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.

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].) 

