# 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