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\begin{document}

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\section*{Dijkgraaf-Witten theory}

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

\hypertarget{chernsimons_theory}{}\paragraph*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory}

[[!include infinity-Chern-Simons theory - contents]]

\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{concise_survey_of_the_ingredients}{Concise survey of the ingredients}\dotfill \pageref*{concise_survey_of_the_ingredients} \linebreak
\noindent\hyperlink{gentle_exposition}{Gentle exposition}\dotfill \pageref*{gentle_exposition} \linebreak
\noindent\hyperlink{details_of_dwtheory_as_an_extended_tqft}{Details of DW-theory as an extended TQFT}\dotfill \pageref*{details_of_dwtheory_as_an_extended_tqft} \linebreak
\noindent\hyperlink{finite_group_cohomology}{Finite Group Cohomology}\dotfill \pageref*{finite_group_cohomology} \linebreak
\noindent\hyperlink{StatesInCodimensionk}{The $k$-vector spaces of states in codimension $k$}\dotfill \pageref*{StatesInCodimensionk} \linebreak
\noindent\hyperlink{relation_to_chernsimons_theory}{Relation to Chern-Simons theory}\dotfill \pageref*{relation_to_chernsimons_theory} \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{Dijkgraaf-Witten theory} in dimension $n$ is the [[TFT|topological]] [[sigma-model]] [[quantum field theory]] whose [[target space]] is the [[classifying space]] of a [[discrete group]] and whose [[background gauge field]] is a [[circle n-bundle with connection]] on $\mathbf{B}G$ (necessarily flat) which is equivalently a [[cocycle]] in the [[group cohomology]] of $G$ with coefficients in the [[circle group]].

Viewed in a broader context and generalizing: Dijkgraaf-Witten theory is the [[schreiber:∞-Chern-Simons theory]] induced from a [[characteristic class]] $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ on a [[discrete ∞-groupoid]] $\mathbf{B}G := Disc B G$. If $G$ here is an ordinary [[discrete group]] this is traditional Dijkgraaf-Witten theory, if $G$ is a discrete [[2-group]] and the background field is a circle 4-bundle, then this is called the [[Yetter model]].

This are the first two steps in filtering of target spaces by [[homotopy type]] [[truncated|truncation]] of [[schreiber:∞-Chern-Simons theory]] .

\hypertarget{concise_survey_of_the_ingredients}{}\subsubsection*{{Concise survey of the ingredients}}\label{concise_survey_of_the_ingredients}

We may think of this as describing the [[quantum mechanics]] of an $(n-1)$-[[brane]] with $n$-dimensional [[worldvolume]] $\Sigma$ propagating on $B G$ and being [[charge]]d under a [[gauge field|higher analog]] of the [[electromagnetic field]]:

a field configuration over $\Sigma$ (a $\Sigma$-shaped \emph{trajectory}) is a [[morphism]] $\phi : \Sigma \to \mathbf{B}G$, hence equivalently a $G$-[[principal bundle]] on $\Sigma$. The [[configuration space]] of fields over $\Sigma$ is the [[groupoid]] of $G$-[[principal bundle]]s over $\Sigma$.

The [[background gauge field]] is a [[morphism]] $\alpha : \mathbf{B}G \to \mathbf{B}^n U(1)$ -- hence a [[characteristic class]] for $G$: a [[cocycle]] of degree $n$ in the [[group cohomology]] of $G$.

The value of the [[Lagrangian]] $L(\phi)$ on a field configuration $\phi$ is the [[characteristic class]] of this bundle with respect to the universal characteristic class of the given [[principal infinity-bundle|circle n-bundle]]:

\begin{displaymath}
L : (\Sigma \stackrel{\phi}{\to} \mathbf{B}G)
  \mapsto
  (\alpha(\phi) : \Sigma \stackrel{\phi}{\to} \mathbf{B}G \stackrel{\alpha}{\to} \mathbf{B}^n U(1)).
\end{displaymath}
This is the [[classical field theory]] input of the model. The [[extended quantum field theory]] defined by this is supposed to be a rule that assigns space of [[state]]s to lower dimensional pieces of $\Sigma$ and to $n$-dimensional $\Sigma$s a propagator.

The space of [[state]]s assigned to a $\Sigma$ of [[dimension]] $n-k$ for $k \in \mathbb{N}$ is the [[k-groupoid]] of [[section]]s of the higher line bundle [[associated infinity-bundle|associated]] to the [[circle n-bundle|circle (n-k)-bundle]] $\tau_\Sigma \alpha$ obtained by [[transgression]] of $\alpha$ to the [[mapping space]] $\mathbf{H}(\Sigma, \mathbf{B}G)$.

