ag.algebraic geometry - The algebro-geometric counterpart of the Dijkgraaf-Witten model

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].)