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\begin{document}

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\section*{quantization of 3d Chern-Simons theory}

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

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

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{something_more}{Something more}\dotfill \pageref*{something_more} \linebreak
\noindent\hyperlink{space_of_chernsimons_quantum_states}{Space of Chern-Simons quantum states}\dotfill \pageref*{space_of_chernsimons_quantum_states} \linebreak
\noindent\hyperlink{space_of_wzwmodel_precorrelators}{Space of WZW-model pre-correlators}\dotfill \pageref*{space_of_wzwmodel_precorrelators} \linebreak
\noindent\hyperlink{ModularFunctorAndEquivariantEllipticCohomology}{Modular functor and equivariant elliptic cohomology}\dotfill \pageref*{ModularFunctorAndEquivariantEllipticCohomology} \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}

This entry discusses the full ([[non-perturbative quantum field theory|non-perturbative]]) [[quantization]] of the [[prequantum field theory|prequantum]] data of standard 3d [[Chern-Simons theory]] (induced from a suitable [[Lie group]] and [[invariant polynomial]]/[[second Chern class]] [[action functional]]) to a [[3d TQFT]].

(For the [[perturbative QFT|perturbative quantization]] of [[Chern-Simons theory]] see \href{Chern-Simons+theory#PerturbativeQuantization}{there}).

Existing literature knows three sectors of this problem, which overlap but do not coincide

\begin{enumerate}%
\item [[path integral quantization]]. This may be made precise sense of in [[perturbation theory]] where it involves lots of interesting structure such as [[analytic torsion]] (\href{Chern-Simons#theory#Witten89}{Witten 89}). However, being just [[perturbation theory]] it is just an approximation to the full answer.


\item [[geometric quantization]] yields the full (non-pertutbative quantization) in [[codimension]] 1, but does not say anything about codimension 0.


\item The [[Reshetikhin-Turaev construction]] produces a [[3d TQFT]] from algebraic data that is naturally associated with the prequantum data defining Chern-Simons theory (such as the category of [[positive energy representations]] of the [[loop group]] of the given [[gauge group]] $G$, or else of a [[quantum group]] \hyperlink{Sawin06}{Sawin 06}), but it is not a priori clear that this 3d quantum field theory is genuinely the result of quantizing the Chern-Simons [[action functional]].



\end{enumerate}
The known relation between the second and the third point here is the following:

That the complex-geometric [[modular functor]] obtained from [[geometric quantization]] of Chern-Simons theory as in (\hyperlink{AxelrodPietraWitten91}{Axelrod-Pietra-Witten 91}, \hyperlink{Hitchin90}{Hitchin 90}) coincides with that of [[conformal blocks]] of the [[WZW model]] was shown in (\hyperlink{Laszlo98}{Laszlo 98},see also \hyperlink{Andersen11}{Andersen 11}, \hyperlink{Andersen12}{Andersen 12}). That this in turn indeed satisfies the required [[sewing law]] (and hence really is a modular functor in the strong sense) was shown in (\hyperlink{TsuchiyaUenoYamada}{Tsuchiya-Ueno-Yamada}). By \href{modular%20functor#TopologicalLift}{deprojectivization} these constructions yield a topological modular functor of the form also obtained from the [[Reshetikhin-Turaev construction]].

These (topological) modular functors are fixed by their [[genus]]-0 data (\hyperlink{AndersenUeno06}{Andersen-Ueno 06}) which is equivalently the datum of a (weakly) [[modular tensor category]]. Hence for matching [[geometric quantization]] of 3d Chern-Simons theory to the [[Reshetikhin-Turaev construction]] one has to match the [[modular tensor categories]] obtained from the [[conformal blocks]] of the [[WZW model]] in genus-0 to that associated with the coresponding [[quantum groups]]. This works (\hyperlink{Ostrik14}{Ostrik 14}).

\hypertarget{something_more}{}\subsection*{{Something more}}\label{something_more}

\begin{quote}%
random notes, needs to be brought into shape


\end{quote}
\hypertarget{space_of_chernsimons_quantum_states}{}\subsubsection*{{Space of Chern-Simons quantum states}}\label{space_of_chernsimons_quantum_states}

Chern-Simons [[action functional]]

\begin{displaymath}
\exp(\tfrac{i}{\hbar}S_{CS}) \colon \mathbf{Fields}_{CS} \longrightarrow \mathbf{B}^{n+1} U(1)_{conn}
\end{displaymath}
given a [[closed manifold]] $\Sigma_n$ then choice of [[complex structure]] $\mathbf{\Sigma}_n$ on $\Sigma_n$ is supposed to naturally induce a [[complex structure]] on the space of (on-shell) fields over $\Sigma_n$

\begin{displaymath}
\mathbf{Fields}_{CS}(\mathbf{\Sigma}_n)
\end{displaymath}
Moreover, [[transgression]] of $\exp(\tfrac{i}{\hbar}S)$ to $\mathbf{Fields}_{CS}(\Sigma)$ is supposed to yield a [[holomorphic line bundle]] with connection with respect to that complex structure

\begin{displaymath}
\exp(\tfrac{i}{\hbar}S(\mathbf{\Sigma}_n))
  \colon
  \mathbf{Fields}_{CS}(\mathbf{\Sigma}_n)
  \longrightarrow
  \mathbf{B}U(1)_{conn} 
  \,.
\end{displaymath}
This is the [[prequantum line bundle]] of the Chern-Simons theory, already equipped with a [[Kähler polarization]].