The propagator on $\Sigma$ of dimension $n$ is given by the [[path integral]] computed with [[measure]] the [[groupoid cardinality]] of $\mathbf{B}G$ and [[integral kernel]] given by the [[action functional]]

\begin{displaymath}
\exp(i S(-)) : G Bund(\Sigma) \to U(1)
\end{displaymath}
that sends a field $\phi$ to the evaluation of $\alpha(\phi)$ on the [[fundamental class]] of $\Sigma$

\begin{displaymath}
\exp(i S(\phi)) = \int_\Sigma \alpha(\phi)
  \,.
\end{displaymath}
\hypertarget{gentle_exposition}{}\subsubsection*{{Gentle exposition}}\label{gentle_exposition}

(\ldots{})

\hypertarget{details_of_dwtheory_as_an_extended_tqft}{}\subsection*{{Details of DW-theory as an extended TQFT}}\label{details_of_dwtheory_as_an_extended_tqft}

The Dijkgraaf-Witten model is an example of (fully) [[extended topological quantum field theory]]. Namely, the above data not only assign an element in $U(1)$ to any closed $n$-dimensional manifold, but also a vector space to any closed $(n-1)$-dimensional manifold, a [[2-vector space]] to any closed $(n-2)$ manifold, and so on, ending with an [[n-vector space]] assigned to the point. Also, manifolds with boundary corresponds to (higher) linear operators between these (higher) vector spaces. According to the [[cobordism hypothesis]], the whole structure of the Dijkgraaf-Witten model as an fully extended TQFT is contained in the datum of the $n$-Vector space it assigns to the point. This is the space of [[section|sections]] of the flat $n$-vector bundle $\mathbf{B}G\to n Vect$ induced by the background field $\mathbf{B}G\to \mathbf{B}^n U(1)$.

\hypertarget{finite_group_cohomology}{}\subsubsection*{{Finite Group Cohomology}}\label{finite_group_cohomology}

Since the target space of Dijkgraaf-Witten theory is the [[delooping]] groupoid $\mathbf{B}G$ of a [[group]] $G$ ([[internalization|internal]] to [[Set]]), any background field given by a morphism $\alpha : \mathbf{B}G \to A$ in [[∞Grpd]] is a [[cocycle]] in the [[group cohomology]] of $G$, as described there.

Here we have a finite (or discrete) group $G$, and a discrete abelian group $A$, and we want to define $H^n(G;A)$. A way of doing this is to realize everything topologically: from $G$ we build the classifying space $\mathcal{B}G$, and from $A$ the [[Eilenberg-MacLane space]] $\mathcal{B}^n A=K(A,n)$. Then we consider the space of maps $hom(\mathcal{B}G,\mathcal{B}^n A)$ (these are our [[cocycle]]s) and take its $\pi_0$.

This way we have a familiar description, in a certain sense (topological spaces, continuous maps, homotopies,..), of the set $H^n(G;A)$. The drawback is that the topological spaces involved here are ``gigantic'' (infinite dimensional [[CW-complex]]es), where we had started with a very ``little'' datum: a finite group. So one can wonder if there is a finite model for the above construction, and the homotopy hypothesis serves it on a silver plate. Namely, since $G$ is discrete, $\mathcal{B}G$ is a [[homotopy n-type|1-type]], and nothing but the [[geometric realization]] of the [[delooping]] [[groupoid]] $\mathbf{B}G$ (boldface $B$ here); similarly $\mathcal{B}^n A$ is the topological geometric realization of the $n$-groupoid $\mathbf{B}^n A$, and the space of cocycles is $hom(B G,B^n A)$. since $G$ is a finite group, $B G$ is a finite groupoid, and so $hom(B G,B^n A)$ is a finite set. This set is the finite model for $hom(\mathcal{B}G,\mathcal{B}^n A)$ we were looking for.

