nLab
Dijkgraaf-Witten theory

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 $\exp (iS):BG\to B^{n}U(1)$

  • 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 $B^{n}U(1)$

  • (for this purpose often what is considered is the simple model of $n\mathrm{Vect}$ given by $B^{n-1}1{\mathrm{dVect}}_{\mathrm{iso}}$).

• parameter spaces $\Sigma ={\Pi }_1(X)$ are skeleta of the fundamental groupoids of $n$-dimensional manifolds $X$.

Therefore

• a field configuration $\Sigma ={\Pi }_1(X)\stackrel{\varphi }{\to }BG$ is a $G$-bundle on $X$ (recall that $G$ is assumed to be a finite group);

• the action ${\Pi }_1(X)\stackrel{\varphi }{\to }BG\stackrel{\exp (iS)}{\to }{B}^{n}U(1)$ 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 $[{\Pi }_1(X),BG]$.

Remarks

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 smooth $n$-groupoids (infinity-stacks on $\mathrm{Diff}$).

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 (arXiv)

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

In a similar vein, so far for $n=1$ it is shown in

• Johan Alm, Quantization as a Kan Extension

that the quantization procedure for DW theory is nothing but the Kan extension of the background field from target space down to parameter space.

An description of extended DW theory as the systematic pull-push quantization induced by a generalized differential cocycle is proposed at

• Nonabelian cocycles and their sigma model QFTs

There in particular a first-principle derivation of the fact that extended DW theory assigns the representation category of the Drinfeld double to the circle is given (following Willerton, but deriving his construction from still more fundamental abstract nonsense).