Accordingly, the [[geometric quantization]] of the CS action functional assigns to $\Sigma_n$ the [[Hilbert space]] $\mathcal{H}_{\mathbf{\Sigma}}$ of holomorphic sections of $\exp(\tfrac{i}{\hbar}S(\mathbf{\Sigma}_n))$.

As the complex structure $\mathbf{\Sigma}_n$ on $\Sigma_n$ varies over the [[moduli stack of complex structures]] $\mathcal{M}_{\Sigma}$, these vector spaces $\mathcal{H}_{\mathbf{\Sigma}_n}$ form a [[vector bundle]] with projective flat connection (the \emph{[[Hitchin connection]]}) on the moduli stack

\begin{displaymath}
\itexarray{
    \mathcal{H}
    \\
    \downarrow
    \\
    \mathcal{M}_{\Sigma}
  }
\end{displaymath}
The assignment

\begin{displaymath}
\Sigma_n \mapsto \mathcal{H}_{\Sigma}
\end{displaymath}
natural in [[diffeomorphisms]] of $\Sigma$ is called the \emph{[[modular functor]]}, this we focus on more \hyperlink{ModularFunctorAndEquivariantEllipticCohomology}{below}

One such section $\Psi$ is to be singled out. For instance if $\exp(\tfrac{i}{\hbar}S(\Sigma_n))$ is a [[theta characteristic]] then there is up to scale a unique holomorphic section. This singling-out is formalized by the [[FRS-formalism]]. See there for more.

\hypertarget{space_of_wzwmodel_precorrelators}{}\subsubsection*{{Space of WZW-model pre-correlators}}\label{space_of_wzwmodel_precorrelators}

Under the [[AdS3-CFT2 and CS-WZW correspondence]] the states of Chern-Simons theory also correspond to [[partition functions]] of the [[gauged WZW model]] and hence to [[generating functions]] for [[correlation functions]] of the actual [[WZW model]].

The [[field (physics)|fields]] $\mathbf{Fields}(\Sigma_n)$ may also be thought of as the [[sources]] of a (higher) [[gauged WZW model]] on $\Sigma$.

The [[holographic principle]] says that the [[quantum state]]/[[wavefunction]]

\begin{displaymath}
\Psi \in \Gamma(\mathbf{Fields}_{CS}(\mathbf{\Sigma}_n), L)
\end{displaymath}
is also the [[generating function]] for the [[correlators]] of the [[WZW model]] on $\Sigma_n$, meaning that its [[functional derivatives]]

\begin{uremark}
\end{uremark}
with respect to the Chern-Simons-[[field (physics)|fields]], hence the WZW [[sources]], are the [[n-point functions]] of the WZW model for [[current algebra]] insertions, as indicated

(e.g. \hyperlink{Gawedzki99}{Gawdzki 99 (4.23), 5.1})

\hypertarget{ModularFunctorAndEquivariantEllipticCohomology}{}\subsubsection*{{Modular functor and equivariant elliptic cohomology}}\label{ModularFunctorAndEquivariantEllipticCohomology}

Discussion of [[equivariant elliptic cohomology]] (see there at \emph{\href{equivariant+elliptic+cohomology#InterpretationInQuantumFieldTheory}{interpretation in QFT}}) shows that the construction of the [[modular functor]] refines from equipping $\Sigma$ with [[complex structure]] to equipping it with [[arithmetic geometry|arithmetic structure]]. Hence for $\Sigma$ a [[torus]] it refines from structures of [[elliptic curves]] over the complex numbers to general arithmetic elliptic curves (over the [[integers]]) and in fact to [[derived elliptic curves]] (over the [[sphere spectrum]]). Hence eventually the theory of [[geometric quantization]] needs to be refined to admit [[polarizations]] in [[arithmetic geometry]]. See at \emph{[[differential cohesion and idelic structure]]} for more on this.

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

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


\item [[Narasimhan-Seshadri theorem]]


\item [[Harder-Narasimhan theorem]]


\item [[equivariant elliptic cohomology]]



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

Basics are recalled for instance in

\begin{itemize}%
\item Fernando Falceto, [[Krzysztof Gaw?dzki]], \emph{Chern-Simons States at Genus One}, Commun.Math.Phys. 159 (1994) 549-580 (\href{http://arxiv.org/abs/hep-th/9211003}{arXiv:hep-th/9211003})


\item [[Krzysztof Gaw?dzki]], \emph{Conformal field theory: a case study} in Y. Nutku, C. Saclioglu, T. Turgut (eds.) \emph{Frontier in Physics} 102, Perseus Publishing (2000) (\href{http://xxx.lanl.gov/abs/hep-th/9904145}{hep-th/9904145})


\item Yasuhiro Abe, \emph{Application of abelian holonomy formalism to the elementary theory of numbers} (\href{http://arxiv.org/abs/1005.4299}{arXiv:1005.4299})



\end{itemize}
The [[geometric quantization]] of 3d CS theory in codimension 1 is due to

\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})


\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}
(see also at \emph{[[Hitchin connection]]}).