To be continued\ldots{}

\hypertarget{StatesInCodimensionk}{}\subsubsection*{{The $k$-vector spaces of states in codimension $k$}}\label{StatesInCodimensionk}

The [[n-vector space|k-vector space]] associated with a closed oriented $(n-k)$-dimensional manifold $X_{n-k}$ admits a simple description in terms of [[section]]s:

The background field $\alpha : \mathbf{B}G \to A$ is [[transgression|transgressed]] to the mapping space $[\Pi(X_{n-k}), \mathbf{B}G]$ by forming the [[internal hom]]

\begin{displaymath}
[\Pi(X_{n-k}), \mathbf{B}G]
  \stackrel{[\Pi(X_{n-k}), \alpha]}{\to}
  [\Pi(X_{n-k}), A]
  \stackrel{\tau_{\leq k}}{\to}
  \tau_{\leq k} [\Pi(X_{n-k}), A]
  \,,
\end{displaymath}
where the last morphism is the projection on the [[truncated|k-truncation]]. This defines a [[cocycle]] on the \emph{space of fields} $[\Pi(X_{n-k}), \mathbf{B}G]$ over $X_{n-k}$, which classifies some [[principal ∞-bundle]] on this space. Given a canonical [[representation]] of the \emph{spaces of phases} $\tau_k [\Pi(X_{n-k}), A]$ on a [[n-vector space|k-vector space]] we obtain the corresponding [[associated bundle]] over the space of fields. The $(k-1)$-category assigned by the [[extended topological quantum field theory]] to the closed $X_{n-k}$ is the category of sections of this $k$-vector bundle.

\begin{prop}
\label{}\hypertarget{}{}
We have

\begin{displaymath}
\tau_k [\Pi(X_{n-k}), \mathbf{B}^n U(1)]
  \simeq
  \mathbf{B}^k U(1)
\end{displaymath}
\end{prop}
\begin{proof}
By general abstract reasoning (recalled at [[cohomology]] and [[fiber sequence]], for instance) we have for the [[homotopy group]]s that

\begin{displaymath}
\pi_i[\Pi(X_{n-k}),\mathbf{B}^n U(1)]
  \simeq 
  H^{n-i}(X_{n-k}, U(1))
  \,.
\end{displaymath}
Now use the [[universal coefficient theorem]], which asserts that we have an [[exact sequence]]

\begin{displaymath}
0
  \to 
  Ext^1(H_{n-i-1}(X_{n-k},\mathbb{Z}),U(1))
  \to 
  H^{n-i}(X_{n-k},U(1))
  \to 
  Hom(H_{n-i}(X_{n-k},\mathbb{Z}),U(1))
  \to 0
  \,.
\end{displaymath}
Since $U(1)$ is an [[injective object|injective]] $\mathbb{Z}$-[[module]] we have

\begin{displaymath}
Ext^1(-,U(1))=0
  \,.
\end{displaymath}
Together this means that we have an [[isomorphism]]

\begin{displaymath}
H^{n-i}(X_{n-k},U(1))
  \simeq 
  Hom_{Ab}(H_{n-i}(X_{n-k},\mathbb{Z}),U(1))
\end{displaymath}
that identifies the [[cohomology group]] in question with the [[internal hom]] in [[Ab]] from the integral [[homology]] group of $X_{n-k}$ to $U(1)$.

For $i\lt k$, the right hand side is zero, and so

\begin{displaymath}
\pi_i[\Pi(X_{n-k}),\mathbf{B}^n U(1)]=0 \;\;\;\;
  for i\lt k
  \,.
\end{displaymath}
For $i=k$, instead, $H_{n-i}(X_{n-k},\mathbb{Z})\simeq \mathbb{Z}$, since $X_{n-k}$ is a closed $(n-k)$-manifold and so

\begin{displaymath}
\pi_k[\Pi(X_{n-k}),\mathbf{B}^n U(1)]\simeq U(1)
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Another proof of the isomorphism $H^{n-k}(X_{n-k},U(1))\cong U(1)$ and of the identities $H^{n-i}(X_{n-k},U(1))=0$ for $i\lt k$ can be obtained as follows. Consider the short exact sequence of locally constant sheaves of abelian groups

\begin{displaymath}
0\to \mathbb{Z}\to \mathbb{R}\to U(1)\to 1.
\end{displaymath}
This induces a long exact sequence in cohomology

\begin{displaymath}
\cdots \to H^{n-i-1}(X_{n-k},U(1))\to H^{n-i}(X_{n-k},\mathbb{Z})\to H^{n-i}(X_{n-k},\mathbb{R}) \to H^{n-i}(X_{n-k},U(1))\to H^{n-i+1}(X_{n-k},\mathbb{Z})\to \cdots
\end{displaymath}
For $i\lt k$ we have $H^{n-i}(X_{n-k},U(1))=0$ by dimensional reasons, while for $i=k$ we find the exact sequence

