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\section*{topological quantum field 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{nontopological_qfts}{Non-topological QFTs}\dotfill \pageref*{nontopological_qfts} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{homotopy_qfts}{Homotopy QFTs}\dotfill \pageref*{homotopy_qfts} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{OriginInPhysics}{Origin in physics}\dotfill \pageref*{OriginInPhysics} \linebreak
\noindent\hyperlink{global_1functorial_tqft}{Global (1-functorial) TQFT}\dotfill \pageref*{global_1functorial_tqft} \linebreak
\noindent\hyperlink{local_functorial_tqft}{Local ($n$-functorial) TQFT}\dotfill \pageref*{local_functorial_tqft} \linebreak


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

A \textbf{topological quantum field theory} is a [[quantum field theory]] which -- as a [[FQFT|functorial quantum field theory]] -- is a functor on a flavor of the [[(∞,n)-category of cobordisms]] $Bord_n^S$, where the [[n-morphism]]s are [[cobordism]]s without any non-topological further structure $S$ -- for instance no [[Riemannian metric]] structure -- but possibly some ``topological structure'', such as [[Spin structure]] or similar.

For more on the general idea and its development, see [[FQFT]] and [[extended topological quantum field theory]].

\begin{remark}
\label{}\hypertarget{}{}
Often topological \emph{quantum} field theories are just called \emph{topological field theories} and accordingly the abbreviation TQFT is reduced to TFT. Strictly speaking this is a misnomer, which is however convenient and very common. It should be noted, however, that TQFTs may have classical counterparts which would better deserve to be called TFTs. But they are not usually.

\end{remark}
\hypertarget{nontopological_qfts}{}\subsection*{{Non-topological QFTs}}\label{nontopological_qfts}

In contrast to topological QFTs, non-topological quantum field theories in the [[FQFT]] description are $n$-functors on $n$-categories $Bord^S_n$ whose morphisms are manifolds with extra $S$-structure, for instance

\begin{itemize}%
\item $S =$ conformal structure $\to$ [[conformal field theory]]


\item $S =$ [[Riemannian manifold|Riemannian structure]] $\to$ ``euclidean QFT''


\item $S =$ [[pseudo-Riemannian metric|pseudo-Riemannian structure]] $\to$ ``relativistic QFT''



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

\begin{itemize}%
\item [[2d TQFT]]

\begin{itemize}%
\item [[TCFT]]


\item [[2d Chern-Simons theory]]



\end{itemize}

\item [[3d TQFT]]

\begin{itemize}%
\item [[Dijkgraaf-Witten theory]]


\item [[Chern-Simons theory]]


\item [[Reshetikhin-Turaev model]]


\item [[Turaev-Viro model]]



\end{itemize}

\item [[4d TQFT]]

\begin{itemize}%
\item [[topological Yang-Mills theory]]


\item [[topologically twisted D=4 super Yang-Mills theory ]]


\item [[4d Chern-Simons theory]]


\item [[Yetter model]]



\end{itemize}


\end{itemize}
\hypertarget{homotopy_qfts}{}\subsection*{{Homotopy QFTs}}\label{homotopy_qfts}

These somehow lie between the previous two types. There is some simple extra structure in the form of a `characteristic map' from the manifolds and bordisms to a `background space' $X$. In many of the simplest examples, this is taken to be the [[classifying space]] of a group, but this is not the only case that can be considered. The topic is explored more fully in [[HQFT]].

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

\begin{itemize}%
\item [[cohomological field theory]], [[TCFT]]


\item [[FQFT]], [[extended topological quantum field theory]]


\item [[topological quantum computation]]



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

See also the references at [[2d TQFT]], [[3d TQFT]] and [[4d TQFT]].

\hypertarget{OriginInPhysics}{}\subsubsection*{{Origin in physics}}\label{OriginInPhysics}

The concept originates in the guise of [[cohomological quantum field theory]] motivated from TQFTs appearing in [[string theory]] in

\begin{itemize}%
\item [[Edward Witten]], \emph{Topological quantum field theory}, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386. (\href{http://projecteuclid.org/euclid.cmp/1104161738}{euclid:1104161738})


\item [[Edward Witten]], \emph{Introduction to cohomological field theory}, International Journal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])


\item Stefan Cordes, [[Gregory Moore]], Sanjaye Ramgoolam, \emph{Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories}, Nucl. Phys. Proc. Suppl.41:184-244,1995 (\href{http://arxiv.org/abs/hep-th/9411210}{arXiv:hep-th/9411210})



