Contents

Idea

Dijkgraaf-Witten theory in dimension $n$ is the topological sigma-model quantum field theory whose

• target space is the groupoid $BG=\{•⟲g\mid g\in G\}$ obtained by delooping from a finite group $G$ ;

• background field is an $n$-functor $\alpha :BG\to B^{n}U(1)$

  • this is the same thing as a $U(1)$-valued group n-cocycle $c$ on $G$;

• or rather the background field is the associated functor $BG\to B^{n}U(1)\to n\mathrm{Vect}$ for the canonical representation of $B^{n}U(1)$ on n-vector spaces

• parameter spaces $\Sigma =\Pi (X)$ are skeleta of the fundamental ∞-groupoids of $n$-dimensional manifolds $X$.

Therefore

• a field configuration $\Sigma =\Pi (X)\stackrel{\varphi }{\to }BG$ is a $G$-principal bundle on $X$ (recall that $G$ is assumed to be a finite group);

• the action $\Pi (X)\stackrel{\varphi }{\to }BG\stackrel{\alpha }{\to }{B}^{n}U(1)$ of this field configuration is the cohomology class $c(\varphi )$ of this bundle under the given group cocycle;

• the weight in the path integral over all $\varphi$ for $n$-dimensional $X$ (i.e. in codimension 0) is the groupoid measure of the functor category $[\Pi (X),BG]$.

Details of DW-theory 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 sections of the flat $n$-vector bundle $BG\to n\mathrm{Vect}$ induced by the background field $BG\to B^{n}U(1)$.

Finite Group Cohomology

Since the target space of Dijkgraaf-Witten theory is the delooping groupoid $BG$ of a group $G$ (internal to Set), any background field given by a morphism $\alpha :BG\to A$ 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 $ℬG$, and from $A$ the Eilenberg-MacLane space ${ℬ}^{n}A=K(A,n)$. Then we consider the space of maps $\mathrm{hom}(ℬG,{ℬ}^{\mathrm{nA}})$ (these are our cocycles) and take its ${\pi }_0$.

This way we have a, in a certain sense familiar (topological spaces, continuous maps, homotopies,..), description of the set $H^n(G;A)$. The drawback is that the topological spaces involved here are “gigantic” (infinite dimensional CW-complexes), 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 hypotesis serves it on a silver plate. Namely, since $G$ is discrete, $ℬG$ is a 1-type, and nothing but the geometric realization of the delooping groupoid $BG$ (boldface $B$ here); similarly ${ℬ}^{n}A$ is the topological geometric realization of the $n$-groupoid $B^{n}A$, and the space of cocycles is $\mathrm{hom}(BG,B^{n}A)$. since $G$ is a finite group, $BG$ is a finite groupoid, and so $\mathrm{hom}(BG,B^{n}A)$ is a finite set. This set is the finite model for $\mathrm{hom}(ℬG,{ℬ}^{n}A)$ we were looking for.

To be continued…

The $k$-vector spaces of states in codimension $k$

The k-vector space associated with a closed $(n-k)$-dimensional manifold $X_{n-k}$ admits a simple description in terms of sections:

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

where the last morphism is the projection on the k-truncation. This defines a cocycle on the space of fields $[\Pi (X_{n-k}),BG]$ over $X_{n-k}$, which classifies some principal ∞-bundle on this space. Given a canonical representation of the spaces of phases ${\tau }_k[\Pi (X_{n-k}),A]$ on a 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.

This means that the transgression of the Dijkgraaf-witten background field

to the space of field configurations $[\Pi (X_{n-k}),BG]$ over $X_{n-k}$ is a cocycle of the form

This classifies a $B^{k-1}U(1)$-principal ∞-bundle $P$ over the space of field configurations, given by the pullback

(Here $E{B}^{k-1}U(1)$ is as described at generalized universal bundle.)

By the canonical $k$-representation $\rho :B^{k}U(1)\to k{\mathrm{Vect}}_{ℂ}$ of $B^{k-1}U(1)$ on complex k-vector spaces, we have associated to this canonically a $k$-vector bundle $E$, which may be realized as the pullback

Here $k{\mathrm{Vect}}_{*}$ is the 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 (k-1)-category of sections of this bundle $E$.

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Relation to Chern-Simons theory

Dijkgraaf-Witten theiry 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 ∞-groupoids (∞-stacks on CartSp).

References

The idea originates, of course, in

• Robbert Dijkgraaf? and Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393.

A first comprehensive structural account od DW theory as a functorial QFT was given in

• Dan Freed, Frank Quinn?, Chern-Simons theory with finite gauge group (arXiv)

A review is given on p. 68 of

• Bruce Bartlett, Categorical aspects of topological quantum field theory (arXiv)

Further conceptual clarifications were established in

• Simon Willerton, The twisted Drinfeld double of a finite group via gerbes and finite groupoids (arXiv)

Recently there have been attempts to understand the structure here more systematically:

Section 3 of

• Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups

proposes a general abstract nonsense way to construct path integral quantizations for finite group theories such as DW.

For more on this see the discussion on the n-Forum.

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