\begin{displaymath}
\cdots \to H^{n-k}(X_{n-k},\mathbb{Z})\to H^{n-k}(X_{n-k},\mathbb{R}) \to H^{n-k}(X_{n-k},U(1))\to 0.
\end{displaymath}
Since $X_{n-k}$ is a closed oriented manifold, we have $H^{n-k}(X_{n-k},\mathbb{Z})=\mathbb{Z}$, $H^{n-k}(X_{n-k},\mathbb{R})=\mathbb{R}$, and the map $H^{n-k}(X_{n-k},\mathbb{Z})\to H^{n-k}(X_{n-k},\mathbb{R})$ is the inclusion of $\mathbb{Z}$ into $\mathbb{R}$. Hence $H^{n-k}(X_{n-k},U(1))\cong \mathbb{R}/\mathbb{Z}\cong U(1)$.

\end{prop}
This means that the [[transgression]] of the Dijkgraaf-Witten background field

\begin{displaymath}
\alpha : \mathbf{B}G \to \mathbf{B}^n U(1)
\end{displaymath}
to the \emph{space of field configurations} $[\Pi(X_{n-k}), \mathbf{B}G]$ over $X_{n-k}$ is a [[cocycle]] of the form

\begin{displaymath}
[\Pi(X_{n-k}), \alpha] : [\Pi(X_{n-k}), \mathbf{B}G] \to \mathbf{B}^k U(1)
  \,.
\end{displaymath}
This classifies a $\mathbf{B}^{k-1} U(1)$-[[principal ∞-bundle]] $P$ over the space of field configurations, given by the [[pullback]]

\begin{displaymath}
\itexarray{
    P &\to & \mathbf{E} \mathbf{B}^{k-1} U(1)
    \\
    \downarrow && \downarrow
     \\
     [\Pi(X_{n-k}), \mathbf{B}G] &\stackrel{[\Pi(X_{n-k}), \rho]}{\to}& \mathbf{B}^k U(1)
  }
  \,.
\end{displaymath}
(Here $\mathbf{E} \mathbf{B}^{k-1} U(1)$ is as described at [[universal principal ∞-bundle]].)

By the canonical $k$-[[representation]] $\rho : \mathbf{B}^k U(1) \to k Vect_{\mathbb{C}}$ of $\mathbf{B}^{k-1}U(1)$ on complex [[n-vector space|k-vector space]]s, we have [[associated bundle|associated]] to this canonically a $k$-vector bundle $E$, which may be realized as the pullback

\begin{displaymath}
\itexarray{
    E &\to & k Vect_*
    \\
    \downarrow && \downarrow
     \\
     [\Pi(X_{n-k}), \mathbf{B}G] &\stackrel{\rho \circ [\Pi(X_{n-k}), \rho]}{\to}& k Vect
  }
  \,.
\end{displaymath}
Here $k Vect_*$ is the [[n-category|k-category]] of pointed $k$-vector bundles, see again [[generalized universal bundle]] for more.

If $X_{n-k}$ is closed then the $k$-vector spaces associated by the TFT to $X_{n-k}$ is the [[n-category|(k-1)-category]] of [[section]]s of this bundle $E$.

\ldots{}

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

Dijkgraaf-Witten theory is to be thought of as the finite group version of [[Chern-Simons theory]]. Chern-Simons theory looks formally just as the above, only that all finite $n$-groupoids appearing here are replaced by [[Lie ∞-groupoid]]s ([[∞-stack]]s on [[CartSp]]).

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

\begin{itemize}%
\item [[schreiber:∞-Chern-Simons theory]]


\item [[higher dimensional Chern-Simons theory]]

\begin{itemize}%
\item [[1d Chern-Simons theory]]

\begin{itemize}%
\item [[1d Dijkgraaf-Witten theory]]

\end{itemize}

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


\item [[3d Chern-Simons theory]]

\begin{itemize}%
\item \textbf{Dijkgraaf-Witten theory}


\item [[2d Wess-Zumino-Witten theory]]



\end{itemize}

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


\item [[5d Chern-Simons theory]]


\item [[6d Chern-Simons theory]]


\item [[7d Chern-Simons theory]]


\item [[infinite-dimensional Chern-Simons theory]]


\item [[AKSZ sigma-model]]



\end{itemize}


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

The idea originates, of course, in

\begin{itemize}%
\item [[Robbert Dijkgraaf]], [[Edward Witten]], \emph{[[DW.pdf:file]]}, Commun. Math. Phys. \textbf{129} (1990), 393,

