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\section*{extended topological quantum field theory}

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

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_category_of_extended_cobordisms}{The category of extended cobordisms}\dotfill \pageref*{the_category_of_extended_cobordisms} \linebreak
\noindent\hyperlink{extended_tqft}{Extended TQFT}\dotfill \pageref*{extended_tqft} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{classes_of_examples_by_dimension}{Classes of examples by dimension}\dotfill \pageref*{classes_of_examples_by_dimension} \linebreak
\noindent\hyperlink{generic_examples}{Generic examples}\dotfill \pageref*{generic_examples} \linebreak
\noindent\hyperlink{specific_examples}{Specific examples}\dotfill \pageref*{specific_examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{construction_of_etqfts}{Construction of ETQFT's}\dotfill \pageref*{construction_of_etqfts} \linebreak
\noindent\hyperlink{classification_of_etqfts}{Classification of ETQFT's}\dotfill \pageref*{classification_of_etqfts} \linebreak
\noindent\hyperlink{relation_of_etqft_to_aqft}{Relation of ETQFT to AQFT}\dotfill \pageref*{relation_of_etqft_to_aqft} \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}

\emph{Extended quantum field theory} (or \emph{multi-tiered quantum field theory}) is the fully [[local quantum field theory|local]] formulation of [[functorial quantum field theory]], formulated in [[higher category theory]]

Whereas a

\begin{itemize}%
\item [[category theory|1-categorical]] [[TQFT]] may be regarded as a rule that allows one to compute topological invariants $Z(\Sigma)$ assigned to $d$-dimensional [[manifold]]s by cutting these manifolds into a sequence $\{\Sigma_i\}$ of $d$-dimensional composable [[cobordism]]s with $(d-1)$-dimensional boundaries $\partial \Sigma_i$, e.g. $\Sigma = \Sigma_2 \coprod_{\partial \Sigma_1 = \partial \Sigma_2} \Sigma_1$, then assigning quantities $Z(\Sigma_i)$ to each of these and then composing these quantities in some way, e.g. as $Z(\Sigma) = Z(\Sigma_2)\circ Z(\Sigma_1)$;

\end{itemize}
we have that

\begin{itemize}%
\item in extended [[TQFT]] $Z(\Sigma)$ may be computed by decomposing $\Sigma$ into $d$-dimensional pieces with piecewise smooth boundaries, whose boundary strata are of arbitrary codimension $k$.

\end{itemize}
For that reason extended QFT is also sometimes called \textbf{local} or \textbf{localized} QFT. In fact, the notion of locality in [[quantum field theory]] is precisely this notion of locality. And, as also discussed at [[FQFT]], this higher dimensional version of locality is naturally encoded in terms of [[higher category theory|n-functoriality]] of $Z$ regarded as a functor on a [[higher category theory|higher category]] of [[cobordism]]s.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{the_category_of_extended_cobordisms}{}\subsubsection*{{The category of extended cobordisms}}\label{the_category_of_extended_cobordisms}

The definition of a $j$-cobordism is recursive. A $(j+1)$-cobordism between $j$-cobordisms is a [[compact space|compact]] [[orientation|oriented]] $(j+1)$-dimensional [[smooth manifold]] with corners whose the boundary is the [[disjoint union]] of the target $j$-cobordism and the orientation reversal of the source $j$-cobordism. (The base case of the recursion is the [[empty set]], thought of as a $(-1)$-dimensional manifold.)

$n Cob_d$ is an $n$-category with smooth compact oriented $(d-n)$-manifolds as objects and cobordisms of cobordisms up to $n$-cobordisms, up to diffeomorphism, as morphisms.

There are various suggestions with more or less detail for a precise definition of a higher category $n Cob_n$ of fully extended $n$-dimensional cobordisms.

A very general (and very natural) one consists in taking a further step in categorification: one takes $n$-cobordisms as $n$-morphisms and smooth homotopy classes of diffeomorphisms beween them as $(n+1)$-morphisms. Next one iterates this; see details at [[(∞,n)-category of cobordisms]].

See

\begin{itemize}%
\item [[extended cobordism]].

\end{itemize}
\hypertarget{extended_tqft}{}\subsubsection*{{Extended TQFT}}\label{extended_tqft}

Fix a [[base ring]] $R$, and let $C$ be the [[symmetric monoidal category|symmetric monoidal]] $n$-category of $n$-$R$-modules.

An $n$-extended $C$-valued TQFT of dimension $d$ is a symmetric $n$-tensor functor $Z: n Cob_d \rightarrow C$ that maps

\begin{itemize}%
\item smooth compact oriented $d$-manifolds to elements of $R$


\item smooth compact oriented $(d-1)$-manifolds to $R$-modules


\item cobordisms of smooth compact oriented $(d-1)$-manifolds to $R$-linear maps between $R$-modules


\item smooth compact oriented $(d-2)$-manifolds to $R$-linear [[additive categories]]


\item cobordisms of smooth compact oriented $(d-2)$-manifolds to functors between $R$-linear categories


\item etc \ldots{}


\item smooth compact oriented $(d-n)$-manifolds to $R$-linear $(n-1)$-categories


\item cobordisms of smooth compact oriented $(d-n)$-manifolds to $(n-1)$-functors between $R$-linear $(n-1)$-categories



\end{itemize}
with compatibility conditions and gluing formulas that must be satisfied\ldots{} For instance, since the functor $Z$ is required to be monoidal, it sends monoidal units to monoidal units. Therefore, the $d$-dimensional [[vacuum]] is mapped to the unit element of $R$, the $(d-1)$-dimensional vacuum to the $R$-module $R$, the $(d-2)$-dimensional vacuum to the category of $R$-modules, etc.

Here $n$ can range between $0$ and $d$. This generalizes to an arbitrary symmetric monoidal category $C$ as codomain category.

