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\section*{Reshetikhin-Turaev construction}

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

\hypertarget{functorial_quantum_field_theory}{}\paragraph*{{Functorial quantum field theory}}\label{functorial_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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_a_boundary_of_the_craneyetter_model}{As a boundary of the Crane-Yetter model}\dotfill \pageref*{as_a_boundary_of_the_craneyetter_model} \linebreak
\noindent\hyperlink{relation_to_chernsimons_theory}{Relation to Chern-Simons theory}\dotfill \pageref*{relation_to_chernsimons_theory} \linebreak
\noindent\hyperlink{relation_to_conformal_field_theory}{Relation to conformal field theory}\dotfill \pageref*{relation_to_conformal_field_theory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The Reshetikhin-Turaev construction is the [[FQFT]] construction of a 3d [[TQFT]] from the data of a [[modular tensor category]] $\mathcal{C}$. It is something like the ``square root'' of the [[Turaev-Viro model]] on $\mathcal{C}$.

In the case that $C$ is a category of [[positive energy representation]]s of a [[loop group]] $\Omega G$ of a [[Lie group]] $G$, then this algebraically defined QFT is thought to be the result of [[quantization]] of [[Chern-Simons theory]] over the group $G$.

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

\hypertarget{as_a_boundary_of_the_craneyetter_model}{}\subsubsection*{{As a boundary of the Crane-Yetter model}}\label{as_a_boundary_of_the_craneyetter_model}

The Reshetikhin-Turaev model is a [[boundary field theory]] of the [[4d TQFT]] [[Crane-Yetter model]] (\hyperlink{BarrettGarciIslasMartins04}{Barrett\&Garci-Islas\&Martins 04, theorem 2}) Related discussion is in [[Freed]] ``[[4-3-2 8-7-6]]''.

\hypertarget{relation_to_chernsimons_theory}{}\subsubsection*{{Relation to Chern-Simons theory}}\label{relation_to_chernsimons_theory}

The RT-construction for group $G$ is expected to be the [[FQFT]] of $G$-[[Chern-Simons theory]], though a fully explicit proof of this via [[quantization]] is currently not in the literature.

See at \emph{[[quantization of Chern-Simons theory]]} for more on this.

\hypertarget{relation_to_conformal_field_theory}{}\subsubsection*{{Relation to conformal field theory}}\label{relation_to_conformal_field_theory}

The [[CFT|Fuchs-Runkel-Schweigert-construction]] builds from the RT-construction explicitly the rational 2-dimensional [[2d CFT]] boundary theory (see at \emph{[[holographic principle]]}).

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

Original articles include

\begin{itemize}%
\item [[Nikolai Reshetikhin|N. Reshetikhin]], [[Vladimir Turaev|V. 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})

\end{itemize}
A standard textbook account is

\begin{itemize}%
\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}).

\end{itemize}
(See the dedicated page \emph{[[Help me! I'm trying to understand Bakalov and Kirillov]]} for help with understanding the computations in this book.)

See also

\begin{itemize}%
\item [[Kevin Walker]], \emph{On Witten's 3-Manifold Invariants}, (\href{http://canyon23.net/math/1991TQFTNotes.pdf}{old version} \href{http://canyon23.net/math/tc.pdf}{draft of new version})

\end{itemize}
Discussion that relates the [[geometric quantization]] of $G$-[[Chern-Simons theory]] to the Reshetikhin-Turaev construction of a 3d-[[TQFT]] from the [[modular tensor category]] induced by $G$ is in

\begin{itemize}%
\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}
and references cited there.

\begin{itemize}%
\item Alain Brugui\`e{}res, Alexis Virelizier, \emph{Hopf diagrams and quantum invariants}, \href{http://arxiv.org/abs/math/0505119}{math.QA/0505119}; \emph{Categorical centers and Reshetikhin-Turaev invariants}, \href{http://arxiv.org/abs/0812.2426}{arxiv/0812.2426}

\end{itemize}
The relation to the [[Crane-Yetter model]] was discussed in

\begin{itemize}%
\item [[John Barrett]], J. Garcia-Islas, [[João Faria Martins]], \emph{Observables in the Turaev-Viro and Crane-Yetter models}, J. Math. Phys. 48:093508, 2007 (\href{http://arxiv.org/abs/math/0411281}{arXiv:math/0411281})

\end{itemize}
[[!redirects Reshetikhin?Turaev construction]] [[!redirects Reshetikhin--Turaev construction]] [[!redirects ????????∞-∞????? construction]] [[!redirects ???????????????? construction]] [[!redirects ????????∞--∞????? construction]] [[!redirects Reshetikin-Turaev model]] [[!redirects Reshetikhin-Turaev model]] [[!redirects Reshetikhin?Turaev model]] [[!redirects Reshetikhin--Turaev model]] [[!redirects Reshetikin-Turaev theory]] [[!redirects Reshetikhin-Turaev theory]] [[!redirects ????????∞-∞????? model]] [[!redirects ???????????????? model]] [[!redirects ????????∞--∞????? model]]

[[!redirects Turaev-Reshetikhin TQFT]] [[!redirects Turaev-Reshetikhin TQFTs]]

[[!redirects Reshetikhin-Turaev invariant]] [[!redirects Reshetikhin-Turaev invariants]]

[[!redirects RT-invariant]] [[!redirects RT-invariants]]



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