\end{itemize}
The discussion of the [[quasi-Hopf algebra]] associated with a [[group cohomology]] 3-[[cocycle]] $c \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$ originates in

\begin{itemize}%
\item [[Robbert Dijkgraaf]], V. Pasquier, P. Roche, \emph{QuasiHopf algebras, group cohomology and orbifold models}, Nucl. Phys. B Proc. Suppl. \textbf{18B} (1990), 60-72; \emph{Quasi-quantum groups related to orbifold models}, Modern quantum field theory (Bombay, 1990), 375--383, World Sci. 1991

\end{itemize}
A review is in

\begin{itemize}%
\item A. Coste, J-M. Maillard, \emph{Representation Theory of Twisted Group Double}, Annales Fond.Broglie 29 (2004) 681-694, (\href{http://arxiv.org/abs/hep-th/0309257}{arXiv:hep-th/0309257})

\end{itemize}
and conceptual clarifications were established in

\begin{itemize}%
\item [[Simon Willerton]], \emph{The twisted Drinfeld double of a finite group via gerbes and finite groupoids}, \href{http://arxiv.org/abs/math.QA/0503266}{arXiv/math.QA/0503266}

\end{itemize}
and, earlier, in an unpublished manuscript of [[Paul Bressler]] (2002-2004). See at \emph{[[Drinfeld double]]} for more on this.

A first comprehensive structural account of DW theory as a [[FQFT|functorial QFT]] was given in

\begin{itemize}%
\item [[Daniel Freed]], [[Frank Quinn]], \emph{Chern-Simons theory with finite gauge group} Commun.Math.Phys. \textbf{156}:435-472, 1993, (\href{http://de.arxiv.org/abs/hep-th/9111004}{arXiv:hep-th/9111004})

\end{itemize}
A review is given on p. 68 of

\begin{itemize}%
\item [[Bruce Bartlett]], \emph{Categorical aspects of topological quantum field theory}, \href{http://de.arxiv.org/abs/math.QA/0512103}{arXiv/math.QA/0512103}

\end{itemize}
First steps towards understand DW theory as an [[extended TQFT]] appear in

\begin{itemize}%
\item [[Daniel Freed]], \emph{Higher Algebraic Structures and Quantization} Commun.Math.Phys. 159 (1994) 343-398 (\href{http://arxiv.org/abs/arXiv:hep-th/9212115}{arXiv:arXiv:hep-th/9212115})

\end{itemize}
Discussion aiming towards a refinement of DW theory to an [[extended TQFT]] is in

\begin{itemize}%
\item Kevin Wray, \emph{Extended topological gauge theories in codimension 0 and higher} (\href{http://math.berkeley.edu/~kwray/papers/thesis.pdf}{pdf})

\end{itemize}
Further conceptual refinement of this is indicated in section 3 and section 8 of

\begin{itemize}%
\item [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]} (2010)

\end{itemize}
This proposes a general abstract way to construct [[path integral]] quantizations for finite group theories such as DW, see also at \emph{[[prequantum field theory]]}. More along these lines is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Finiteness and ambidexterity in $K(n)$-local stable homotopy theory}, talk at \emph{Notre Dame Graduate Summer School on Topology and Field Theories} and \emph{Harvard lecture} 2012 (\href{http://www.youtube.com/watch?v=eQayYLDw1VA}{video part 1}, \href{http://www.youtube.com/watch?v=OEShrQyvmS4}{part 2}, \href{http://www.youtube.com/watch?v=nOIcdn1iUR4}{part 3} \href{http://www.youtube.com/watch?v=ZwnClYedaYM}{part 4}, \href{http://www.math.northwestern.edu/~celliott/notre_dame_notes/Lurie_notes.pdf}{pdf lecture notes} by Chris Elliott)

\end{itemize}
See also

\begin{itemize}%
\item [[Thomas Nikolaus]], \emph{Equivariant Dijkgraaf-Witten theory}, a talk at Muenster (\href{http://www.math.uni-hamburg.de/home/nikolaus/Muenster.pdf}{pdf})

\end{itemize}
For more on this see the discussion on the \href{http://nforum.mathforge.org/discussion/1046/path-category-vs-cobordisms-for-bundles/?Focus=8337#Comment_8337}{n-Forum}.

[[!redirects Dijkgraaf--Witten theory]] [[!redirects Dijkgraaf?Witten theory]]

[[!redirects Dijkgraaf-Witten model]] [[!redirects Dijkgraaf-Witten models]]



\end{document}