# The algebro-geometric counterpart of the Dijkgraaf-Witten model

Can the Dikgraaf-Witten 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 3-dimensional topological quantum field theories and complex analytic 2-dimensional modular functors, but I'm unaware of rigorous results in this direction.

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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 Chern-Simons theory with a finite group from the complex-analytic 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 G-covers" as a direct image under the forgetful map from moduli space of G-covers to the usual moduli space. –  A. Pascal Aug 13 '10 at 7:27
Thanks, I was unaware of this note. Do you know whether the forthcoming papers announced there have then actually been written? –  domenico fiorenza Aug 13 '10 at 8:33
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. –  A. Pascal Aug 13 '10 at 9:01
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) –  domenico fiorenza Aug 13 '10 at 10:39

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 non-topological 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 G-bundles, and by Abramovich, Graber, & Vistoli, who figured out how to deal with D-M stacks.

(You can also carry out these constructions in K-theory for finite-dimensional Lie groups. See, for example, Frenkel, Teleman, & [cough].)

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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. –  domenico fiorenza Aug 13 '10 at 16:37
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. –  Jeff Harvey Dec 3 '10 at 14:44
In addition to these references, check out "Consistent Orientations of Moduli Spaces" by Freed-Hopkins-Teleman, particularly section 4 of that paper. –  Kevin H. Lin Dec 5 '10 at 1:07