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

\hypertarget{classes_of_examples_by_dimension}{}\subsubsection*{{Classes of examples by dimension}}\label{classes_of_examples_by_dimension}

$n=1$ gives ordinary [[TQFT]].

The most common case is when $R = \mathbb{C}$ (the [[complex numbers]]), giving unitary ETQFT.

The most common cases for $C$ are

\begin{itemize}%
\item $C = n Hilb(R)$, the category of $n$-[[n-Hilbert space|Hilbert spaces]] over a topological field $R$. As far as we know this is only defined up to $n=2$.


\item $C = n Vect(R)$, the category of $n$-[[n-vector space|vector spaces]] over a field $R$.


\item $C = n Mod(R)$, the (conjectured?) category of $n$-[[n-module|modules]] over a commutative ring $R$.



\end{itemize}
3d: [[Turaev-Viro model]]

\hypertarget{generic_examples}{}\subsubsection*{{Generic examples}}\label{generic_examples}

By the [[cobordism hypothesis]]-theorem every [[fully dualizable object]] in a symmetric monoidal $(\infty,n)$-category with duals provides an example.

\hypertarget{specific_examples}{}\subsubsection*{{Specific examples}}\label{specific_examples}

\begin{itemize}%
\item [[Levin-Wen model]]


\item \href{Topological+Quantum+Field+Theories+from+Compact+Lie+Groups#3dCSFullyExtended}{Chern-Simons theory as a fully extended TQFT}


\item \ldots{}many more\ldots{}



\end{itemize}
See also at \emph{[[TCFT]]}.

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

\hypertarget{construction_of_etqfts}{}\subsubsection*{{Construction of ETQFT's}}\label{construction_of_etqfts}

\begin{itemize}%
\item By generators and relations


\item By path integrals (this is Daniel Freed's approach)


\item By modular tensor n-categories?



\end{itemize}
\hypertarget{classification_of_etqfts}{}\subsubsection*{{Classification of ETQFT's}}\label{classification_of_etqfts}

Assume $Z: n Cob_d \rightarrow n Vect(R)$ is an extended TQFT. Since $Z$ maps the $(d-1)$-dimensional vacuum to $R$ as an $R$-vector space, by functoriality $Z$ is forced to map a $d$-dimensional closed manifold to an element of $R$. Iterating this argument, one is naturally led to conjecture that, under the correct categorical hypothesis, the behaviour of $Z$ is enterely determined by its behaviour on $(d-n)$-dimensional manifolds. See details at [[cobordism hypothesis]].

\hypertarget{relation_of_etqft_to_aqft}{}\subsubsection*{{Relation of ETQFT to AQFT}}\label{relation_of_etqft_to_aqft}

See

\begin{itemize}%
\item [[topological chiral homology]].

\end{itemize}
also

\begin{itemize}%
\item \emph{\href{http://arxiv.org/abs/0806.1079}{AQFT from n-functorial QFT}}.

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

\begin{itemize}%
\item [[local quantum field theory]]


\item [[extended Lagrangian]]



\end{itemize}
More on extended QFTs is also at

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

\end{itemize}
[[!include Isbell duality - table]]

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

\begin{itemize}%
\item [[Dan Freed]], Remarks on Chern-Simons theory


\item Daniel Freed, Quantum Groups from Path Integrals. \href{http://arxiv.org/abs/q-alg/9501025}{arXiv}


\item Daniel Freed, Higher Algebraic Structures and Quantization. \href{http://arxiv.org/abs/hep-th/9212115}{arXiv}


\item [[John Baez]] and [[James Dolan]], Higher-dimensional Algebra and Topological Quantum Field Theory. \href{http://arxiv.org/abs/q-alg/9503002}{arXiv}


\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}. \href{http://arxiv.org/abs/0905.0465}{arXiv}



\end{itemize}
With an eye towards the full extension of [[Chern-Simons theory]]:

\begin{itemize}%
\item [[Dan Freed]], \emph{Remarks on Fully Extended 3-Dimensional Topological Field Theories} (2011) (\href{http://www.ma.utexas.edu/users/dafr/stringsmath_np.pdf}{pdf})


\item [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]} , in P. R. Kotiuga (ed.) \emph{A celebration of the mathematical legacy of Raoul Bott} AMS (2010) (\href{http://arxiv.org/abs/0905.0731}{arXiv})


\item [[Dan Freed]], \emph{[[4-3-2 8-7-6]]}, talk at \emph{\href{https://people.maths.ox.ac.uk/tillmann/ASPECTS.html}{ASPECTS of Topology}} Dec 2012



\end{itemize}
For TQFTs appearing in [[solid state physics]] in the context of [[topological order]]:

\begin{itemize}%
\item [[Daniel Freed]], [[Gregory Moore]], \emph{Twisted equivariant matter}, \href{http://arxiv.org/abs/1208.5055}{arxiv/1208.5055} (uses [[equivariant K-theory]] to classify free fermion gapped phases with symmetry)


\item [[Daniel Freed]], \emph{Short-range entanglement and invertible field theories} (\href{http://arxiv.org/abs/1406.7278}{arXiv:1406.7278})



\end{itemize}
[[!redirects EQFT]] [[!redirects extended TQFT]]

[[!redirects extended quantum field theory]] [[!redirects extended QFT]]

[[!redirects extended quantum field theories]] [[!redirects extended topological quantum field theories]] [[!redirects extended QFTs]]

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

[[!redirects multi-tiered field theory]] [[!redirects multi-tiered field theories]]

[[!redirects multi-tiered quantum field theory]] [[!redirects multi-tiered quantum field theories]]

[[!redirects extended TQFTs]]

[[!redirects extended TFT]] [[!redirects extended TFTs]]



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