [[Aise Johan de Jong]], \emph{On Hitchin's Connection} (\href{http://arxiv.org/abs/alg--geom/9701007}{arXiv:alg--geom/9701007})

\end{itemize}
A nice review and new concise account is in

\begin{itemize}%
\item [[Eduard Looijenga]], \emph{From WZW models to Modular Functors} (\href{http://arxiv.org/abs/1009.2245}{arXiv:1009.2245})


\item Johan Martens [[Jørgen Andersen]], notes by S\o{}ren J\o{}rgensen, section 1.2.1 and 4 of \emph{Topological quantum field theories and moduli spaces}, 2011 (\href{http://maths.fuglede.dk/noter/tqftms.pdf}{pdf})



\end{itemize}
Discussion for [[deformation quantization]] instead of [[geometric quantization]] is in

\begin{itemize}%
\item [[Jørgen Andersen]], \emph{Hitchin's connection, Toeplitz operators and symmetry invariant deformation quantization} (\href{http://arxiv.org/abs/math/0611126}{arXiv:math/0611126})

\end{itemize}
This also reproduces the original construciton in the context of [[Chern-Simons theory]] in

\begin{itemize}%
\item [[Scott Axelrod]], S. Della Pietra, [[Edward Witten]], \emph{Geometric quantization of Chern-Simons gauge theory}, Jour. Diff. Geom. 33 (1991), 787-902. (\href{http://projecteuclid.org/euclid.jdg/1214446565}{EUCLID})

\end{itemize}
More details (on [[metaplectic correction]]) and generalization to connections over more general manifolds are in

\begin{itemize}%
\item Peter Scheinost, [[Martin Schottenloher]], \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}
Similar generalization away from [[moduli spaces of flat connections]] to general [[symplectic manifolds]] with [[Kähler structure]] on them is also in

\begin{itemize}%
\item Lauridsen, \emph{Aspects of quantum mathematics -- Hitchin connections and the AJ conjecture}, PhD thesis Aarhus 2010 (\href{http://pure.au.dk/portal/files/41741925/imf_phd_2010_mrl.pdf}{pdf})

\end{itemize}
Relation between the Hitchin connection and bundles of [[conformal blocks]] is discussed in

\begin{itemize}%
\item [[Yves Laszlo]] \emph{Hitchin's and WZW connection are the same}, J. Differential Geom. 49 (1998), no. 3, 547--576 (\href{http://www.emis.de/journals/NYJM/JDG/archive/vol.49/3_5.pdf}{pdf})

\end{itemize}
For more on this see at \emph{[[quantization of 3d Chern-Simons theory]]}.

[[!redirects Hitchin connections]]



\end{document}