Can the DikgraafWitten model for a finite gauge group $G$ [Robbert Dijkgraaf and Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393] be described in terms of the geometry of moduli spaces $\overline{\mathcal{M}}_{g,n,\beta}([*//G])$ of stable maps to the stack $[*//G]$? I strongly suspect the answer is yes, in view of the classical relation between 3dimensional topological quantum field theories and complex analytic 2dimensional modular functors, but I'm unaware of rigorous results in this direction.

1$\begingroup$ I would look at math.QA/0310087 by Alexader Kirillov Jr. Here is the abstract: In this note, we give a description of the modular functor associated to the ChernSimons theory with a finite group from the complexanalytic point of view, i.e. as a vector bundle with a flat connection on the moduli space of punctured curves. We show that it can be obtained from the trivial local system on the moduli space of "admissible Gcovers" as a direct image under the forgetful map from moduli space of Gcovers to the usual moduli space. $\endgroup$ – A. Pascal Aug 13 '10 at 7:27

$\begingroup$ Thanks, I was unaware of this note. Do you know whether the forthcoming papers announced there have then actually been written? $\endgroup$ – domenico fiorenza Aug 13 '10 at 8:33

$\begingroup$ This paper of Kirillov and Prince suggests yet another forthcominig paper, where such details will be worked out: arXiv:0807.0939. I'm guessing that Prince was a student of Kirillov's and was given this as a thesis problem. $\endgroup$ – A. Pascal Aug 13 '10 at 9:01

2$\begingroup$ cool!.. and since there's no better way to carefully read them than latexing them.. (I'll post a link here as latexing is complete) $\endgroup$ – domenico fiorenza Aug 13 '10 at 10:39

2$\begingroup$ Here they are: math.ucr.edu/~alex/Ncafe_postings/DBZ_lecture1.pdf math.ucr.edu/~alex/Ncafe_postings/DBZ_lecture2.pdf math.ucr.edu/~alex/Ncafe_postings/DBZ_lecture3.pdf $\endgroup$ – Thomas Nikolaus Dec 3 '10 at 15:23
This has been done, in a variety of related ways. A lot of the difficulty is in defining an appropriate notion of a "stable" map to [pt/G].
The earliest mathematical work I know of is Chen & Ruan's "orbifold cohomology", which is done in the symplectic category. (Caveats: Abramovich's lecture notes on orbifold GW theory quote a 1996 letter from Kontsevich, who outlines a lot of the basic ideas in 2 pages. Also, string theorists were looking at nontopological sigma models to orbifolds at least as far back as Dixon, Harvey, Vafa, & Witten's 1985 papers.)
In algebraic geometry, this stuff has been studied by Jarvis, Kaufmann, & Kimura, who focused on Gbundles, and by Abramovich, Graber, & Vistoli, who figured out how to deal with DM stacks.
(You can also carry out these constructions in Ktheory for finitedimensional Lie groups. See, for example, Frenkel, Teleman, & [cough].)

$\begingroup$ Thanks a lot for the references. I'm actually more interested in the tqft aspects than in the rigorous definition of the moduli stack of stable maps to a DM stack. Kirillov's paper math.QA/0310087 very much goes in the direction I'm interested, but it is extremely sketchy. $\endgroup$ – domenico fiorenza Aug 13 '10 at 16:37

$\begingroup$ I don't have enough rep to edit the above answer, so AJ, perhaps you can do it for me and change Dixon, Harvey, Vafa, & Strominger to Dixon, Harvey, Vafa, & Witten. $\endgroup$ – Jeff Harvey Dec 3 '10 at 14:44

$\begingroup$ In addition to these references, check out "Consistent Orientations of Moduli Spaces" by FreedHopkinsTeleman, particularly section 4 of that paper. $\endgroup$ – Kevin H. Lin Dec 5 '10 at 1:07