\end{itemize}
and in the discussion of [[Chern-Simons theory]] (``Schwarz-type TQFT'') in

\begin{itemize}%
\item [[Edward Witten]], \emph{Quantum Field Theory and the Jones Polynomial} Commun. Math. Phys. 121 (3) (1989) 351--399. MR0990772 (\href{http://projecteuclid.org/euclid.cmp/1104178138}{project EUCLID})

\end{itemize}
\hypertarget{global_1functorial_tqft}{}\subsubsection*{{Global (1-functorial) TQFT}}\label{global_1functorial_tqft}

The [[FQFT]]-[[axioms]] for global (i.e. 1-functorial) TQFTs are due to

\begin{itemize}%
\item [[Michael Atiyah]], \emph{Topological quantum field theories}, Publications Math\'e{}matiques de l'IH\'E{}S 68 (68): 175--186, (1989) (\href{http://www.numdam.org/item?id=PMIHES_1988__68__175_0}{Numdam})

\end{itemize}
Exposition of the conceptual ingrediants includes

\begin{itemize}%
\item [[John Baez]], \emph{Quantum Quandaries: a Category-Theoretic Perspective} (\href{http://arxiv.org/abs/quant-ph/0404040}{arXiv:quant-ph/0404040})

\end{itemize}
and more technical lecture notes include

\begin{itemize}%
\item [[Daniel Freed]], \emph{Lectures on topological quantum field theory}, 1992 (\href{http://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf}{pdf})


\item [[Kevin Walker]], \emph{TQFTs}, 2006 (\href{http://canyon23.net/math/tc.pdf}{pdf})



\end{itemize}
An introduction specifically to [[2d TQFTs]] is in

\begin{itemize}%
\item [[Joachim Kock]], \emph{Frobenius algebras and 2D topological quantum field theories}, No. 59 of LMSST, Cambridge University Press, 2003., (full information \href{http://mat.uab.es/~kock/TQFT.html}{here}).

\end{itemize}
See also the references at \emph{[[HQFT]]}.

\hypertarget{local_functorial_tqft}{}\subsubsection*{{Local ($n$-functorial) TQFT}}\label{local_functorial_tqft}

The local [[FQFT]] formulation (i.e. [[n-functor|n-functorial]]) together with the [[cobordism hypothesis]] was suggested in

\begin{itemize}%
\item [[John Baez]], [[James Dolan]], \emph{Higher dimensional algebra and Topological Quantum Field Theory} J.Math.Phys. 36 (1995) 6073-6105 (\href{http://arxiv.org/abs/q-alg/9503002}{arXiv:q-alg/9503002})

\end{itemize}
and formalized and proven in

\begin{itemize}%
\item [[Jacob Lurie]], [[On the Classification of Topological Field Theories]]; \emph{TQFT and the Cobordism Hypothesis} (\href{http://www.ma.utexas.edu/video/dafr/lurie/}{video}, \href{http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/perspectives_TQFT_notes.html}{notes})

\end{itemize}
This also shows how [[TCFT]] fits in, which formalizes the original proposal of 2d [[cohomological quantum field theory]].

Lecture notes include

\begin{itemize}%
\item [[Constantin Teleman]], \emph{Five lectures on topological field theory}, 2014 (\href{http://math.berkeley.edu/~teleman/math/barclect.pdf}{pdf})

\end{itemize}
A discussion amplifying the aspects of [[higher category theory]] is in

\begin{itemize}%
\item [[Anton Kapustin]], \emph{Topological field theory, higher categories, and their applications}, survey for ICM 2010, (\href{http://arxiv.org/abs/1004.2307}{arxiv/1004.2307})

\end{itemize}
See also

\begin{itemize}%
\item Mark Feshbach, Alexander A. Voronov, \emph{A higher category of cobordisms and topological quantum field theory}, \href{http://arxiv.org/abs/1108.3349}{arxiv/1108.3349}

\end{itemize}
Indication of local [[quantization]] in the context of [[infinity-Dijkgraaf-Witten theory]] is in

\begin{itemize}%
\item [[Daniel Freed]], [[Michael Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]}

\end{itemize}
[[!redirects topological quantum field theories]] [[!redirects TQFT]] [[!redirects TQFTs]] [[!redirects TFT]] [[!redirects TFTs]]

[[!redirects topological field theory]] [[!redirects topological field theories]]



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