The [[3d TQFT]] candidate for quantum CS theory in the form of the [[Reshetikhin-Turaev construction]] and the corresponding [[modular tensor category]] data is discussed in

\begin{itemize}%
\item Reshetikhin; Turaev, \emph{Invariants of 3-manifolds via link polynomials and quantum groups}. Invent. Math. 103 (1991), no. 3, 547--597. (\href{http://mathlab.snu.ac.kr/~top/quantum/article/Reshetikhin01.pdf}{pdf})


\item B. Bakalov \& [[Alexandre Kirillov]], \emph{Lectures on tensor categories and modular functors} AMS, University Lecture Series, (2000) (\href{http://www.math.sunysb.edu/~kirillov/tensor/tensor.html}{web}).


\item [[Stephen Sawin]], \emph{Quantum groups at roots of unity and modularity} J. Knot Theory Ramifications 15 (2006), no. 10, 1245--1277 (\href{http://arxiv.org/abs/math/0308281}{arXiv:0308281})


\item [[Victor Ostrik]], \emph{\href{http://mathoverflow.net/a/178304/381}{MO comment August 2014}}



\end{itemize}
The relation between the [[modular functor]] obtained from the [[conformal blocks]] of the [[WZW model]] and from geometric quantization of CS theory 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})


\item Tsuchiya; K Ueno; Yamada, \emph{Conformal field theory on universal family of stable curves with gauge symmetries}, Integrable systems in quantum field theory and statistical mechanics, 459--566.


\item [[Jørgen Andersen]], K. Ueno, \emph{Modular functors are determined by their genus zero data}, Journal of Quantum Topology (\href{http://arxiv.org/abs/math/0611087}{arXiv:math/0611087})



\end{itemize}
Discussion specific to [[special unitary group|special unitary]] [[gauge group]] is in

\begin{itemize}%
\item [[Jørgen Andersen]], K. Ueno, \emph{Abelian Conformal Field theories and Determinant Bundles}, International Journal of Mathematics, 18 919 - 993 (2007).


\item [[Jørgen Andersen]], K. Ueno, \emph{Geometric Construction of Modular Functors from Conformal Field Theory}, Journal of Knot theory and its Ramifications, 16 127 -- 202, (2007).


\item [[Jørgen Andersen]], K. Ueno, \emph{Construction of the Reshetikhin-Turaev TQFT via Conformal Field Theory} (\href{http://arxiv.org/abs/1110.5027}{arXiv:1110.5027})


\item [[Jørgen Andersen]], \emph{A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications} (\href{http://arxiv.org/abs/1206.2785}{arXiv:1206.2785})



\end{itemize}
For discussion of the state of the proof see also

\begin{itemize}%
\item MO, \emph{\href{http://mathoverflow.net/questions/86792/why-hasnt-anyone-proved-that-the-two-standard-approaches-to-quantizing-chern-si}{Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?}}

\end{itemize}
and in particular \href{http://mathoverflow.net/a/185401/381}{this reply} by [[Andre Henriques]].

Detailed review in the case of abelian Chern-Simons theory includes

\begin{itemize}%
\item Spencer D. Stirling, \emph{Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories} (\href{http://arxiv.org/abs/0807.2857}{arXiv:http://arxiv.org/abs/0807.2857v1})

\end{itemize}
Discussion in terms of [[Weyl quantization]] of [[Wilson lines]] and details on the role of [[theta functions]] is in

Discussion of [[quantization of Chern-Simons theory]] in terms of [[Weyl quantization]] and [[skein relations]] is in

\begin{itemize}%
\item [[Jørgen Andersen]], \emph{Deformation quantization and geometric quantization of abelian moduli spaces}, Commun. Math. Phys., 255 (2005), 727--745


\item [[Razvan Gelca]], [[Alejandro Uribe]], \emph{The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same}, Commun.Math.Phys. 233 (2003) 493-512 (\href{http://arxiv.org/abs/math-ph/0201059}{arXiv:math-ph/0201059})


\item [[Razvan Gelca]], [[Alejandro Uribe]], \emph{From classical theta functions to topological quantum field theory} (\href{http://arxiv.org/abs/1006.3252}{arXiv:1006.3252}, \href{http://www.math.ttu.edu/~rgelca/berk.pdf}{slides pdf})


\item [[Razvan Gelca]], [[Alejandro Uribe]], \emph{Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$} (\href{http://arxiv.org/abs/1007.2010}{arXiv:1007.2010})



\end{itemize}
Another approach is

\begin{itemize}%
\item [[Daniel Freed]], [[Mike Hopkins]], [[Constantin Teleman]], [[Jacob Lurie]], \emph{\href{http://ncatlab.org/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups#3dCSFullyExtended}{TQFT from compact Lie groups -- 3d Chern-Simons as a fully extended TQFT}}.

\end{itemize}
[[!redirects quantization of Chern-Simons theory]]



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